Finite Group Databasebeta

← #30613145

Subgroups — #30613145

Order 15616.

670 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462620C211yesmaximal, characteristic, complemented
2#21462561C211yesmaximal, characteristic
3#21461456C211yesmaximal, characteristic, complemented
4#21461454C211yesmaximal, characteristic, complemented
5#21461429C211yesmaximal, characteristic, complemented
6#21461427C211yesmaximal, characteristic, complemented
7#21461423C211yesmaximal, characteristic, complemented
8#21461422C211yesmaximal, characteristic, complemented
9#21461404C211yesmaximal, characteristic, complemented
10#21461402C211yesmaximal, characteristic, complemented
11#21461386C211yesmaximal, characteristic, complemented
12#21461164C211yesmaximal, characteristic
13C2×#16840612C211yesmaximal, characteristic, complemented
14#21461020C211yesmaximal, characteristic, complemented
15#21455847C211yesmaximal, characteristic
16C2×#14519864—21no—
17#16840814—21no—
18#16840814—21no—
19#16840812C2211yescharacteristic
20#16840812C2211yescharacteristic
21#16840811C2211yescharacteristic
22#16840811C2211yescharacteristic
23#16840797C2211yescharacteristic
24#16840796C2211yescharacteristic
25#16840794—21no—
26#16840791—21no—
27#16840790C2211yescharacteristic
28D4×#92420C2211yescharacteristic
29#16840765—21no—
30#16840765C2211yescharacteristic
31C2×C4⋊#92420C2211yescharacteristic
32#16840735—21no—
33C4×#11758217—21no—
34C4×#11758217C2211yescharacteristic
35C2×C4×#92420C2211yescharacteristic
36#16840624C2211yescharacteristic
37#16840624C2211yescharacteristic
38#16840623C2211yescharacteristic
39#16840623C2211yescharacteristic
40#16840612C2211yescharacteristic
41#16840612C2211yescharacteristic
42#16840612C2211yescharacteristic
43#16840612C2211yescharacteristic
44#16840610C2211yescharacteristic
45#16840610C2211yescharacteristic
46D4⋊#92420C2211yescharacteristic
47D4⋊#92420C2211yescharacteristic
48#92420⋊C8C2211yescharacteristic
49#92420⋊C8C2211yescharacteristic
50C8×#92420C2211yescharacteristic
51C8×#92420C2211yescharacteristic
52#16840307C2211yescharacteristic
53C2×#14519742C2211yescharacteristic
54#16840080C411yescharacteristic
55#16840046C411yescharacteristic
56#16840046C411yescharacteristic
57#16839961C411yescharacteristic
58#16839961C411yescharacteristic
59#16839903C411yescharacteristic
60#16839900C411yescharacteristic
61#16839853C2211yescharacteristic
62#16839853C2211yescharacteristic
63#16839849C2211yescharacteristic
64#16839849C2211yescharacteristic
65#16839812C2211yescharacteristic
66C61×#481C411yescharacteristic
67#14519864—44no—
68C2×#11758213—21no—
69C2×#11758217—21no—
70C2×#11758217—21no—
71C2×#11758217D411yescharacteristic
72C2×#11758217D411yescharacteristic
73C2×D61⋊C8—21no—
74C22×#92420—21no—
75C22×#92420—21no—
76C22×#92420—21no—
77#14519844—21no—
78#14519844—21no—
79#14519844—21no—
80C4⋊#92420—21no—
81C4⋊#92420—21no—
82C4⋊#92420—21no—
83C4⋊#92420C2311yescharacteristic
84C4⋊#92420C2311yescharacteristic
85C4⋊#92420C2311yescharacteristic
86C4⋊#92420C2311yescharacteristic
87#14519834—21no—
88#14519834—21no—
89C22⋊#92420—21no—
90C22⋊#92420—21no—
91#14519832—21no—
92#14519832—21no—
93#14519832D411yescharacteristic
94#14519832D411yescharacteristic
95C4×#92420—21no—
96C4×#92420—21no—
97C4×#92420—21no—
98C4×#92420C2311yescharacteristic
99C4×#92420C2311yescharacteristic
100C4×#92420C2311yescharacteristic
101C4×#92420C2311yescharacteristic
102C2×#11758224—21no—
103C22×Dic122C2×C411yescharacteristic
104C2×#11758190C2×C411yescharacteristic
105C2×#11758190C2×C411yescharacteristic
106C2×#11758200C2×C411yescharacteristic
107C2×#11758200C2×C411yescharacteristic
108#14519776C2×C411yescharacteristic
109#14519773C2×C411yescharacteristic
110#14519763—21no—
111#14519751C2×C411yescharacteristic
112#14519749—21no—
113#14519742—21no—
114#14519742—21no—
115#14519742C2×C411yescharacteristic
116#14519742C2×C411yescharacteristic
117#14519742C2311yescharacteristic
118#14519742C2311yescharacteristic
119D4×Dic61C2311yescharacteristic
120#14519739C2311yescharacteristic
121#14519737—21no—
122C2×#11758196—21no—
123C2×C4×Dic61—21no—
124C2×C4×Dic61C2311yescharacteristic
125#14519708C2×C411yescharacteristic
126#14519708C2×C411yescharacteristic
127#14519706C2×C411yescharacteristic
128#14519706C2×C411yescharacteristic
129#14519702C2×C411yescharacteristic
130#14519702C2×C411yescharacteristic
131#14519700C2×C411yescharacteristic
132#14519700C2×C411yescharacteristic
133#14519697C2×C411yescharacteristic
134#14519697C2×C411yescharacteristic
135C22⋊#92418C2×C411yescharacteristic
136C8×Dic61C2311yescharacteristic
137C8×Dic61C2311yescharacteristic
138#121×C61C2×C411yescharacteristic
139C2×C4⋊C4×C61C2×C411yescharacteristic
140#106×C61C2×C411yescharacteristic
141#106×C61C2×C411yescharacteristic
142#102×C61C2×C411yescharacteristic
143#102×C61C2×C411yescharacteristic
144C22⋊C8×C61C2×C411yescharacteristic
145C2×D4×C61—21no—
146C2×D4×C61C22×C411yescharacteristic
147C22×C4×C61—21no—
148C22×C4×C61C22×C411yescharacteristic
149C2×C22⋊D61—21no—
150C2×C22⋊D61—21no—
151C2×C22⋊D61—21no—
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61C2×D411yescharacteristic
155C22×Dic61#4111yescharacteristic
156#11758224—42no—
157#11758224—42no—
158C2×D244C22×C411yescharacteristic
159C2×C4×D61—21no—
160C2×Dic122—22no—
161C2×Dic122—21no—
162C2×Dic122—21no—
163C2×Dic122—41no—
164C2×Dic122C22×C411yescharacteristic
165C2×Dic122C22×C411yescharacteristic
166#11758217—22no—
167#11758217—42no—
168#11758217—42no—
169#11758217SD1614yescomplemented
170C2×#92420—21no—
171C2×#92420—21no—
172C2×#92420—21no—
173C2×#92420—21no—
174C2×#92420—21no—
175C2×#92420—21no—
176C2×#92420—42no—
177C2×#92420—42no—
178C2×#92420—42no—
179C2×#92420C2×D411yescharacteristic
180C2×#92420C2×D411yescharacteristic
181C2×#92420C2×D411yescharacteristic
182C2×#92420C2×D411yescharacteristic
183#11758213—42no—
184D61⋊C8—42no—
185C2×C8×C61C22×C411yescharacteristic
186C2×C8×C61C22×C411yescharacteristic
187C4⋊C4×C61—21no—
188C4⋊C4×C61C22×C411yescharacteristic
189C4⋊C4×C61C22×C411yescharacteristic
190C4⋊C4×C61C22×C411yescharacteristic
191C22⋊C4×C61—21no—
192C22⋊Dic61—21no—
193#11758200—22no—
194#11758200—41no—
195#11758198—21no—
196#11758198—21no—
197C4⋊Dic61C22×C411yescharacteristic
198#11758196—21no—
199#11758196—21no—
200#11758196—21no—
201#11758196—21no—
202#11758196—21no—
203C4×Dic61—21no—
204C4×Dic61—21no—
205C4×Dic61—21no—
206C4×Dic61—21no—
207C4×Dic61C22×C411yescharacteristic
208C4×Dic61C2×D411yescharacteristic
209C4×Dic61#4111yescharacteristic
210C4×Dic61C2411yescharacteristic
211C2×#92418C22×C411yescharacteristic
212C2×#92418C22×C411yescharacteristic
213#11758190—22no—
214#11758190—41no—
215C23×C61—21no—
216C23×C61C4×D411yescharacteristic
217C22×D61—21no—
218D4×C61—21no—
219D4×C61—21no—
220D4×C61—42no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—81no—
227C2×C4×C61C4×D411yescharacteristic
228C2×C4×C61C23×C411yescharacteristic
229C22⋊D61—42no—
230C22⋊D61—42no—
231C22⋊D61—81no—
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—41no—
236C2×Dic61—42no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239C2×Dic61C4×D411yescharacteristic
240C2×Dic61C4×D411yescharacteristic
241C2×Dic61C2×SD1612yes—
242C2×Dic61#13512yes—
243C2×Dic61C22×D411yescharacteristic
244C2×Dic61C2×#4111yescharacteristic
245D244—21no—
246D244—21no—
247C4×D61—41no—
248C4×D61—41no—
249Dic122—21no—
250Dic122—21no—
251Dic122—21no—
252Dic122—42no—
253Dic122—42no—
254Dic122#13112yescomplemented
255#92420—22no—
256#92420—21no—
257#92420—21no—
258#92420—41no—
259#92420—41no—
260#92420—41no—
261#92420—41no—
262#92420—41no—
263#92420C2×SD1612yes—
264#92420C2×SD1612yes—
265C8×C61—21no—
266C8×C61—21no—
267#92418—21no—
268#92418—21no—
269#17997—611nomaximal, Sylow
270C22×C61—21no—
271C22×C61—21no—
272C22×C61—41no—
273C22×C61—41no—
274C22×C61—41no—
275C22×C61C4×SD1612yes—
276C22×C61C2×C4×D411yescharacteristic
277C2×D61—41no—
278C2×D61—41no—
279C2×D61—41no—
280C4×C61—21no—
281C4×C61—41no—
282C4×C61—41no—
283C4×C61—41no—
284C4×C61C2×#13111yescharacteristic
285C4×C61#56811yescharacteristic
286Dic61—22no—
287Dic61—21no—
288Dic61—21no—
289Dic61—41no—
290Dic61—41no—
291Dic61C4×SD1612yes—
292Dic61C22×SD1611yescharacteristic
293Dic61C2×#13511yescharacteristic
294#3103—611no—
295Q8×M16—611no—
296#2935—611no—
297#1590—611no—
298#1590—611no—
299#1583—611no—
300#1583—611no—
301#1582—611no—
302#1582—611no—
303C8×SD16—611no—
304C8×SD16—611no—
305#1570—611no—
306#1492—611no—
307C2×Q8⋊C8—611no—
308C8×M16—611no—
309C2×C61—22no—
310C2×C61—81no—
311C2×C61#293311yescharacteristic
312C2×C61C2×C4×SD1611yescharacteristic
313C2×C61#295611yescharacteristic
314D61—81no—
315C2×#131—1221no—
316#532—611no—
317C2×C4×Q8—611no—
318#481—611no—
319#481#692311yescharacteristic
320#475—611no—
321#475—611no—
322#460—611no—
323#460—611no—
324#448—611no—
325#448—611no—
326#446—611no—
327#446—611no—
328#446—1221no—
329#446—1221no—
330C8×Q8—611no—
331C8×Q8—611no—
332C4×SD16—611no—
333C4×SD16—611no—
334C4×SD16—611no—
335C4×SD16—611no—
336#436—611no—
337#436—611no—
338#435—1221no—
339#435—1221no—
340C8×D4—611no—
341C8×D4—611no—
342#431—611no—
343#423—611no—
344#423—1221no—
345C2×C4⋊C8—611no—
346#408—1221no—
347C4×M16—611no—
348C4×M16—1221no—
349C2×C4×C8—611no—
350#334—611no—
351#334—611no—
352#334—611no—
353#334—611no—
354Q8⋊C8—611no—
355Q8⋊C8—611no—
356Q8⋊C8—611no—
357Q8⋊C8—611no—
358D4⋊C8—611no—
359D4⋊C8—611no—
360D4⋊C8—611no—
361D4⋊C8—611no—
362#322—611no—
363#322—611no—
364C82—611no—
365C82—611no—
366C61#1799711yescharacteristic, complemented, Sylow
367C2×#41—1221no—
368C22×Q8—611no—
369C2×SD16—611no—
370C2×SD16—611no—
371C2×SD16—611no—
372C2×SD16—611no—
373#131—2444no—
374C2×M16—611no—
375C2×M16—611no—
376C2×M16—1221no—
377C2×M16—1221no—
378C2×M16—1221no—
379C22×C8—1221no—
380C22×C8—1221no—
381C22×C8—1221no—
382C22×C8—1221no—
383C4⋊Q8—611no—
384#127—611no—
385#124—1221no—
386#121—611no—
387#121—1221no—
388#121C2×#692311yescharacteristic
389C4×Q8—611no—
390C4×Q8—611no—
391C4×Q8—611no—
392C4×Q8—611no—
393C4×Q8—1221no—
394C4×Q8—1221no—
395C4×D4—611no—
396C4×D4—611no—
397C4×D4—1221no—
398C2×C4⋊C4—1221no—
399C2×C4⋊C4C2×#692311yescharacteristic
400C2×C42—611no—
401C2×C42—1221no—
402#106—611no—
403#106—611no—
404#106C2×#692311yescharacteristic
405#106C2×#692311yescharacteristic
406C4⋊C8—611no—
407C4⋊C8—611no—
408C4⋊C8—611no—
409C4⋊C8—611no—
410C4⋊C8—611no—
411C4⋊C8—611no—
412C4⋊C8—1221no—
413C4⋊C8—1221no—
414C4⋊C8—1221no—
415C4⋊C8—1221no—
416C4⋊C8—1221no—
417C4⋊C8—1221no—
418#103—611no—
419#103—611no—
420#103—611no—
421#103—611no—
422#102—611no—
423#102—611no—
424#102C2×#692311yescharacteristic
425#102C2×#692311yescharacteristic
426C22⋊C8—611no—
427C22⋊C8—1221no—
428C22⋊C8—1221no—
429C22⋊C8—1221no—
430C22⋊C8—1221no—
431C22⋊C8C2×#692311yescharacteristic
432#97—611no—
433#97—611no—
434#97—1221no—
435#97—1221no—
436C4×C8—611no—
437C4×C8—611no—
438C4×C8—611no—
439C4×C8—611no—
440C4×C8—611no—
441C4×C8—611no—
442C4×C8—1221no—
443C4×C8—1221no—
444C4×C8—1221no—
445#41—2442no—
446#41—2442no—
447C2×Q8—611no—
448C2×Q8—611no—
449C2×Q8—1221no—
450C2×Q8—1222no—
451C2×Q8—1221no—
452C2×Q8—2441no—
453C2×D4—21no—
454C2×D4—611no—
455C2×D4—1221no—
456C2×D4—1221no—
457C2×D4—1221no—
458C2×D4C22×#692311yescharacteristic
459C22×C4—21no—
460C22×C4—611no—
461C22×C4—611no—
462C22×C4—1221no—
463C22×C4—1221no—
464C22×C4—1221no—
465C22×C4C22×#692311yescharacteristic
466SD16—1222no—
467SD16—1222no—
468SD16—2441no—
469SD16—2441no—
470M16—614no—
471M16—1222no—
472M16—2442no—
473M16—2442no—
474M16—2442no—
475C2×C8—611no—
476C2×C8—611no—
477C2×C8—611no—
478C2×C8—611no—
479C2×C8—611no—
480C2×C8—611no—
481C2×C8—1221no—
482C2×C8—1221no—
483C2×C8—1221no—
484C2×C8—1221no—
485C2×C8—1221no—
486C2×C8—1221no—
487C2×C8—2442no—
488C2×C8—2442no—
489C2×C8—2442no—
490C2×C8—2442no—
491C2×C8C22×#692311yescharacteristic
492C2×C8C22×#692311yescharacteristic
493C4⋊C4—21no—
494C4⋊C4—611no—
495C4⋊C4—1221no—
496C4⋊C4—1221no—
497C4⋊C4—1221no—
498C4⋊C4—1221no—
499C4⋊C4—1221no—
500C4⋊C4C22×#692311yescharacteristic
501C4⋊C4C22×#692311yescharacteristic
502C4⋊C4C22×#692311yescharacteristic
503C22⋊C4—21no—
504C22⋊C4—1221no—
505C22⋊C4—1221no—
506C22⋊C4—1221no—
507C42—611no—
508C42—611no—
509C42—611no—
510C42—611no—
511C42—1221no—
512C42—1221no—
513C42—1221no—
514C42—1221no—
515C23—21no—
516C23—1221no—
517C23D4×#692311yescharacteristic
518Q8—612no—
519Q8—1221no—
520Q8—1221no—
521Q8—1221no—
522Q8—2442no—
523Q8—2442no—
524D4—21no—
525D4—21no—
526D4—42no—
527D4—1221no—
528D4—1221no—
529D4—4881no—
530D4—2442no—
531D4—2442no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—81no—
538C2×C4—612no—
539C2×C4—612no—
540C2×C4—611no—
541C2×C4—611no—
542C2×C4—611no—
543C2×C4—611no—
544C2×C4—1221no—
545C2×C4—1221no—
546C2×C4—1221no—
547C2×C4—2441no—
548C2×C4—2441no—
549C2×C4—2441no—
550C2×C4—2441no—
551C2×C4—2441no—
552C2×C4—2442no—
553C2×C4D4×#692311yescharacteristic
554C2×C4C23×#692311yescharacteristic
555C8—21no—
556C8—21no—
557C8—612no—
558C8—612no—
559C8—1221no—
560C8—1221no—
561C8—1222no—
562C8—1221no—
563C8—1221no—
564C8—2441no—
565C8—2441no—
566C8—2441no—
567C8—2441no—
568C8—2441no—
569C22—21no—
570C22—21no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—2441no—
575C22—2441no—
576C22—2441no—
577C22SD16×#692312yes—
578C22C2×D4×#692311yescharacteristic
579C4—21no—
580C4—41no—
581C4—41no—
582C4—41no—
583C4—611no—
584C4—611no—
585C4—612no—
586C4—1222no—
587C4—1221no—
588C4—1221no—
589C4—2441no—
590C4—2441no—
591C4D61⋊#13111yescharacteristic
592C4#1684084111yescharacteristic
593C2—22no—
594C2—81no—
595C2—4881no—
596C2#2146255311yescharacteristic
597C2C2×SD16×#692311yescharacteristic
598C2#2146259311yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#481C4
socleC22×C61C2×C4×D4
prime radical#16840307C22
cosocleC4×Dic61C24
hypercenter#481#6923
hypocenterC61#17997
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
1. #21462620C2semidirect
3. #21461456C2semidirect
4. #21461454C2semidirect
5. #21461429C2semidirect
6. #21461427C2semidirect
7. #21461423C2semidirect
8. #21461422C2semidirect
9. #21461404C2semidirect
10. #21461402C2semidirect
11. #21461386C2semidirect
13. C2×#16840612C2semidirect
14. #21461020C2semidirect
169. #11758217SD16semidirect
254. Dic122#131semidirect
366. C61#17997semidirect

Maximal subgroups

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

subgroupindex
#214626202
#214625612
#214614562
#214614542
#214614292
#214614272
#214614232
#214614222
#214614042
#214614022
#214613862
#214611642
C2×#168406122
#214610202
#214558472
#1799761