Finite Group Databasebeta

← #29702037

Subgroups — #29702037

Order 15616.

850 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840042C211yesmaximal, characteristic
2#21448908C211yesmaximal, characteristic, complemented
3#21446791C211yesmaximal, characteristic, complemented
4#21446585C211yesmaximal, characteristic, complemented
5#21446554C211yesmaximal, characteristic, complemented
6#21446549C211yesmaximal, characteristic, complemented
7#21446408C211yesmaximal, characteristic, complemented
8#21446387C211yesmaximal, characteristic, complemented
9#21446080C211yesmaximal, characteristic
10#21446066C211yesmaximal, characteristic
11#21445694C211yesmaximal, characteristic, complemented
12#21445644C211yesmaximal, characteristic
13#21445644C211yesmaximal, characteristic
14C4×#14519698C211yesmaximal, characteristic
15C61×#2349C211yesmaximal, characteristic
16C2×#14519777C2211yescharacteristic
17#16840042C418yescomplemented
18C2×#14519710C2211yescharacteristic
19C2×#14519710C2211yescharacteristic
20C2×#14519710C2211yescharacteristic
21C2×#14519708C2211yescharacteristic
22C2×C4⋊#92418C2211yescharacteristic
23C2×#14519698C2211yescharacteristic
24C2×#14519698C2211yescharacteristic
25C2×#14519698C2211yescharacteristic
26C2×#14519698C2211yescharacteristic
27C2×#14519688C2211yescharacteristic
28C2×#14519688C2211yescharacteristic
29C2×C4×#92418C2211yescharacteristic
30#16839666—21no—
31#16839666—21no—
32#16839660C2211yescharacteristic
33#16839660C2211yescharacteristic
34#16839658—21no—
35C4⋊C4×Dic61—21no—
36C4×C4⋊Dic61C2211yescharacteristic
37C4×C4⋊Dic61C2211yescharacteristic
38C4×C4⋊Dic61C2211yescharacteristic
39#16839524C2211yescharacteristic
40#16839524C2211yescharacteristic
41#16839524C2211yescharacteristic
42#16839524C2211yescharacteristic
43#16839524C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839517C2211yescharacteristic
48#16839517C2211yescharacteristic
49C2×C4×D4×C61C2211yescharacteristic
50C2×C4×D4×C61C2211yescharacteristic
51C2×C4×D4×C61C2211yescharacteristic
52C61×#389C2211yescharacteristic
53C61×#389C2211yescharacteristic
54C61×#385—21no—
55C61×#385—21no—
56C61×#382C2211yescharacteristic
57C61×#380—21no—
58C4×C4⋊C4×C61C2211yescharacteristic
59C2×#95×C61—21no—
60C22×Dic122—21no—
61C22×#92418C2311yescharacteristic
62C22×#92418C2311yescharacteristic
63C22×#92418C2311yescharacteristic
64C22×#92418C2311yescharacteristic
65#14519777—22no—
66#14519777C2×C414yescomplemented
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×#11758196—21no—
73C2×#11758196—21no—
74C2×C4×Dic61—21no—
75C2×C4×Dic61—21no—
76C2×C4×Dic61—21no—
77#14519710—41no—
78#14519710C2×C414yescomplemented
79#14519710C2×C414yescomplemented
80#14519708—41no—
81C4⋊#92418C2×C414yescomplemented
82#14519698—22no—
83#14519698C2×C414yes—
84#14519698C2×C414yes—
85#14519698Q814yescomplemented
86#14519688—22no—
87#14519688—41no—
88C4×#92418—41no—
89#14519684—21no—
90#14519684—21no—
91#14519684—21no—
92#14519684—21no—
93#14519684—21no—
94C22×D4×C61C2311yescharacteristic
95C23×C4×C61—21no—
96C23×C4×C61—21no—
97C23×C4×C61—21no—
98C4×D4×C61—22no—
99C4×D4×C61—22no—
100C4×D4×C61—22no—
101C4×D4×C61—41no—
102C4×D4×C61—41no—
103C4×D4×C61C2×C414yes—
104C2×C4⋊C4×C61—21no—
105C2×C4⋊C4×C61—21no—
106C2×C4⋊C4×C61D411yescharacteristic
107C2×C4⋊C4×C61D411yescharacteristic
108C2×C4⋊C4×C61C2311yescharacteristic
109C2×C4⋊C4×C61C2311yescharacteristic
110C2×C4⋊C4×C61C2311yescharacteristic
111C2×C22⋊C4×C61—21no—
112C2×C22⋊C4×C61—21no—
113C2×C22⋊C4×C61—21no—
114C2×C42×C61C2311yescharacteristic
115C2×C42×C61C2311yescharacteristic
116C2×C42×C61C2311yescharacteristic
117#95×C61—21no—
118#95×C61—21no—
119#95×C61—21no—
120#95×C61—21no—
121#95×C61D411yescharacteristic
122#95×C61D411yescharacteristic
123C24×C61—21no—
124C2×D4×C61—21no—
125C2×D4×C61—21no—
126C2×D4×C61—21no—
127C2×D4×C61C22×C412yes—
128C2×D4×C61C2×Q812yes—
129C2×D4×C61#4112yes—
130C22×C4×C61—21no—
131C22×C4×C61—24no—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—42no—
135C22×C4×C61—42no—
136C22×C4×C61—42no—
137C22×C4×C61—42no—
138C22×C4×C61—81no—
139C22×C4×C61—81no—
140C22×C4×C61—81no—
141C22×C4×C61—81no—
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2411yescharacteristic
149C22×Dic61—21no—
150C22×Dic61—21no—
151C22×Dic61—21no—
152C22×Dic61—21no—
153C2×Dic122—24no—
154C2×Dic122—81no—
155C2×Dic122—81no—
156C4⋊C4×C61—22no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—41no—
163C4⋊C4×C61—81no—
164C4⋊C4×C61—81no—
165C4⋊C4×C61C22×C412yes—
166C22⋊C4×C61—24no—
167C22⋊C4×C61—42no—
168C22⋊C4×C61—42no—
169C42×C61—21no—
170C42×C61—21no—
171C42×C61—21no—
172C42×C61—21no—
173C42×C61C22×C412yes—
174C42×C61#4112yes—
175C4⋊Dic61—24no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61C22×C412yes—
182C4⋊Dic61C22×C412yes—
183C4⋊Dic61C2×Q812yes—
184#11758196—81no—
185#11758196—81no—
186C4×Dic61—42no—
187C4×Dic61—81no—
188C4×Dic61—81no—
189C2×#92418—21no—
190C2×#92418—21no—
191C2×#92418—21no—
192C2×#92418—21no—
193C2×#92418—21no—
194C2×#92418—21no—
195C2×#92418—21no—
196C2×#92418—21no—
197C2×#92418—21no—
198C2×#92418C22×C412yes—
199C2×#92418C22×C412yes—
200C2×#92418C2×Q812yes—
201C23×C61—22no—
202C23×C61—22no—
203C23×C61—22no—
204C23×C61—41no—
205C23×C61—41no—
206C23×C61—41no—
207C23×C61—41no—
208C23×C61C22×D411yescharacteristic
209D4×C61—22no—
210D4×C61—22no—
211D4×C61—22no—
212D4×C61C4×Q814yes—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—22no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—84no—
238C2×C4×C61—84no—
239C2×C4×C61—84no—
240C2×C4×C61C4×D412yes—
241C2×C4×C61C4×D412yes—
242C2×C4×C61#13712yes—
243C2×C4×C61C23×C411yescharacteristic
244C2×C4×C61C22×Q811yescharacteristic
245C2×C4×C61C2×#4111yescharacteristic
246C2×C4×C61C2×#4111yescharacteristic
247C2×C4×C61H(5,𝔽2)11yescharacteristic
248C2×C4×C612−1+411yescharacteristic
249C2×Dic61—22no—
250C2×Dic61—22no—
251C2×Dic61—22no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261Dic122—84no—
262#92418—22no—
263#92418—22no—
264#92418—22no—
265#92418—41no—
266#22979—611nomaximal, Sylow
267C22×C61—24no—
268C22×C61—42no—
269C22×C61—42no—
270C22×C61—42no—
271C22×C61—42no—
272C22×C61—42no—
273C22×C61—42no—
274C22×C61C2×C4×D411yescharacteristic
275C22×C61#54611yescharacteristic
276C22×C61D4×Q811yescharacteristic
277C22×C61C2×#13711yescharacteristic
278C22×C61#57511yescharacteristic
279C22×C61#57611yescharacteristic
280C22×C61#57611yescharacteristic
281C4×C61—22no—
282C4×C61—22no—
283C4×C61—22no—
284C4×C61—42no—
285C4×C61—41no—
286C4×C61—41no—
287C4×C61—41no—
288C4×C61—81no—
289C4×C61—81no—
290C4×C61C2×C4×Q811yescharacteristic
291C4×C61#52011yescharacteristic
292C4×C61#55211yescharacteristic
293C4×C61#55411yescharacteristic
294Dic61—42no—
295Dic61—42no—
296Dic61—42no—
297Dic61—81no—
298C2×#474—611no—
299#2369—611no—
300#2349D6111yescharacteristic
301#1993—611no—
302#1949—611no—
303#1943—611no—
304#1942—611no—
305#1921—611no—
306#1918—611no—
307#1864—611no—
308#1862—611no—
309#1771—611no—
310#1762—611no—
311#1762—611no—
312C4×#102—611no—
313C2×C61—44no—
314C2×C61#236011yescharacteristic
315C2×C61C4×#13711yescharacteristic
316C2×C61#294811yescharacteristic
317C2×C61#331411yescharacteristic
318C2×C61#331911yescharacteristic
319C2×C61#338511yescharacteristic
320C2×C61#338911yescharacteristic
321C2×C4⋊Q8—611no—
322C2×C4×D4C2×D6111yescharacteristic
323C2×C4×D4C2×D6111yescharacteristic
324C2×C4×D4C2×D6111yescharacteristic
325#474—618no—
326C2×#107—611no—
327C2×#107—611no—
328C2×#107—611no—
329C2×#106—611no—
330C2×C4⋊C8—611no—
331C2×#102—611no—
332C2×#102—611no—
333C2×#102—611no—
334C2×#102—611no—
335C2×#97—611no—
336C2×#97—611no—
337C2×C4×C8—611no—
338#391—1221no—
339#391—1221no—
340#389—611no—
341#389—611no—
342#389—1221no—
343#389C2×D6111yescharacteristic
344#389C2×D6111yescharacteristic
345#385—21no—
346#385—21no—
347#382C2×D6111yescharacteristic
348#380—21no—
349C4×C4⋊C4—611no—
350C4×C4⋊C4—611no—
351C4×C4⋊C4—611no—
352C4×C4⋊C4—1221no—
353C4×C4⋊C4C2×D6111yescharacteristic
354C2×#95—21no—
355#340—611no—
356#340—611no—
357#340—611no—
358#340—611no—
359#340—611no—
360#340—611no—
361#340—611no—
362#340—611no—
363#336—611no—
364#336—611no—
365C61#2297911yescharacteristic, complemented, Sylow
366C22×Q8—1221no—
367C22×D4C22×D6111yescharacteristic
368C23×C4—21no—
369C23×C4—21no—
370C23×C4—21no—
371C22×C8—611no—
372C22×C8—611no—
373C22×C8—611no—
374C22×C8—611no—
375C4⋊Q8—614no—
376C4⋊Q8—1222no—
377C4×D4—22no—
378C4×D4—22no—
379C4×D4—22no—
380C4×D4—41no—
381C4×D4—41no—
382C4×D4C4×D6114yes—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—611no—
386C2×C4⋊C4—611no—
387C2×C4⋊C4—611no—
388C2×C4⋊C4—611no—
389C2×C4⋊C4—1221no—
390C2×C4⋊C4—1221no—
391C2×C4⋊C4—1221no—
392C2×C4⋊C4C22⋊D6111yescharacteristic
393C2×C4⋊C4C22⋊D6111yescharacteristic
394C2×C4⋊C4C22×D6111yescharacteristic
395C2×C4⋊C4C22×D6111yescharacteristic
396C2×C4⋊C4C22×D6111yescharacteristic
397C2×C22⋊C4—21no—
398C2×C22⋊C4—21no—
399C2×C22⋊C4—21no—
400C2×C42—1221no—
401C2×C42—1221no—
402C2×C42—1221no—
403C2×C42C22×D6111yescharacteristic
404C2×C42C22×D6111yescharacteristic
405C2×C42C22×D6111yescharacteristic
406#107—614no—
407#107—614no—
408#107—2441no—
409#106—2441no—
410C4⋊C8—614no—
411#102—614no—
412#102—614no—
413#102—614no—
414#102—1222no—
415#97—1222no—
416#97—2441no—
417C4×C8—2441no—
418#95—21no—
419#95—21no—
420#95—21no—
421#95—21no—
422#95—1221no—
423#95—1221no—
424#95—1221no—
425#95—1221no—
426#95—1221no—
427#95C22⋊D6111yescharacteristic
428#95C22⋊D6111yescharacteristic
429C24—21no—
430C2×Q8—1224no—
431C2×Q8—4881no—
432C2×Q8—4881no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4C2×C4×D6112yes—
437C2×D4#1175822412yes—
438C2×D4Q8×D6112yes—
439C22×C4—21no—
440C22×C4—24no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—81no—
448C22×C4—81no—
449C22×C4—81no—
450C22×C4—81no—
451C22×C4—1221no—
452C22×C4—1221no—
453C22×C4—1221no—
454C22×C4—1221no—
455C22×C4C2×C22⋊D6111yescharacteristic
456C22×C4C2×C22⋊D6111yescharacteristic
457C22×C4C2×C22⋊D6111yescharacteristic
458C22×C4C2×C22⋊D6111yescharacteristic
459C22×C4C2×C22⋊D6111yescharacteristic
460C22×C4C2×C22⋊D6111yescharacteristic
461C22×C4C23×D6111yescharacteristic
462C2×C8—612no—
463C2×C8—612no—
464C2×C8—612no—
465C2×C8—1221no—
466C2×C8—1221no—
467C2×C8—1221no—
468C2×C8—1221no—
469C2×C8—1221no—
470C2×C8—1221no—
471C2×C8—1221no—
472C2×C8—1221no—
473C2×C8—1221no—
474C4⋊C4—21no—
475C4⋊C4—22no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—41no—
481C4⋊C4—81no—
482C4⋊C4—81no—
483C4⋊C4—612no—
484C4⋊C4—612no—
485C4⋊C4—612no—
486C4⋊C4—1221no—
487C4⋊C4—1221no—
488C4⋊C4—1221no—
489C4⋊C4—1224no—
490C4⋊C4—1221no—
491C4⋊C4—1221no—
492C4⋊C4—4881no—
493C4⋊C4—4881no—
494C4⋊C4C2×C4×D6112yes—
495C22⋊C4—24no—
496C22⋊C4—42no—
497C22⋊C4—42no—
498C42—21no—
499C42—21no—
500C42—21no—
501C42—21no—
502C42—4881no—
503C42—4881no—
504C42—2442no—
505C42C2×C4×D6112yes—
506C42#1175822212yes—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23C22×C22⋊D6111yescharacteristic
515Q8—4884no—
516D4—22no—
517D4—22no—
518D4—22no—
519D4#1451974214yes—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—84no—
545C2×C4—84no—
546C2×C4—84no—
547C2×C4—1222no—
548C2×C4—1222no—
549C2×C4—1222no—
550C2×C4—2441no—
551C2×C4—2441no—
552C2×C4—2441no—
553C2×C4—2441no—
554C2×C4—2441no—
555C2×C4—2441no—
556C2×C4—2441no—
557C2×C4—2441no—
558C2×C4—2441no—
559C2×C4C4×C22⋊D6112yes—
560C2×C4C4×C22⋊D6112yes—
561C2×C4#1451980712yes—
562C2×C4C22×C4×D6111yescharacteristic
563C2×C4C2×Q8×D6111yescharacteristic
564C2×C4C2×#1175822411yescharacteristic
565C2×C4C2×#1175822211yescharacteristic
566C2×C4#1451982211yescharacteristic
567C2×C4#1451982511yescharacteristic
568C8—1222no—
569C8—1222no—
570C8—1222no—
571C8—2441no—
572C22—24no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22C2×C4×C22⋊D6111yescharacteristic
580C22#1684040611yescharacteristic
581C22Q8×C22⋊D6111yescharacteristic
582C22C2×#1451980711yescharacteristic
583C22#1684056011yescharacteristic
584C22#1684056511yescharacteristic
585C22#1684056511yescharacteristic
586C4—22no—
587C4—22no—
588C4—22no—
589C4—41no—
590C4—41no—
591C4—42no—
592C4—41no—
593C4—81no—
594C4—81no—
595C4—4881no—
596C4—2442no—
597C4—2442no—
598C4—2442no—
599C4C2×#1451974211yescharacteristic
600C4#1684024311yescharacteristic
601C4#1684044711yescharacteristic
602C4#1684046711yescharacteristic
603C2—44no—
604C2#2144882411yescharacteristic
605C2C4×#1451980711yescharacteristic
606C2#2145466911yescharacteristic
607C2#2145790011yescharacteristic
608C2#2145797511yescharacteristic
609C2#2145872911yescharacteristic
610C2#2145877311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2349C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
cosocleC22×C4×C61C24
hypercenter#2349D61
hypocenterC61#22979
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2349

Complements

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

NH ≅ G/Ntype
2. #21448908C2semidirect
3. #21446791C2semidirect
4. #21446585C2semidirect
5. #21446554C2semidirect
6. #21446549C2semidirect
7. #21446408C2semidirect
8. #21446387C2semidirect
11. #21445694C2semidirect
17. #16840042C4semidirect
66. #14519777C2×C4semidirect
78. #14519710C2×C4semidirect
79. #14519710C2×C4semidirect
81. C4⋊#92418C2×C4semidirect
85. #14519698Q8semidirect
365. C61#22979semidirect

Maximal subgroups

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

subgroupindex
C2×#168400422
#214489082
#214467912
#214465852
#214465542
#214465492
#214464082
#214463872
#214460802
#214460662
#214456942
#214456442
C4×#145196982
C61×#23492
#2297961