Finite Group Databasebeta

← #29709299

Subgroups — #29709299

Order 15616.

956 conjugacy classes of subgroups.

The subgroup lattice has 720 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839872C211yesmaximal, characteristic, complemented
2C2×C4×#11758199C211yesmaximal, characteristic, complemented
3#21448463C211yesmaximal, characteristic, complemented
4#21447301C211yesmaximal, characteristic, complemented
5#21447233C211yesmaximal, characteristic, complemented
6#21447207C211yesmaximal, characteristic, complemented
7#21447193C211yesmaximal, characteristic, complemented
8#21447129C211yesmaximal, characteristic, complemented
9#21447129C211yesmaximal, characteristic, complemented
10#21447050C211yesmaximal, characteristic, complemented
11#21446890C211yesmaximal, characteristic
12#21445676C211yesmaximal, characteristic, complemented
13#21445644C211yesmaximal, characteristic, complemented
14C4×#14519698C211yesmaximal, characteristic, complemented
15C61×#2328C211yesmaximal, characteristic, complemented
16C22×#11758199C2211yescharacteristic
17C22×#11758199C2211yescharacteristic
18C2×C4×D244C2211yescharacteristic
19#16839872—24no—
20C4×#11758199—24no—
21C2×#14519710C2211yescharacteristic
22C2×#14519708C2211yescharacteristic
23C2×C4⋊#92418C2211yescharacteristic
24C2×C4⋊#92418C2211yescharacteristic
25C2×#14519698C2211yescharacteristic
26C2×#14519698C2211yescharacteristic
27C2×#14519698C2211yescharacteristic
28C2×#14519698C2211yescharacteristic
29C2×#14519698C2211yescharacteristic
30C2×#14519698C2211yescharacteristic
31C2×#14519697C2211yescharacteristic
32C2×#14519697C2211yescharacteristic
33C2×#14519688C2211yescharacteristic
34C2×C4×#92418C2211yescharacteristic
35#16839664C2211yescharacteristic
36#16839664C2211yescharacteristic
37#16839660C2211yescharacteristic
38#16839660C2211yescharacteristic
39#16839651—21no—
40#16839651—21no—
41#16839624—21no—
42#16839621—21no—
43C4×C4⋊Dic61C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839517C2211yescharacteristic
49#16839517C2211yescharacteristic
50C2×#127×C61C2211yescharacteristic
51C2×C4×D4×C61C2211yescharacteristic
52C2×C4×D4×C61C2211yescharacteristic
53C61×#390C2211yescharacteristic
54C61×#390C2211yescharacteristic
55C61×#386—21no—
56C61×#386—21no—
57C61×#386—21no—
58C61×#386—21no—
59C61×#382C2211yescharacteristic
60C61×#376C2211yescharacteristic
61C22×D244C2311yescharacteristic
62C22×C4×D61—21no—
63C2×#11758199—22no—
64C2×#11758199—22no—
65C2×#11758199—22no—
66C2×#11758199—22no—
67C2×#11758199—22no—
68C2×#11758199—22no—
69C22×#92418C2311yescharacteristic
70C22×#92418C2311yescharacteristic
71C22×#92418C2311yescharacteristic
72C22×#92418C2311yescharacteristic
73C4×D244—22no—
74C4×D244D414yescomplemented
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×#11758198—21no—
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×C4×Dic61—21no—
86#14519710—22no—
87#14519708—22no—
88C4⋊#92418—22no—
89C4⋊#92418—22no—
90#14519698—22no—
91#14519698—22no—
92#14519698—22no—
93#14519698—22no—
94#14519698—22no—
95#14519698—22no—
96#14519697—22no—
97#14519697—22no—
98#14519688—22no—
99C4×#92418—22no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103C22×D4×C61—21no—
104C22×D4×C61—21no—
105C22×D4×C61C2311yescharacteristic
106C22×D4×C61C2311yescharacteristic
107C23×C4×C61—21no—
108C23×C4×C61—21no—
109#127×C61—22no—
110#127×C61—22no—
111C4×D4×C61—22no—
112C4×D4×C61—22no—
113C4×D4×C61—22no—
114C4×D4×C61—22no—
115C2×C4⋊C4×C61C2311yescharacteristic
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C22⋊C4×C61—21no—
118C2×C22⋊C4×C61—21no—
119C2×C22⋊C4×C61—21no—
120C2×C22⋊C4×C61—21no—
121C2×C22⋊C4×C61—21no—
122C2×C22⋊C4×C61—21no—
123C2×C42×C61C2311yescharacteristic
124C2×C42×C61C2311yescharacteristic
125C2×C42×C61C2311yescharacteristic
126#95×C61—21no—
127#95×C61—21no—
128#95×C61D411yescharacteristic
129#95×C61D411yescharacteristic
130#95×C61D411yescharacteristic
131#95×C61D411yescharacteristic
132C24×C61—21no—
133C24×C61—21no—
134C23×D61—21no—
135C2×D4×C61—24no—
136C2×D4×C61—24no—
137C2×D4×C61—21no—
138C2×D4×C61—21no—
139C2×D4×C61—21no—
140C2×D4×C61—21no—
141C2×D4×C61—21no—
142C2×D4×C61—21no—
143C2×D4×C61—21no—
144C2×D4×C61—21no—
145C2×D4×C61—21no—
146C2×D4×C61—21no—
147C2×D4×C61—81no—
148C2×D4×C61—81no—
149C2×D4×C61—81no—
150C2×D4×C61—81no—
151C2×D4×C61#4112yes—
152C2×D4×C61#4112yes—
153C22×C4×C61—21no—
154C22×C4×C61—21no—
155C22×C4×C61—42no—
156C22×C4×C61—42no—
157C22×C4×C61—81no—
158C22×C4×C61—81no—
159C22×C4×C61—81no—
160C22×C4×C61—81no—
161C22×C4×C61C2×D411yescharacteristic
162C22×C4×C61C2×D411yescharacteristic
163C22×C4×C61C2×D411yescharacteristic
164C22×C4×C61C2×D411yescharacteristic
165C22×C4×C61C2×D411yescharacteristic
166C22×C4×C61C2×D411yescharacteristic
167C22×C4×C61C2411yescharacteristic
168C22×Dic61—21no—
169C22×Dic61—21no—
170C22×Dic61—21no—
171C2×D244—21no—
172C2×D244—21no—
173C2×D244—21no—
174C2×D244—21no—
175C2×D244—21no—
176C2×D244C2×D412yes—
177C2×C4×D61—24no—
178C2×C4×D61—81no—
179C2×C4×D61—81no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C22⋊C4×C61—42no—
185C22⋊C4×C61—42no—
186C22⋊C4×C61—81no—
187C22⋊C4×C61—81no—
188C22⋊C4×C61—81no—
189C22⋊C4×C61—81no—
190C42×C61—21no—
191C42×C61—21no—
192C42×C61—21no—
193C42×C61—21no—
194C42×C61C2×D412yes—
195C42×C61#4112yes—
196#11758199—28no—
197#11758199—28no—
198#11758198—24no—
199#11758198—81no—
200#11758198—81no—
201#11758198—81no—
202#11758198—81no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—21no—
208C4⋊Dic61C2×D412yes—
209#11758196—81no—
210#11758196—81no—
211C4×Dic61—81no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418—21no—
221C2×#92418—21no—
222C2×#92418#4112yes—
223C2×#92418#4112yes—
224C23×C61—22no—
225C23×C61—22no—
226C23×C61—41no—
227C23×C61—41no—
228C23×C61—41no—
229C23×C61—41no—
230C23×C61—41no—
231C23×C61—41no—
232C23×C61—41no—
233C23×C61—41no—
234C23×C61—41no—
235C23×C61—41no—
236C23×C61—41no—
237C23×C61—41no—
238C23×C61C22×D411yescharacteristic
239C22×D61—22no—
240C22×D61—41no—
241C22×D61—41no—
242C22×D61—41no—
243C22×D61—41no—
244C22×D61—41no—
245C22×D61—41no—
246D4×C61—22no—
247D4×C61—22no—
248D4×C61—22no—
249D4×C61—22no—
250D4×C61—41no—
251D4×C61—41no—
252D4×C61—41no—
253D4×C61—41no—
254D4×C61—84no—
255D4×C61—84no—
256C2×C4×C61—21no—
257C2×C4×C61—21no—
258C2×C4×C61—21no—
259C2×C4×C61—21no—
260C2×C4×C61—21no—
261C2×C4×C61—21no—
262C2×C4×C61—21no—
263C2×C4×C61—21no—
264C2×C4×C61—21no—
265C2×C4×C61—21no—
266C2×C4×C61—21no—
267C2×C4×C61—21no—
268C2×C4×C61—21no—
269C2×C4×C61—21no—
270C2×C4×C61—21no—
271C2×C4×C61—21no—
272C2×C4×C61—41no—
273C2×C4×C61—41no—
274C2×C4×C61—41no—
275C2×C4×C61—41no—
276C2×C4×C61—41no—
277C2×C4×C61—41no—
278C2×C4×C61—84no—
279C2×C4×C61—84no—
280C2×C4×C61#12112yes—
281C2×C4×C61#12112yes—
282C2×C4×C61C22×D411yescharacteristic
283C2×C4×C61C2×#4111yescharacteristic
284C2×C4×C61C2×#4111yescharacteristic
285C2×C4×C61C2×#4111yescharacteristic
286C2×C4×C61C2×#4111yescharacteristic
287C2×C4×C61C2×#4111yescharacteristic
288C2×Dic61—22no—
289C2×Dic61—41no—
290C2×Dic61—41no—
291C2×Dic61—41no—
292C2×Dic61—41no—
293C2×Dic61—41no—
294C2×Dic61—41no—
295C2×Dic61—41no—
296C2×Dic61—41no—
297D244—22no—
298D244—22no—
299D244—41no—
300D244—41no—
301C4×D61—84no—
302#92418—22no—
303#92418—22no—
304#92418—41no—
305#92418—41no—
306#23688—611nomaximal, Sylow
307C22×C61—42no—
308C22×C61—42no—
309C22×C61—42no—
310C22×C61—42no—
311C22×C61—42no—
312C22×C61—42no—
313C22×C61—81no—
314C22×C61—81no—
315C22×C61—81no—
316C22×C61—81no—
317C22×C61—81no—
318C22×C61—81no—
319C22×C61—81no—
320C22×C61—81no—
321C22×C61C2×#12111yescharacteristic
322C22×C61#52511yescharacteristic
323C22×C61#53211yescharacteristic
324C22×C61#57611yescharacteristic
325C22×C61#57611yescharacteristic
326C22×C61#57611yescharacteristic
327C22×C61#57711yescharacteristic
328C2×D61—42no—
329C2×D61—42no—
330C2×D61—42no—
331C2×D61—81no—
332C2×D61—81no—
333C2×D61—81no—
334C2×D61—81no—
335C4×C61—22no—
336C4×C61—22no—
337C4×C61—41no—
338C4×C61—41no—
339C4×C61—41no—
340C4×C61—41no—
341C4×C61—81no—
342C4×C61—81no—
343C4×C61#52511yescharacteristic
344C4×C61#52511yescharacteristic
345C4×C61#52911yescharacteristic
346C4×C61#52911yescharacteristic
347Dic61—42no—
348Dic61—81no—
349Dic61—81no—
350C2×#442—611no—
351C2×C4×D8—611no—
352#2328D6111yescharacteristic, complemented
353#2319—611no—
354#2085—611no—
355#2073—611no—
356#2067—611no—
357#2065—611no—
358#2054—611no—
359#2054—611no—
360#2041—611no—
361#2011—611no—
362#1768—611no—
363#1762—611no—
364C4×#102—611no—
365C2×C61—82no—
366C2×C61—82no—
367C2×C61#239211yescharacteristic
368C2×C61#304111yescharacteristic
369C2×C61#304511yescharacteristic
370C2×C61#311811yescharacteristic
371C2×C61#312211yescharacteristic
372C2×C61#317611yescharacteristic
373C2×C61#317911yescharacteristic
374D61—82no—
375C22×D8—611no—
376C22×D8—611no—
377C2×#127C2×D6111yescharacteristic
378C2×C4×D4—611no—
379C2×C4×D4C2×D6111yescharacteristic
380C2×C4×D4C2×D6111yescharacteristic
381#442—1224no—
382C4×D8—1224no—
383C2×#107—611no—
384C2×#106—611no—
385C2×C4⋊C8—611no—
386C2×C4⋊C8—611no—
387C2×#102—611no—
388C2×#102—611no—
389C2×#102—611no—
390C2×#102—611no—
391C2×#102—611no—
392C2×#102—611no—
393C2×#102—611no—
394C2×#102—611no—
395C2×#97—611no—
396C2×C4×C8—611no—
397#390—611no—
398#390—611no—
399#390C2×D6111yescharacteristic
400#390C2×D6111yescharacteristic
401#389—611no—
402#389—611no—
403#388—1221no—
404#388—1221no—
405#386—21no—
406#386—21no—
407#386—21no—
408#386—21no—
409#382C2×D6111yescharacteristic
410#381—1221no—
411#380—1221no—
412C4×C4⋊C4—611no—
413#376C2×D6111yescharacteristic
414#340—611no—
415#340—611no—
416#340—611no—
417#340—611no—
418#336—611no—
419#336—611no—
420C61#2368811yescharacteristic, complemented, Sylow
421C22×D4—21no—
422C22×D4—21no—
423C22×D4—611no—
424C22×D4C22×D6111yescharacteristic
425C22×D4C22×D6111yescharacteristic
426C23×C4—21no—
427C23×C4—21no—
428C23×C4—1221no—
429C2×D8—1222no—
430C2×D8—1222no—
431C2×D8—1222no—
432C2×D8—1222no—
433C2×D8—1222no—
434C2×D8—1222no—
435C22×C8—611no—
436C22×C8—611no—
437C22×C8—611no—
438C22×C8—611no—
439#127—22no—
440#127—22no—
441C4×D4—22no—
442C4×D4—22no—
443C4×D4—22no—
444C4×D4—22no—
445C4×D4—614no—
446C4×D4—1222no—
447C2×C4⋊C4—611no—
448C2×C4⋊C4—611no—
449C2×C4⋊C4—611no—
450C2×C4⋊C4—1221no—
451C2×C4⋊C4—1221no—
452C2×C4⋊C4C22×D6111yescharacteristic
453C2×C4⋊C4C22×D6111yescharacteristic
454C2×C22⋊C4—21no—
455C2×C22⋊C4—21no—
456C2×C22⋊C4—21no—
457C2×C22⋊C4—21no—
458C2×C22⋊C4—21no—
459C2×C22⋊C4—21no—
460C2×C22⋊C4—1221no—
461C2×C22⋊C4—1221no—
462C2×C22⋊C4—1221no—
463C2×C22⋊C4—1221no—
464C2×C22⋊C4—1221no—
465C2×C42—1221no—
466C2×C42C22×D6111yescharacteristic
467C2×C42C22×D6111yescharacteristic
468C2×C42C22×D6111yescharacteristic
469#107—1222no—
470#106—1222no—
471C4⋊C8—1222no—
472C4⋊C8—1222no—
473#102—1222no—
474#102—1222no—
475#102—1222no—
476#102—1222no—
477#102—1222no—
478#102—1222no—
479#102—1222no—
480#102—1222no—
481#97—1222no—
482C4×C8—1222no—
483#95—21no—
484#95—21no—
485#95—1221no—
486#95—1221no—
487#95—1221no—
488#95C22⋊D6111yescharacteristic
489#95C22⋊D6111yescharacteristic
490#95C22⋊D6111yescharacteristic
491#95C22⋊D6111yescharacteristic
492C24—21no—
493C24—21no—
494C24—1221no—
495C2×D4—24no—
496C2×D4—24no—
497C2×D4—21no—
498C2×D4—21no—
499C2×D4—21no—
500C2×D4—21no—
501C2×D4—21no—
502C2×D4—21no—
503C2×D4—21no—
504C2×D4—21no—
505C2×D4—21no—
506C2×D4—21no—
507C2×D4—81no—
508C2×D4—81no—
509C2×D4—81no—
510C2×D4—81no—
511C2×D4—612no—
512C2×D4—1221no—
513C2×D4—1221no—
514C2×D4—1221no—
515C2×D4—1221no—
516C2×D4—1221no—
517C2×D4#1175822212yes—
518C2×D4#1175822212yes—
519C22×C4—21no—
520C22×C4—21no—
521C22×C4—42no—
522C22×C4—42no—
523C22×C4—81no—
524C22×C4—81no—
525C22×C4—81no—
526C22×C4—81no—
527C22×C4—1221no—
528C22×C4—1224no—
529C22×C4—1221no—
530C22×C4—1221no—
531C22×C4—4881no—
532C22×C4—4881no—
533C22×C4C2×C22⋊D6111yescharacteristic
534C22×C4C2×C22⋊D6111yescharacteristic
535C22×C4C2×C22⋊D6111yescharacteristic
536C22×C4C2×C22⋊D6111yescharacteristic
537C22×C4C2×C22⋊D6111yescharacteristic
538C22×C4C2×C22⋊D6111yescharacteristic
539C22×C4C23×D6111yescharacteristic
540D8—1228no—
541D8—1228no—
542C2×C8—612no—
543C2×C8—612no—
544C2×C8—1221no—
545C2×C8—1221no—
546C2×C8—1221no—
547C2×C8—1221no—
548C2×C8—1221no—
549C2×C8—1221no—
550C2×C8—1221no—
551C2×C8—1221no—
552C2×C8—1221no—
553C2×C8—1221no—
554C4⋊C4—21no—
555C4⋊C4—21no—
556C4⋊C4—21no—
557C4⋊C4—21no—
558C4⋊C4—612no—
559C4⋊C4—1221no—
560C4⋊C4—1221no—
561C4⋊C4—1221no—
562C4⋊C4—1221no—
563C4⋊C4—1221no—
564C4⋊C4—4881no—
565C4⋊C4—4881no—
566C22⋊C4—42no—
567C22⋊C4—42no—
568C22⋊C4—81no—
569C22⋊C4—81no—
570C22⋊C4—81no—
571C22⋊C4—81no—
572C22⋊C4—1224no—
573C22⋊C4—4881no—
574C22⋊C4—4881no—
575C22⋊C4—4881no—
576C22⋊C4—4881no—
577C42—21no—
578C42—21no—
579C42—21no—
580C42—21no—
581C42—4881no—
582C42D4×D6112yes—
583C42#1175822412yes—
584C23—22no—
585C23—22no—
586C23—41no—
587C23—41no—
588C23—41no—
589C23—41no—
590C23—41no—
591C23—41no—
592C23—41no—
593C23—41no—
594C23—41no—
595C23—41no—
596C23—41no—
597C23—41no—
598C23—1222no—
599C23—2441no—
600C23—2441no—
601C23—2441no—
602C23—2441no—
603C23—2441no—
604C23—2441no—
605C23C22×C22⋊D6111yescharacteristic
606D4—22no—
607D4—22no—
608D4—22no—
609D4—22no—
610D4—41no—
611D4—41no—
612D4—41no—
613D4—41no—
614D4—84no—
615D4—84no—
616D4—1222no—
617D4—1222no—
618D4—2441no—
619D4—2441no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—84no—
643C2×C4—84no—
644C2×C4—1222no—
645C2×C4—4884no—
646C2×C4—2441no—
647C2×C4—2441no—
648C2×C4—2441no—
649C2×C4—2441no—
650C2×C4—2441no—
651C2×C4—2441no—
652C2×C4—2441no—
653C2×C4—2441no—
654C2×C4#1451974812yes—
655C2×C4#1451974812yes—
656C2×C4C2×D4×D6111yescharacteristic
657C2×C4#41×D6111yescharacteristic
658C2×C4#41×D6111yescharacteristic
659C2×C4C2×#1175822411yescharacteristic
660C2×C4C2×#1175822211yescharacteristic
661C2×C4C2×#1175822211yescharacteristic
662C8—1222no—
663C8—1222no—
664C8—2441no—
665C8—2441no—
666C22—42no—
667C22—42no—
668C22—42no—
669C22—42no—
670C22—42no—
671C22—42no—
672C22—81no—
673C22—81no—
674C22—81no—
675C22—81no—
676C22—81no—
677C22—81no—
678C22—81no—
679C22—81no—
680C22—4881no—
681C22—4881no—
682C22—4881no—
683C22—4881no—
684C22—2442no—
685C22—2442no—
686C22—2442no—
687C22C2×#1451974811yescharacteristic
688C22#1684027311yescharacteristic
689C22#1684030811yescharacteristic
690C22#1684056511yescharacteristic
691C22#1684056511yescharacteristic
692C22#1684056511yescharacteristic
693C22#1684057311yescharacteristic
694C4—22no—
695C4—22no—
696C4—41no—
697C4—41no—
698C4—41no—
699C4—41no—
700C4—81no—
701C4—81no—
702C4—4881no—
703C4—4881no—
704C4—2442no—
705C4#1684027111yescharacteristic
706C4#1684027111yescharacteristic
707C4#1684029211yescharacteristic
708C4#1684029211yescharacteristic
709C2—82no—
710C2—82no—
711C2—4882no—
712C2#2144905711yescharacteristic
713C2#2145531811yescharacteristic
714C2#2145535411yescharacteristic
715C2#2145600311yescharacteristic
716C2#2145606211yescharacteristic
717C2#2145646311yescharacteristic
718C2#2145650411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2328C2
socleC23×C61C22×D4
prime radical#21446890C2
hypercenter#2328D61
hypocenterC61#23688
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ #376 ◁ #2328

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#16839872C2semidirect
2. C2×C4×#11758199C2semidirect
3. #21448463C2semidirect
4. #21447301C2semidirect
5. #21447233C2semidirect
6. #21447207C2semidirect
7. #21447193C2semidirect
8. #21447129C2semidirect
9. #21447129C2semidirect
10. #21447050C2semidirect
12. #21445676C2semidirect
13. #21445644C2semidirect
14. C4×#14519698C2semidirect
15. C61×#2328C2semidirect
74. C4×D244D4semidirect
352. #2328D61semidirect
420. C61#23688semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168398722
C2×C4×#117581992
#214484632
#214473012
#214472332
#214472072
#214471932
#214471292
#214470502
#214468902
#214456762
#214456442
C4×#145196982
C61×#23282
#2368861