Finite Group Databasebeta

← #29709327

Subgroups — #29709327

Order 15616.

908 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839860C211yesmaximal, characteristic, complemented
2C2×C4×#11758201C211yesmaximal, characteristic, complemented
3#21448544C211yesmaximal, characteristic
4#21447300C211yesmaximal, characteristic, complemented
5#21447237C211yesmaximal, characteristic, complemented
6#21447204C211yesmaximal, characteristic, complemented
7#21447190C211yesmaximal, characteristic, complemented
8#21447134C211yesmaximal, characteristic, complemented
9#21447134C211yesmaximal, characteristic, complemented
10#21447053C211yesmaximal, characteristic
11#21446906C211yesmaximal, characteristic
12#21445676C211yesmaximal, characteristic, complemented
13#21445647C211yesmaximal, characteristic, complemented
14C4×#14519697C211yesmaximal, characteristic, complemented
15C61×#2320C211yesmaximal, characteristic, complemented
16C22×#11758201C2211yescharacteristic
17C22×#11758201C2211yescharacteristic
18C2×#14519774C2211yescharacteristic
19C2×C4×D244C2211yescharacteristic
20C2×C4×D244C2211yescharacteristic
21#16839860—24no—
22C4×#11758201—24no—
23C2×#14519710C2211yescharacteristic
24C2×#14519708C2211yescharacteristic
25C2×C4⋊#92418C2211yescharacteristic
26C2×C4⋊#92418C2211yescharacteristic
27C2×#14519701C2211yescharacteristic
28C2×#14519701C2211yescharacteristic
29C2×#14519697C2211yescharacteristic
30C2×#14519697C2211yescharacteristic
31C2×#14519697C2211yescharacteristic
32C2×#14519697C2211yescharacteristic
33C2×#14519697C2211yescharacteristic
34C2×#14519697C2211yescharacteristic
35C2×#14519688C2211yescharacteristic
36C2×C4×#92418C2211yescharacteristic
37#16839664C2211yescharacteristic
38#16839664C2211yescharacteristic
39#16839640—21no—
40#16839640—21no—
41#16839640—21no—
42#16839640—21no—
43#16839628C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839517C2211yescharacteristic
49#16839517C2211yescharacteristic
50C2×C4×Q8×C61C2211yescharacteristic
51C61×#391C2211yescharacteristic
52C61×#391C2211yescharacteristic
53C61×#389C2211yescharacteristic
54C61×#389C2211yescharacteristic
55C61×#387—21no—
56C61×#387—21no—
57C61×#383—21no—
58C61×#382—21no—
59C4×C4⋊C4×C61C2211yescharacteristic
60C61×#376C2211yescharacteristic
61C22×D244—21no—
62C22×D244—21no—
63C22×D244C2311yescharacteristic
64C22×D244C2311yescharacteristic
65C22×C4×D61—21no—
66C22×C4×D61—21no—
67C2×#11758201—22no—
68C2×#11758201—22no—
69C2×#11758201—22no—
70C2×#11758201—22no—
71C2×#11758201—22no—
72C2×#11758201—22no—
73C22×#92418C2311yescharacteristic
74C22×#92418C2311yescharacteristic
75C22×#92418C2311yescharacteristic
76C22×#92418C2311yescharacteristic
77#14519774—22no—
78#14519774—22no—
79C4×D244—22no—
80C4×D244—22no—
81C4×D244—22no—
82C4×D244—22no—
83C2×C4⋊Dic61C2311yescharacteristic
84C2×C4⋊Dic61C2311yescharacteristic
85C2×#11758198—21no—
86C2×#11758198—21no—
87C2×#11758198—21no—
88C2×#11758198—21no—
89C2×#11758198—21no—
90C2×#11758198—21no—
91#14519710—22no—
92#14519708—22no—
93C4⋊#92418—22no—
94C4⋊#92418—22no—
95#14519701—22no—
96#14519701—22no—
97#14519697—22no—
98#14519697—22no—
99#14519697—22no—
100#14519697—22no—
101#14519697—22no—
102#14519697—22no—
103#14519688—22no—
104C4×#92418—22no—
105#14519684—21no—
106#14519684—21no—
107C22×Q8×C61C2311yescharacteristic
108C4×Q8×C61—22no—
109C4×Q8×C61D414yescomplemented
110C2×C4⋊C4×C61—21no—
111C2×C4⋊C4×C61—21no—
112C2×C4⋊C4×C61—21no—
113C2×C4⋊C4×C61C2311yescharacteristic
114C2×C4⋊C4×C61C2311yescharacteristic
115C2×C4⋊C4×C61C2311yescharacteristic
116C2×C42×C61—21no—
117C2×C42×C61—21no—
118C2×C42×C61C2311yescharacteristic
119C2×C42×C61C2311yescharacteristic
120C2×C42×C61C2311yescharacteristic
121#95×C61—21no—
122#95×C61—21no—
123#95×C61—21no—
124#95×C61—21no—
125#95×C61—21no—
126#95×C61—21no—
127#95×C61—21no—
128#95×C61D411yescharacteristic
129#95×C61D411yescharacteristic
130#95×C61D411yescharacteristic
131#95×C61D411yescharacteristic
132C23×D61—21no—
133C23×D61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61—21no—
138C2×Q8×C61—21no—
139C2×Q8×C61C2×D412yes—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2411yescharacteristic
151C22×Dic61—21no—
152C22×Dic61—21no—
153C2×D244—21no—
154C2×D244—21no—
155C2×D244—21no—
156C2×D244—21no—
157C2×D244—21no—
158C2×D244—21no—
159C2×D244—21no—
160C2×D244—21no—
161C2×D244—21no—
162C2×D244—21no—
163C2×D244—24no—
164C2×D244—24no—
165C2×D244—81no—
166C2×D244—81no—
167C2×D244—81no—
168C2×D244—81no—
169C2×D244#4112yes—
170C2×D244#4112yes—
171C2×C4×D61—42no—
172C2×C4×D61—42no—
173C2×C4×D61—81no—
174C2×C4×D61—81no—
175C2×C4×D61—81no—
176C2×C4×D61—81no—
177C4⋊C4×C61—24no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—81no—
184C4⋊C4×C61—81no—
185C4⋊C4×C61C2×D412yes—
186C42×C61—24no—
187C42×C61—21no—
188C42×C61—21no—
189C42×C61—21no—
190C42×C61—21no—
191C42×C61—81no—
192C42×C61C2×D412yes—
193C42×C61#4112yes—
194#11758201—28no—
195#11758201—28no—
196#11758198—42no—
197#11758198—42no—
198#11758198—81no—
199#11758198—81no—
200#11758198—81no—
201#11758198—81no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418#4112yes—
217C2×#92418#4112yes—
218C23×C61C22×D411yescharacteristic
219C22×D61—22no—
220C22×D61—22no—
221C22×D61—41no—
222C22×D61—41no—
223C22×D61—41no—
224C22×D61—41no—
225C22×D61—41no—
226C22×D61—41no—
227C22×D61—41no—
228C22×D61—41no—
229C22×D61—41no—
230C22×D61—41no—
231C22×D61—41no—
232C22×D61—41no—
233Q8×C61—22no—
234Q8×C61—22no—
235Q8×C61—41no—
236Q8×C61—41no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—22no—
242C2×C4×C61—22no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—21no—
251C2×C4×C61—21no—
252C2×C4×C61—21no—
253C2×C4×C61—21no—
254C2×C4×C61—21no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61—41no—
260C2×C4×C61—41no—
261C2×C4×C61—41no—
262C2×C4×C61—41no—
263C2×C4×C61—41no—
264C2×C4×C61—41no—
265C2×C4×C61#12112yes—
266C2×C4×C61#12112yes—
267C2×C4×C61C22×D411yescharacteristic
268C2×C4×C61C2×#4111yescharacteristic
269C2×C4×C61C2×#4111yescharacteristic
270C2×C4×C61C2×#4111yescharacteristic
271C2×C4×C61C2×#4111yescharacteristic
272C2×C4×C61C2×#4111yescharacteristic
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279D244—22no—
280D244—22no—
281D244—22no—
282D244—22no—
283D244—41no—
284D244—41no—
285D244—41no—
286D244—41no—
287D244—84no—
288D244—84no—
289C4×D61—84no—
290C4×D61—84no—
291#92418—22no—
292#92418—22no—
293#92418—41no—
294#92418—41no—
295#23690—611nomaximal, Sylow
296C22×C61C2×#12111yescharacteristic
297C22×C61#52511yescharacteristic
298C22×C61#53211yescharacteristic
299C22×C61#57611yescharacteristic
300C22×C61#57711yescharacteristic
301C22×C61#57711yescharacteristic
302C22×C61#57711yescharacteristic
303C2×D61—42no—
304C2×D61—42no—
305C2×D61—42no—
306C2×D61—42no—
307C2×D61—42no—
308C2×D61—42no—
309C2×D61—81no—
310C2×D61—81no—
311C2×D61—81no—
312C2×D61—81no—
313C2×D61—81no—
314C2×D61—81no—
315C2×D61—81no—
316C2×D61—81no—
317C4×C61—22no—
318C4×C61—22no—
319C4×C61—41no—
320C4×C61—41no—
321C4×C61—42no—
322C4×C61—41no—
323C4×C61—41no—
324C4×C61—42no—
325C4×C61—81no—
326C4×C61—81no—
327C4×C61#52511yescharacteristic
328C4×C61#52511yescharacteristic
329C4×C61#52911yescharacteristic
330C4×C61#52911yescharacteristic
331Dic61—81no—
332Dic61—81no—
333C2×#440—611no—
334C2×C4×SD16—611no—
335#2328—611no—
336#2320D6111yescharacteristic, complemented
337#2085—611no—
338#2074—611no—
339#2067—611no—
340#2065—611no—
341#2055—611no—
342#2055—611no—
343#2041—611no—
344#2014—611no—
345#1768—611no—
346#1762—611no—
347C4×#102—611no—
348C2×C61#239211yescharacteristic
349C2×C61#304311yescharacteristic
350C2×C61#304511yescharacteristic
351C2×C61#311811yescharacteristic
352C2×C61#312311yescharacteristic
353C2×C61#317411yescharacteristic
354C2×C61#317911yescharacteristic
355D61—82no—
356D61—82no—
357C22×SD16—611no—
358C22×SD16—611no—
359C2×#127—611no—
360C2×C4×Q8C2×D6111yescharacteristic
361C2×C4×D4—611no—
362C2×C4×D4—611no—
363#440—1224no—
364C4×SD16—1224no—
365C2×#107—611no—
366C2×#106—611no—
367C2×C4⋊C8—611no—
368C2×C4⋊C8—611no—
369C2×#103—611no—
370C2×#103—611no—
371C2×#102—611no—
372C2×#102—611no—
373C2×#102—611no—
374C2×#102—611no—
375C2×#102—611no—
376C2×#102—611no—
377C2×#97—611no—
378C2×C4×C8—611no—
379#391C2×D6111yescharacteristic
380#391C2×D6111yescharacteristic
381#390—611no—
382#390—611no—
383#389C2×D6111yescharacteristic
384#389C2×D6111yescharacteristic
385#387—21no—
386#387—21no—
387#386—1221no—
388#386—1221no—
389#386—1221no—
390#386—1221no—
391#383—21no—
392#382—21no—
393#382—611no—
394C4×C4⋊C4C2×D6111yescharacteristic
395#376C2×D6111yescharacteristic
396#340—611no—
397#340—611no—
398#340—611no—
399#340—611no—
400#336—611no—
401#336—611no—
402C61#2369011yescharacteristic, complemented, Sylow
403C22×Q8C22×D6111yescharacteristic
404C22×D4—611no—
405C22×D4—611no—
406C22×D4—1221no—
407C22×D4—1221no—
408C23×C4—1221no—
409C23×C4—1221no—
410C2×SD16—1222no—
411C2×SD16—1222no—
412C2×SD16—1222no—
413C2×SD16—1222no—
414C2×SD16—1222no—
415C2×SD16—1222no—
416C22×C8—611no—
417C22×C8—611no—
418C22×C8—611no—
419C22×C8—611no—
420#127—1222no—
421#127—1222no—
422C4×Q8—22no—
423C4×Q8D24414yescomplemented
424C4×D4—1222no—
425C4×D4—1222no—
426C4×D4—1222no—
427C4×D4—1222no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—21no—
430C2×C4⋊C4—21no—
431C2×C4⋊C4—611no—
432C2×C4⋊C4—611no—
433C2×C4⋊C4C22×D6111yescharacteristic
434C2×C4⋊C4C22×D6111yescharacteristic
435C2×C4⋊C4C22×D6111yescharacteristic
436C2×C22⋊C4—1221no—
437C2×C22⋊C4—1221no—
438C2×C22⋊C4—1221no—
439C2×C22⋊C4—1221no—
440C2×C22⋊C4—1221no—
441C2×C22⋊C4—1221no—
442C2×C42—21no—
443C2×C42—21no—
444C2×C42C22×D6111yescharacteristic
445C2×C42C22×D6111yescharacteristic
446C2×C42C22×D6111yescharacteristic
447#107—1222no—
448#106—1222no—
449C4⋊C8—1222no—
450C4⋊C8—1222no—
451#103—1222no—
452#103—1222no—
453#102—1222no—
454#102—1222no—
455#102—1222no—
456#102—1222no—
457#102—1222no—
458#102—1222no—
459#97—1222no—
460C4×C8—1222no—
461#95—21no—
462#95—21no—
463#95—21no—
464#95—21no—
465#95—21no—
466#95—21no—
467#95—21no—
468#95—1221no—
469#95—1221no—
470#95C22⋊D6111yescharacteristic
471#95C22⋊D6111yescharacteristic
472#95C22⋊D6111yescharacteristic
473#95C22⋊D6111yescharacteristic
474C24—1221no—
475C24—1221no—
476C2×Q8—21no—
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8C2×D24412yes—
482C2×D4—612no—
483C2×D4—612no—
484C2×D4—1224no—
485C2×D4—1224no—
486C2×D4—1221no—
487C2×D4—1221no—
488C2×D4—1221no—
489C2×D4—1221no—
490C2×D4—1221no—
491C2×D4—1221no—
492C2×D4—1221no—
493C2×D4—1221no—
494C2×D4—1221no—
495C2×D4—1221no—
496C2×D4—4881no—
497C2×D4—4881no—
498C2×D4—4881no—
499C2×D4—4881no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—1221no—
505C22×C4—1221no—
506C22×C4—4881no—
507C22×C4—4881no—
508C22×C4—4881no—
509C22×C4—4881no—
510C22×C4—2442no—
511C22×C4—2442no—
512C22×C4C2×C22⋊D6111yescharacteristic
513C22×C4C2×C22⋊D6111yescharacteristic
514C22×C4C2×C22⋊D6111yescharacteristic
515C22×C4C2×C22⋊D6111yescharacteristic
516C22×C4C2×C22⋊D6111yescharacteristic
517C22×C4C2×C22⋊D6111yescharacteristic
518C22×C4C23×D6111yescharacteristic
519SD16—1228no—
520SD16—1228no—
521C2×C8—612no—
522C2×C8—612no—
523C2×C8—1221no—
524C2×C8—1221no—
525C2×C8—1221no—
526C2×C8—1221no—
527C2×C8—1221no—
528C2×C8—1221no—
529C2×C8—1221no—
530C2×C8—1221no—
531C2×C8—1221no—
532C2×C8—1221no—
533C4⋊C4—21no—
534C4⋊C4—24no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—21no—
539C4⋊C4—81no—
540C4⋊C4—81no—
541C4⋊C4—1221no—
542C4⋊C4—1221no—
543C4⋊C4—1221no—
544C4⋊C4—1221no—
545C4⋊C4C2×D24412yes—
546C22⋊C4—4881no—
547C22⋊C4—4881no—
548C22⋊C4—4881no—
549C22⋊C4—4881no—
550C22⋊C4—2442no—
551C22⋊C4—2442no—
552C42—21no—
553C42—21no—
554C42—21no—
555C42—21no—
556C42—24no—
557C42—81no—
558C42C2×D24412yes—
559C42#1175822212yes—
560C23—1222no—
561C23—1222no—
562C23—2441no—
563C23—2441no—
564C23—2441no—
565C23—2441no—
566C23—2441no—
567C23—2441no—
568C23—2441no—
569C23—2441no—
570C23—2441no—
571C23—2441no—
572C23—2441no—
573C23—2441no—
574C23C22×C22⋊D6111yescharacteristic
575Q8—22no—
576Q8—22no—
577Q8—41no—
578Q8—41no—
579D4—1222no—
580D4—1222no—
581D4—1222no—
582D4—1222no—
583D4—4884no—
584D4—4884no—
585D4—2441no—
586D4—2441no—
587D4—2441no—
588D4—2441no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—4884no—
618C2×C4—4884no—
619C2×C4—2441no—
620C2×C4—2441no—
621C2×C4—2441no—
622C2×C4—2441no—
623C2×C4—2441no—
624C2×C4—2441no—
625C2×C4#1451975112yes—
626C2×C4#1451975112yes—
627C2×C4C22×D24411yescharacteristic
628C2×C4#41×D6111yescharacteristic
629C2×C4#41×D6111yescharacteristic
630C2×C4#41×D6111yescharacteristic
631C2×C4#41×D6111yescharacteristic
632C2×C4C2×#1175822211yescharacteristic
633C8—1222no—
634C8—1222no—
635C8—2441no—
636C8—2441no—
637C22—4881no—
638C22—4881no—
639C22—4881no—
640C22—4881no—
641C22—4881no—
642C22—4881no—
643C22—4881no—
644C22—4881no—
645C22—2442no—
646C22—2442no—
647C22—2442no—
648C22—2442no—
649C22—2442no—
650C22—2442no—
651C22C2×#1451975111yescharacteristic
652C22#1684026911yescharacteristic
653C22#1684030811yescharacteristic
654C22#1684056511yescharacteristic
655C22#1684057011yescharacteristic
656C22#1684057011yescharacteristic
657C22#1684057011yescharacteristic
658C4—22no—
659C4—22no—
660C4—41no—
661C4—42no—
662C4—41no—
663C4—41no—
664C4—41no—
665C4—42no—
666C4—81no—
667C4—81no—
668C4—4881no—
669C4—4881no—
670C4#1684027411yescharacteristic
671C4#1684027411yescharacteristic
672C4#1684029011yescharacteristic
673C4#1684029011yescharacteristic
674C2—4882no—
675C2—4882no—
676C2#2144905511yescharacteristic
677C2#2145534011yescharacteristic
678C2#2145535811yescharacteristic
679C2#2145599711yescharacteristic
680C2#2145606611yescharacteristic
681C2#2145644811yescharacteristic
682C2#2145649611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2320C2
socleC23×C61C22×D4
prime radicalC2×#14519774C22
hypercenter#2320D61
hypocenterC61#23690
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ #376 ◁ #2320

Complements

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

NH ≅ G/Ntype
1. C2×#16839860C2semidirect
2. C2×C4×#11758201C2semidirect
4. #21447300C2semidirect
5. #21447237C2semidirect
6. #21447204C2semidirect
7. #21447190C2semidirect
8. #21447134C2semidirect
9. #21447134C2semidirect
12. #21445676C2semidirect
13. #21445647C2semidirect
14. C4×#14519697C2semidirect
15. C61×#2320C2semidirect
109. C4×Q8×C61D4semidirect
336. #2320D61semidirect
402. C61#23690semidirect
423. C4×Q8D244semidirect

Maximal subgroups

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

subgroupindex
C2×#168398602
C2×C4×#117582012
#214485442
#214473002
#214472372
#214472042
#214471902
#214471342
#214470532
#214469062
#214456762
#214456472
C4×#145196972
C61×#23202
#2369061