Finite Group Databasebeta

← #30613204

Subgroups — #30613204

Order 15616.

670 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
1#21462651C211yesmaximal, characteristic, complemented
2#21462560C211yesmaximal, characteristic, complemented
3#21461469C211yesmaximal, characteristic, complemented
4#21461455C211yesmaximal, characteristic, complemented
5#21461448C211yesmaximal, characteristic, complemented
6#21461441C211yesmaximal, characteristic, complemented
7#21461425C211yesmaximal, characteristic, complemented
8#21461420C211yesmaximal, characteristic, complemented
9#21461409C211yesmaximal, characteristic, complemented
10#21461399C211yesmaximal, characteristic, complemented
11#21461396C211yesmaximal, characteristic, complemented
12#21461175C211yesmaximal, characteristic, complemented
13#21461119C211yesmaximal, characteristic, complemented
14#21461037C211yesmaximal, characteristic, complemented
15#21455865C211yesmaximal, characteristic
16C2×D61⋊M16—21no—
17#16840813C2211yescharacteristic, complemented
18Q8×#92420C2211yescharacteristic, complemented
19#16840797C2211yescharacteristic
20#16840797C2211yescharacteristic
21#16840797C2211yescharacteristic, complemented
22#16840795—21no—
23#16840794—21no—
24#16840794—21no—
25#16840791—21no—
26#16840790C2211yescharacteristic
27#16840790C2211yescharacteristic
28D4×#92420C2211yescharacteristic, complemented
29#16840786—21no—
30#16840781C2211yescharacteristic
31#16840780C2211yescharacteristic
32#16840766C2211yescharacteristic
33#16840733—21no—
34#16840726—21no—
35C4×#11758213C2211yescharacteristic
36#16840626C2211yescharacteristic, complemented
37#16840626C2211yescharacteristic, complemented
38#16840623C2211yescharacteristic, complemented
39#16840623C2211yescharacteristic, complemented
40Q8⋊#92420C2211yescharacteristic, complemented
41Q8⋊#92420C2211yescharacteristic, complemented
42#16840610C2211yescharacteristic
43#16840610C2211yescharacteristic
44D4⋊#92420C2211yescharacteristic
45D4⋊#92420C2211yescharacteristic
46D4⋊#92420C2211yescharacteristic, complemented
47D4⋊#92420C2211yescharacteristic, complemented
48#92420⋊C8C2211yescharacteristic, complemented
49#92420⋊C8C2211yescharacteristic, complemented
50#16840603C2211yescharacteristic, complemented
51C8×#92420C2211yescharacteristic, complemented
52#16840291C2211yescharacteristic
53C4×D4×D61C2211yescharacteristic
54#16840096C411yescharacteristic
55#16840079C411yescharacteristic
56#16840063C411yescharacteristic
57#16840062C411yescharacteristic
58#16839974C411yescharacteristic
59#16839911C411yescharacteristic
60#16839894C411yescharacteristic
61SD16×Dic61C2211yescharacteristic
62#16839847C2211yescharacteristic
63D8×Dic61C2211yescharacteristic
64#16839842C2211yescharacteristic
65#16839814C2211yescharacteristic
66C61×#463C411yescharacteristic
67C2×#11758213—21no—
68C2×#11758213D411yescharacteristic
69C2×#11758213D411yescharacteristic
70D61⋊M16—44no—
71C2×#11758217—21no—
72C2×#11758217—21no—
73C2×#11758217—21no—
74C2×D61⋊C8—21no—
75C2×D61⋊C8—21no—
76C22×#92420—21no—
77#14519844—21no—
78#14519844—21no—
79C4⋊#92420—21no—
80C4⋊#92420—21no—
81C4⋊#92420C2311yescharacteristic
82C4⋊#92420C2311yescharacteristic
83C4⋊#92420C2311yescharacteristic
84C4⋊#92420C2311yescharacteristic
85#14519834—21no—
86#14519834—21no—
87#14519834—21no—
88#14519834—21no—
89C22⋊#92420—21no—
90C22⋊#92420—21no—
91C22⋊#92420—21no—
92C22⋊#92420—21no—
93#14519832—21no—
94#14519832D411yescharacteristic
95#14519832D411yescharacteristic
96C4×#92420—21no—
97C4×#92420—21no—
98C4×#92420C2311yescharacteristic
99C4×#92420C2311yescharacteristic
100C4×#92420C2311yescharacteristic
101C4×#92420C2311yescharacteristic
102C2×D4×D61C2×C411yescharacteristic
103C22×C4×D61—21no—
104C2×#11758201C2×C411yescharacteristic
105C2×#11758199C2×C411yescharacteristic
106#14519770—21no—
107#14519769C2×C411yescharacteristic
108#14519759—21no—
109#14519753C2×C411yescharacteristic
110#14519748C2×C411yescharacteristic
111Q8×Dic61C2311yescharacteristic
112C4×D244C2×C411yescharacteristic
113D4×Dic61C2311yescharacteristic
114D4×Dic61C2311yescharacteristic
115#14519739C2311yescharacteristic
116C4×C22⋊D61—21no—
117#14519737—21no—
118#14519737—21no—
119#14519733—21no—
120C4⋊C4×D61C2×C411yescharacteristic
121C22⋊C4×D61—21no—
122C42×D61C2311yescharacteristic
123#14519711C2×C411yescharacteristic
124#14519710C2×C411yescharacteristic
125#14519709C2×C411yescharacteristic
126#14519708C2×C411yescharacteristic
127C4⋊#92418C2×C411yescharacteristic
128#14519701C2×C411yescharacteristic
129#14519699C2×C411yescharacteristic
130#14519699C2×C411yescharacteristic
131#14519698C2×C411yescharacteristic
132#14519698C2×C411yescharacteristic
133#14519698C2×C411yescharacteristic
134#14519689C2×C411yescharacteristic
135#14519689C2×C411yescharacteristic
136C8×Dic61C2311yescharacteristic
137C8×Dic61C2311yescharacteristic
138C2×SD16×C61C2×C411yescharacteristic
139C2×D8×C61C2×C411yescharacteristic
140#124×C61C2×C411yescharacteristic
141C4×D4×C61C2×C411yescharacteristic
142C4⋊C8×C61C2×C411yescharacteristic
143#103×C61C2×C411yescharacteristic
144#102×C61C2×C411yescharacteristic
145C23×D61—21no—
146C2×Q8×C61C22×C411yescharacteristic
147C2×D4×C61C22×C411yescharacteristic
148C2×D4×C61C22×C411yescharacteristic
149C22×C4×C61—21no—
150C2×C22⋊D61—21no—
151C2×C22⋊D61—21no—
152C22×Dic61—21no—
153C22×Dic61—21no—
154D4×D61—22no—
155D4×D61—41no—
156C2×D244C22×C411yescharacteristic
157C2×C4×D61—21no—
158C2×C4×D61—21no—
159C2×C4×D61—21no—
160C2×C4×D61—42no—
161C2×C4×D61—42no—
162C2×C4×D61C22×C411yescharacteristic
163C2×C4×D61C2×D411yescharacteristic
164C2×C4×D61#4111yescharacteristic
165#11758217—42no—
166#11758217—42no—
167#11758217—42no—
168C2×#92420—21no—
169C2×#92420—21no—
170C2×#92420—21no—
171C2×#92420—21no—
172C2×#92420—21no—
173C2×#92420—21no—
174C2×#92420—42no—
175C2×#92420C2×D411yescharacteristic
176C2×#92420C2×D411yescharacteristic
177C2×#92420C2×D411yescharacteristic
178C2×#92420C2×D411yescharacteristic
179#11758213—22no—
180#11758213—41no—
181#11758213—42no—
182D61⋊C8—42no—
183D61⋊C8—42no—
184SD16×C61—41no—
185D8×C61—41no—
186C2×C8×C61C22×C411yescharacteristic
187C2×C8×C61C22×C411yescharacteristic
188C4⋊C4×C61C22×C411yescharacteristic
189C22⋊C4×C61—21no—
190C22⋊C4×C61—21no—
191C22⋊C4×C61—21no—
192C42×C61C22×C411yescharacteristic
193C22⋊Dic61—21no—
194C22⋊Dic61—21no—
195C22⋊Dic61—21no—
196#11758201—22no—
197#11758199—22no—
198#11758198—21no—
199#11758198—21no—
200#11758198—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61C22×C411yescharacteristic
203C4⋊Dic61C22×C411yescharacteristic
204C4⋊Dic61C22×C411yescharacteristic
205#11758196—21no—
206#11758196—21no—
207#11758196—21no—
208#11758196—21no—
209C4×Dic61—21no—
210C4×Dic61C2×D411yescharacteristic
211C4×Dic61#4111yescharacteristic
212C4×Dic61C2411yescharacteristic
213C2×#92418C22×C411yescharacteristic
214C2×#92418C22×C411yescharacteristic
215C23×C61—21no—
216C23×C61—21no—
217C22×D61—21no—
218C22×D61—42no—
219C22×D61—42no—
220C22×D61—81no—
221C22×D61C4×D411yescharacteristic
222Q8×C61—21no—
223Q8×C61—21no—
224D4×C61—21no—
225D4×C61—21no—
226D4×C61—21no—
227D4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—42no—
231C2×C4×C61C4×D411yescharacteristic
232C2×C4×C61C4×D411yescharacteristic
233C2×C4×C61C23×C411yescharacteristic
234C22⋊D61—42no—
235C22⋊D61—81no—
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61—21no—
239C2×Dic61—41no—
240C2×Dic61—41no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243C2×Dic61C4×D411yescharacteristic
244C2×Dic61C22×D411yescharacteristic
245C2×Dic61C2×#4111yescharacteristic
246D244—21no—
247D244#13112yes—
248C4×D61—21no—
249C4×D61—21no—
250C4×D61—21no—
251C4×D61—21no—
252C4×D61—41no—
253C4×D61—42no—
254C4×D61—42no—
255C4×D61—41no—
256C4×D61—41no—
257#92420—21no—
258#92420—21no—
259#92420—21no—
260#92420—41no—
261#92420—41no—
262#92420—41no—
263#92420—41no—
264#92420—41no—
265#92420—41no—
266#92420AGL(1,ℤ/8)12yes—
267C8×C61—21no—
268C8×C61—21no—
269#92418—21no—
270#92418—21no—
271#18006—611nomaximal, Sylow
272C22×C61—41no—
273C22×C61—41no—
274C22×C61—41no—
275C22×C61—41no—
276C22×C61—41no—
277C22×C61—41no—
278C22×C61C2×C4×D411yescharacteristic
279C2×D61—21no—
280C2×D61—22no—
281C2×D61—21no—
282C2×D61—21no—
283C2×D61—41no—
284C2×D61—41no—
285C2×D61—84no—
286C4×C61—21no—
287C4×C61—21no—
288C4×C61—41no—
289C4×C61—41no—
290C4×C61C2×#13111yescharacteristic
291C4×C61#56811yescharacteristic
292Dic61—21no—
293Dic61—21no—
294Dic61—41no—
295Dic61—41no—
296Dic61—41no—
297Dic61C2×AGL(1,ℤ/8)11yescharacteristic
298Dic61#57511yescharacteristic
299#3105—611no—
300#2969—611no—
301D4×M16—611no—
302#1594—611no—
303#1590—611no—
304#1588—611no—
305#1586—611no—
306#1582—611no—
307#1581—611no—
308SD16⋊C8—611no—
309C8×D8—611no—
310#1574—611no—
311#1495—611no—
312#1478—611no—
313#1453—611no—
314C2×C61—81no—
315C2×C61—81no—
316C2×C61#293311yescharacteristic
317C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
318C2×C61#295711yescharacteristic
319D61—41no—
320D61—42no—
321C22×M16—1221no—
322#529—611no—
323C2×C4×D4—611no—
324#484—611no—
325#481—611no—
326#478—611no—
327#478—611no—
328#463—611no—
329#463#692311yescharacteristic
330#450—611no—
331#447—611no—
332#446—611no—
333C8×Q8—611no—
334C4×SD16—611no—
335C4×SD16—611no—
336C4×D8—611no—
337C4×D8—611no—
338#436—611no—
339#436—611no—
340#436—611no—
341#436—1221no—
342#435—1221no—
343#435—1221no—
344#435—1221no—
345C8×D4—611no—
346C8×D4—611no—
347C8×D4—611no—
348#433—1221no—
349#431—611no—
350#431—611no—
351#431—611no—
352#423—611no—
353#407—1221no—
354#405—1221no—
355C4×M16—611no—
356#335—611no—
357#335—611no—
358#334—611no—
359#334—611no—
360Q8⋊C8—611no—
361Q8⋊C8—611no—
362D4⋊C8—611no—
363D4⋊C8—611no—
364D4⋊C8—611no—
365D4⋊C8—611no—
366D4⋊C8—611no—
367D4⋊C8—611no—
368#322—611no—
369#322—611no—
370#322—611no—
371C82—611no—
372C61#1800611yescharacteristic, complemented, Sylow
373C22×D4—611no—
374C23×C4—1221no—
375C2×SD16—611no—
376C2×SD16C2×#692311yescharacteristic
377C2×D8—611no—
378C2×D8C2×#692311yescharacteristic
379C2×M16—611no—
380C2×M16—611no—
381C2×M16—1221no—
382C2×M16—1221no—
383C2×M16—1221no—
384C2×M16—1221no—
385C2×M16—2444no—
386C22×C8—1221no—
387C22×C8—1221no—
388C22×C8—1221no—
389#126—1221no—
390#125—611no—
391#124C2×#692311yescharacteristic
392#123—1221no—
393C22⋊Q8—611no—
394#121—611no—
395C4×Q8—611no—
396C4×D4—611no—
397C4×D4—611no—
398C4×D4—611no—
399C4×D4—611no—
400C4×D4—1221no—
401C4×D4—1221no—
402C4×D4—1221no—
403C4×D4C2×#692311yescharacteristic
404#117—1221no—
405C2×C4⋊C4—611no—
406C2×C22⋊C4—1221no—
407C2×C42—611no—
408#107—611no—
409#107—611no—
410#106—611no—
411#106—611no—
412C4⋊C8—611no—
413C4⋊C8—611no—
414C4⋊C8—611no—
415C4⋊C8—611no—
416C4⋊C8—611no—
417C4⋊C8—1221no—
418C4⋊C8—1221no—
419C4⋊C8—1221no—
420C4⋊C8—1221no—
421C4⋊C8C2×#692311yescharacteristic
422#103—611no—
423#103C2×#692311yescharacteristic
424#102—611no—
425#102—611no—
426#102—611no—
427#102—611no—
428#102—611no—
429#102C2×#692311yescharacteristic
430C22⋊C8—611no—
431C22⋊C8—611no—
432C22⋊C8—1221no—
433C22⋊C8—1221no—
434C22⋊C8—1221no—
435C22⋊C8—1221no—
436C22⋊C8—1221no—
437C22⋊C8—1221no—
438C22⋊C8—1221no—
439C22⋊C8—1221no—
440#97—611no—
441#97—611no—
442#97—1221no—
443C4×C8—611no—
444C4×C8—611no—
445C4×C8—611no—
446C4×C8—611no—
447C4×C8—611no—
448C4×C8—611no—
449C4×C8—1221no—
450C4×C8—1221no—
451C24—1221no—
452C2×Q8C22×#692311yescharacteristic
453C2×D4—611no—
454C2×D4—1221no—
455C2×D4—1222no—
456C2×D4—1221no—
457C2×D4—2441no—
458C2×D4C22×#692311yescharacteristic
459C2×D4C22×#692311yescharacteristic
460C22×C4—21no—
461C22×C4—611no—
462C22×C4—611no—
463C22×C4—611no—
464C22×C4—1221no—
465C22×C4—1221no—
466C22×C4—1221no—
467C22×C4—1221no—
468C22×C4—1221no—
469C22×C4—2442no—
470C22×C4—2442no—
471SD16—41no—
472SD16—1222no—
473D8—41no—
474D8—1222no—
475M16—1222no—
476M16—2442no—
477M16—2441no—
478M16—2442no—
479M16—2442no—
480M16—2442no—
481C2×C8—611no—
482C2×C8—611no—
483C2×C8—611no—
484C2×C8—611no—
485C2×C8—611no—
486C2×C8—611no—
487C2×C8—1221no—
488C2×C8—1221no—
489C2×C8—1221no—
490C2×C8—1221no—
491C2×C8—1221no—
492C2×C8—1221no—
493C2×C8—2442no—
494C2×C8—2442no—
495C2×C8—2442no—
496C2×C8C22×#692311yescharacteristic
497C2×C8C22×#692311yescharacteristic
498C4⋊C4—611no—
499C4⋊C4—611no—
500C4⋊C4—611no—
501C4⋊C4—1221no—
502C4⋊C4—1221no—
503C4⋊C4—1221no—
504C4⋊C4—1221no—
505C4⋊C4—1221no—
506C4⋊C4C22×#692311yescharacteristic
507C22⋊C4—21no—
508C22⋊C4—21no—
509C22⋊C4—21no—
510C22⋊C4—1221no—
511C22⋊C4—1221no—
512C22⋊C4—1221no—
513C22⋊C4—1221no—
514C22⋊C4—1221no—
515C22⋊C4—1221no—
516C42—611no—
517C42—611no—
518C42—611no—
519C42—1221no—
520C42C22×#692311yescharacteristic
521C23—21no—
522C23—21no—
523C23—611no—
524C23—1221no—
525C23—4881no—
526C23—2442no—
527C23—2442no—
528Q8—21no—
529Q8—21no—
530D4—21no—
531D4—21no—
532D4—21no—
533D4—21no—
534D4—612no—
535D4—1221no—
536D4—4881no—
537D4—2442no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—42no—
541C2×C4—611no—
542C2×C4—611no—
543C2×C4—611no—
544C2×C4—1221no—
545C2×C4—1221no—
546C2×C4—1221no—
547C2×C4—1221no—
548C2×C4—1221no—
549C2×C4—1221no—
550C2×C4—1221no—
551C2×C4—2441no—
552C2×C4—2442no—
553C2×C4—2441no—
554C2×C4—2441no—
555C2×C4—2442no—
556C2×C4—2441no—
557C2×C4—2441no—
558C2×C4—2441no—
559C2×C4—2441no—
560C2×C4D4×#692311yescharacteristic
561C2×C4D4×#692311yescharacteristic
562C2×C4C23×#692311yescharacteristic
563C8—21no—
564C8—21no—
565C8—612no—
566C8—1221no—
567C8—1221no—
568C8—1221no—
569C8—1221no—
570C8—1221no—
571C8—2441no—
572C8—2441no—
573C8—2441no—
574C8—2441no—
575C8—2441no—
576C8—2441no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22—1221no—
584C22—1222no—
585C22—1221no—
586C22—1221no—
587C22—4884no—
588C22—2441no—
589C22—2441no—
590C22C2×D4×#692311yescharacteristic
591C4—21no—
592C4—21no—
593C4—41no—
594C4—41no—
595C4—611no—
596C4—611no—
597C4—1221no—
598C4—1221no—
599C4—2441no—
600C4—2441no—
601C4—2441no—
602C4D61⋊#13111yescharacteristic
603C4#1684084011yescharacteristic
604C2—81no—
605C2—81no—
606C2—2441no—
607C2—2442no—
608C2#2146255011yescharacteristic
609C2AGL(1,ℤ/8)×#692311yescharacteristic
610C2#2146259811yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#463C4
socleC22×C61C2×C4×D4
prime radical#16839894C4
cosocleC4×Dic61C24
hypercenter#463#6923
hypocenterC61#18006
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. #21462651C2semidirect
2. #21462560C2semidirect
3. #21461469C2semidirect
4. #21461455C2semidirect
5. #21461448C2semidirect
6. #21461441C2semidirect
7. #21461425C2semidirect
8. #21461420C2semidirect
9. #21461409C2semidirect
10. #21461399C2semidirect
11. #21461396C2semidirect
12. #21461175C2semidirect
13. #21461119C2semidirect
14. #21461037C2semidirect
17. #16840813C22semidirect
18. Q8×#92420C22semidirect
21. #16840797C22semidirect
28. D4×#92420C22semidirect
36. #16840626C22semidirect
37. #16840626C22semidirect
38. #16840623C22semidirect
39. #16840623C22semidirect
40. Q8⋊#92420C22semidirect
41. Q8⋊#92420C22semidirect
46. D4⋊#92420C22semidirect
47. D4⋊#92420C22semidirect
48. #92420⋊C8C22semidirect
49. #92420⋊C8C22semidirect
50. #16840603C22semidirect
51. C8×#92420C22semidirect
372. C61#18006semidirect

Maximal subgroups

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

subgroupindex
#214626512
#214625602
#214614692
#214614552
#214614482
#214614412
#214614252
#214614202
#214614092
#214613992
#214613962
#214611752
#214611192
#214610372
#214558652
#1800661