Finite Group Databasebeta

← #30612710

Subgroups — #30612710

Order 15616.

598 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462735C211yesmaximal, characteristic
2#21462724C211yesmaximal, characteristic, complemented
3#21461445C211yesmaximal, characteristic, complemented
4#21461436C211yesmaximal, characteristic, complemented
5#21461433C211yesmaximal, characteristic, complemented
6#21461424C211yesmaximal, characteristic, complemented
7#21461413C211yesmaximal, characteristic, complemented
8SD16⋊#92420C211yesmaximal, characteristic, complemented
9#21461406C211yesmaximal, characteristic, complemented
10SD16×#92420C211yesmaximal, characteristic, complemented
11#21461393C211yesmaximal, characteristic, complemented
12#21461151C211yesmaximal, characteristic
13#21461128C211yesmaximal, characteristic, complemented
14#21461041C211yesmaximal, characteristic, complemented
15#21455887C211yesmaximal, characteristic
16#16840814—21no—
17#16840814—21no—
18#16840813C2211yescharacteristic
19#16840813C2211yescharacteristic
20#16840811C2211yescharacteristic
21#16840811C2211yescharacteristic
22Q8×#92420C2211yescharacteristic
23Q8×#92420C2211yescharacteristic
24#16840789—21no—
25#16840789—21no—
26D4×#92420C2211yescharacteristic
27D4×#92420C2211yescharacteristic
28#16840787—21no—
29#16840787—21no—
30#16840784C2211yescharacteristic
31#16840781C2211yescharacteristic
32#16840780C2211yescharacteristic
33#16840779—21no—
34C2×#14519844—21no—
35D61⋊(C4⋊C8)C2211yescharacteristic
36#16840626C2211yescharacteristic
37#16840625C2211yescharacteristic
38#16840624C2211yescharacteristic
39#16840623C2211yescharacteristic
40#16840612C2211yescharacteristic
41#16840612C2211yescharacteristic
42Q8⋊#92420C2211yescharacteristic
43Q8⋊#92420C2211yescharacteristic
44Q8⋊#92420C2211yescharacteristic
45Q8⋊#92420C2211yescharacteristic
46D4⋊#92420C2211yescharacteristic
47D4⋊#92420C2211yescharacteristic
48#92420⋊C8C2211yescharacteristic
49#92420⋊C8C2211yescharacteristic
50#16840603C2211yescharacteristic
51C8×#92420C2211yescharacteristic
52#16840317C2211yescharacteristic
53C4×#11758224C2211yescharacteristic
54#16840101C411yescharacteristic
55#16840101C411yescharacteristic
56#16840048C411yescharacteristic
57#16840042C411yescharacteristic
58#16839969C411yescharacteristic
59#16839921C411yescharacteristic
60#16839921C411yescharacteristic
61#16839854C2211yescharacteristic
62#16839854C2211yescharacteristic
63SD16×Dic61C2211yescharacteristic
64SD16×Dic61C2211yescharacteristic
65#16839814C2211yescharacteristic
66C61×#461C411yescharacteristic
67C2×D61⋊C8—21no—
68C22×#92420—21no—
69C22×#92420—21no—
70#14519844—21no—
71#14519844—21no—
72#14519844—21no—
73#14519844—21no—
74#14519844—21no—
75#14519844D411yescharacteristic
76#14519844D411yescharacteristic
77C4⋊#92420—21no—
78C4⋊#92420—21no—
79C4⋊#92420—21no—
80C4⋊#92420C2311yescharacteristic
81C4⋊#92420C2311yescharacteristic
82C4⋊#92420C2311yescharacteristic
83C4⋊#92420C2311yescharacteristic
84#14519834—21no—
85#14519834—21no—
86#14519834D411yescharacteristic
87#14519834D411yescharacteristic
88C22⋊#92420—21no—
89C22⋊#92420—21no—
90#14519832—21no—
91#14519832—21no—
92#14519832—21no—
93C4×#92420—21no—
94C4×#92420—21no—
95C4×#92420—21no—
96C4×#92420—21no—
97C4×#92420C2311yescharacteristic
98C4×#92420C2311yescharacteristic
99C4×#92420C2311yescharacteristic
100C4×#92420C2311yescharacteristic
101C2×#11758224C2×C411yescharacteristic
102C2×#11758202C2×C411yescharacteristic
103C2×#11758202C2×C411yescharacteristic
104#14519777C2×C411yescharacteristic
105#14519768—21no—
106#14519757—21no—
107#14519753C2×C411yescharacteristic
108#14519753C2×C411yescharacteristic
109Q8×Dic61C2311yescharacteristic
110Q8×Dic61C2311yescharacteristic
111C4×Dic122C2×C411yescharacteristic
112#14519742—21no—
113#14519742C2311yescharacteristic
114D4×Dic61C2311yescharacteristic
115C4×C22⋊D61—21no—
116#14519737—21no—
117#14519735—21no—
118#14519735C2×C411yescharacteristic
119#14519733—21no—
120C2×C4×Dic61—21no—
121C42×D61C2311yescharacteristic
122#14519710C2×C411yescharacteristic
123#14519710C2×C411yescharacteristic
124#14519709C2×C411yescharacteristic
125#14519709C2×C411yescharacteristic
126C4⋊#92418C2×C411yescharacteristic
127#14519702C2×C411yescharacteristic
128#14519702C2×C411yescharacteristic
129#14519701C2×C411yescharacteristic
130#14519701C2×C411yescharacteristic
131#14519698C2×C411yescharacteristic
132#14519698C2×C411yescharacteristic
133#14519689C2×C411yescharacteristic
134#14519689C2×C411yescharacteristic
135C8×Dic61C2311yescharacteristic
136C8×Dic61C2311yescharacteristic
137C2×SD16×C61C2×C411yescharacteristic
138C2×SD16×C61C2×C411yescharacteristic
139C4⋊Q8×C61C2×C411yescharacteristic
140C4×D4×C61C2×C411yescharacteristic
141C4⋊C8×C61C2×C411yescharacteristic
142#103×C61C2×C411yescharacteristic
143#103×C61C2×C411yescharacteristic
144C2×Q8×C61C22×C411yescharacteristic
145C2×Q8×C61C22×C411yescharacteristic
146C2×D4×C61C22×C411yescharacteristic
147C22×C4×C61—21no—
148C2×C22⋊D61—21no—
149C22×Dic61—21no—
150C22×Dic61—21no—
151#11758224—22no—
152#11758224—22no—
153C2×C4×D61C22×C411yescharacteristic
154C2×C4×D61C2×D411yescharacteristic
155C2×C4×D61#4111yescharacteristic
156C2×Dic122—21no—
157C2×Dic122C22×C411yescharacteristic
158C2×#92420—21no—
159C2×#92420—21no—
160C2×#92420—21no—
161C2×#92420—21no—
162C2×#92420—21no—
163C2×#92420—21no—
164C2×#92420—81no—
165C2×#92420—81no—
166C2×#92420C2×D411yescharacteristic
167C2×#92420C2×D411yescharacteristic
168C2×#92420C2×D411yescharacteristic
169C2×#92420C2×D411yescharacteristic
170D61⋊C8—81no—
171SD16×C61—22no—
172SD16×C61—22no—
173C2×C8×C61C22×C411yescharacteristic
174C2×C8×C61C22×C411yescharacteristic
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61C22×C411yescharacteristic
178C22⋊C4×C61—21no—
179C42×C61C22×C411yescharacteristic
180C22⋊Dic61—21no—
181#11758202—22no—
182#11758202—22no—
183#11758198—21no—
184#11758198—21no—
185#11758198—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61C22×C411yescharacteristic
190C4⋊Dic61C22×C411yescharacteristic
191C4⋊Dic61C22×C411yescharacteristic
192#11758196—21no—
193#11758196—21no—
194#11758196—21no—
195#11758196—21no—
196C4×Dic61—21no—
197C4×Dic61—21no—
198C4×Dic61—21no—
199C4×Dic61—21no—
200C4×Dic61—21no—
201C4×Dic61C2×D411yescharacteristic
202C4×Dic61#4111yescharacteristic
203C4×Dic61C2411yescharacteristic
204C2×#92418C22×C411yescharacteristic
205C2×#92418C22×C411yescharacteristic
206C23×C61—21no—
207C22×D61C4×D411yescharacteristic
208Q8×C61—21no—
209Q8×C61—21no—
210Q8×C61—21no—
211Q8×C61—21no—
212D4×C61—21no—
213D4×C61#13112yes—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—81no—
218C2×C4×C61C4×D411yescharacteristic
219C2×C4×C61C4×D411yescharacteristic
220C2×C4×C61C23×C411yescharacteristic
221C22⋊D61—24no—
222C2×Dic61—24no—
223C2×Dic61—21no—
224C2×Dic61—21no—
225C2×Dic61—21no—
226C2×Dic61—21no—
227C2×Dic61—41no—
228C2×Dic61—41no—
229C2×Dic61C4×D411yescharacteristic
230C2×Dic61C22×D411yescharacteristic
231C2×Dic61C2×#4111yescharacteristic
232C4×D61—21no—
233C4×D61—21no—
234C4×D61—21no—
235C4×D61—21no—
236C4×D61—41no—
237Dic122—21no—
238Dic122—81no—
239Dic122#13112yes—
240#92420—21no—
241#92420—21no—
242#92420—21no—
243#92420—21no—
244#92420—41no—
245#92420—41no—
246#92420—41no—
247#92420—41no—
248#92420—41no—
249#92420—41no—
250C8×C61—21no—
251C8×C61—21no—
252#92418—21no—
253#92418—21no—
254#17921—611nomaximal, Sylow
255C22×C61—22no—
256C22×C61—41no—
257C22×C61—41no—
258C22×C61C2×C4×D411yescharacteristic
259C2×D61—21no—
260C2×D61—21no—
261C2×D61—21no—
262C4×C61—21no—
263C4×C61—21no—
264C4×C61—41no—
265C4×C61—41no—
266C4×C61—41no—
267C4×C61C2×#13111yescharacteristic
268C4×C61C2×#13111yescharacteristic
269Dic61—22no—
270Dic61—21no—
271Dic61—21no—
272Dic61—41no—
273Dic61—41no—
274Dic61—41no—
275Dic61#57711yescharacteristic
276Dic61#57811yescharacteristic
277#3107—611no—
278#2988—611no—
279#2985—611no—
280#1587—611no—
281#1585—611no—
282#1584—611no—
283#1582—611no—
284#1579—611no—
285SD16⋊C8—611no—
286C8×Q16—611no—
287C8×SD16—611no—
288#1573—611no—
289#1488—611no—
290#1481—611no—
291#1454—611no—
292C2×C61—42no—
293C2×C61#292911yescharacteristic
294C2×C61#294711yescharacteristic
295C2×C61#295711yescharacteristic
296D61—41no—
297#533—611no—
298C4×#41—611no—
299#485—611no—
300#485—611no—
301#475—611no—
302#474—611no—
303#462—611no—
304#461#692311yescharacteristic
305#452—611no—
306#452—611no—
307#446—611no—
308#446—611no—
309#446—1221no—
310#446—1221no—
311C8×Q8—611no—
312C8×Q8—611no—
313C8×Q8—611no—
314C8×Q8—611no—
315C4×Q16—611no—
316C4×Q16—611no—
317C4×SD16—611no—
318C4×SD16—611no—
319C8×D4—611no—
320C8×D4—611no—
321C8×D4—1221no—
322C8×D4—1221no—
323#433—1221no—
324#433—1221no—
325#432—611no—
326#431—611no—
327#431—611no—
328#431—611no—
329#431—1221no—
330C2×C4⋊C8—611no—
331C2×C4⋊C8—1221no—
332#335—611no—
333#335—611no—
334#334—611no—
335#334—611no—
336Q8⋊C8—611no—
337Q8⋊C8—611no—
338Q8⋊C8—611no—
339Q8⋊C8—611no—
340Q8⋊C8—611no—
341Q8⋊C8—611no—
342D4⋊C8—611no—
343D4⋊C8—611no—
344#322—611no—
345#322—611no—
346#322—611no—
347C82—611no—
348C61#1792111yescharacteristic, complemented, Sylow
349C2×#41—611no—
350C2×Q16—611no—
351C2×Q16—611no—
352C2×SD16C2×#692311yescharacteristic
353C2×SD16C2×#692311yescharacteristic
354C22×C8—1221no—
355C22×C8—1221no—
356C22×C8—1221no—
357C4⋊Q8—611no—
358C4⋊Q8C2×#692311yescharacteristic
359#125—1221no—
360C22⋊Q8—611no—
361C22⋊Q8—611no—
362C22⋊Q8—1221no—
363C4×Q8—611no—
364C4×Q8—611no—
365C4×Q8—611no—
366C4×Q8—611no—
367C4×Q8—1221no—
368C4×D4—611no—
369C4×D4—1221no—
370C4×D4—1221no—
371C4×D4C2×#692311yescharacteristic
372#117—611no—
373#117—1221no—
374#117—1221no—
375C2×C42—611no—
376C2×C42—1221no—
377#107—611no—
378#107—611no—
379#106—611no—
380#106—611no—
381C4⋊C8—611no—
382C4⋊C8—611no—
383C4⋊C8—611no—
384C4⋊C8—611no—
385C4⋊C8—611no—
386C4⋊C8—611no—
387C4⋊C8—611no—
388C4⋊C8—1221no—
389C4⋊C8—1221no—
390C4⋊C8—1221no—
391C4⋊C8—1221no—
392C4⋊C8—1221no—
393C4⋊C8—1221no—
394C4⋊C8—1221no—
395C4⋊C8—1221no—
396C4⋊C8C2×#692311yescharacteristic
397#103—611no—
398#103—611no—
399#103—611no—
400#103—611no—
401#103C2×#692311yescharacteristic
402#103C2×#692311yescharacteristic
403#102—611no—
404#102—611no—
405C22⋊C8—611no—
406C22⋊C8—611no—
407C22⋊C8—611no—
408C22⋊C8—611no—
409C22⋊C8—1221no—
410C22⋊C8—1221no—
411C22⋊C8—1221no—
412C22⋊C8—1221no—
413#97—1221no—
414#97—1221no—
415#97—1221no—
416C4×C8—611no—
417C4×C8—611no—
418C4×C8—611no—
419C4×C8—611no—
420C4×C8—611no—
421C4×C8—611no—
422C4×C8—1221no—
423C4×C8—1221no—
424C4×C8—1221no—
425C4×C8—1221no—
426#41—1222no—
427#41—1222no—
428C2×Q8—611no—
429C2×Q8—1221no—
430C2×Q8C22×#692311yescharacteristic
431C2×Q8C22×#692311yescharacteristic
432C2×D4—1221no—
433C2×D4C22×#692311yescharacteristic
434C22×C4—21no—
435C22×C4—611no—
436C22×C4—611no—
437C22×C4—611no—
438C22×C4—1221no—
439C22×C4—1221no—
440Q16—1222no—
441Q16—1222no—
442SD16—22no—
443SD16—22no—
444C2×C8—611no—
445C2×C8—611no—
446C2×C8—611no—
447C2×C8—611no—
448C2×C8—611no—
449C2×C8—611no—
450C2×C8—1221no—
451C2×C8—1221no—
452C2×C8—1221no—
453C2×C8—1221no—
454C2×C8—1221no—
455C2×C8—1221no—
456C2×C8—4881no—
457C2×C8—4881no—
458C2×C8—4881no—
459C2×C8C22×#692311yescharacteristic
460C2×C8C22×#692311yescharacteristic
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—611no—
464C4⋊C4—611no—
465C4⋊C4—611no—
466C4⋊C4—1221no—
467C4⋊C4—1221no—
468C4⋊C4—1221no—
469C4⋊C4—1221no—
470C4⋊C4—1221no—
471C4⋊C4—1221no—
472C4⋊C4—1221no—
473C4⋊C4C22×#692311yescharacteristic
474C22⋊C4—21no—
475C22⋊C4—1221no—
476C22⋊C4—1221no—
477C22⋊C4—1221no—
478C22⋊C4—1221no—
479C42—611no—
480C42—611no—
481C42—611no—
482C42—1221no—
483C42—1221no—
484C42—1221no—
485C42—1221no—
486C42—1221no—
487C42C22×#692311yescharacteristic
488C23—21no—
489C23—611no—
490Q8—21no—
491Q8—21no—
492Q8—21no—
493Q8—21no—
494Q8—612no—
495Q8—1221no—
496Q8—4881no—
497D4—21no—
498D4—1224no—
499D4#1451986412yes—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—81no—
504C2×C4—611no—
505C2×C4—611no—
506C2×C4—611no—
507C2×C4—1221no—
508C2×C4—1221no—
509C2×C4—1224no—
510C2×C4—1221no—
511C2×C4—1221no—
512C2×C4—1221no—
513C2×C4—1221no—
514C2×C4—1221no—
515C2×C4—1221no—
516C2×C4—2441no—
517C2×C4—2441no—
518C2×C4—2441no—
519C2×C4D4×#692311yescharacteristic
520C2×C4D4×#692311yescharacteristic
521C2×C4C23×#692311yescharacteristic
522C8—21no—
523C8—21no—
524C8—1221no—
525C8—1221no—
526C8—1221no—
527C8—1221no—
528C8—1221no—
529C8—1221no—
530C8—2441no—
531C8—2441no—
532C8—2441no—
533C8—2441no—
534C8—2441no—
535C8—2441no—
536C22—22no—
537C22—41no—
538C22—41no—
539C22—1221no—
540C22—1221no—
541C22—1221no—
542C22C2×D4×#692311yescharacteristic
543C4—21no—
544C4—21no—
545C4—41no—
546C4—41no—
547C4—41no—
548C4—611no—
549C4—611no—
550C4—1221no—
551C4—1222no—
552C4—1221no—
553C4—2441no—
554C4—2441no—
555C4—2441no—
556C4C2×#1451986411yescharacteristic
557C4D61⋊#13111yescharacteristic
558C2—42no—
559C2—2441no—
560C2#2146253311yescharacteristic
561C2#2146257711yescharacteristic
562C2#2146259711yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#461C4
socleC22×C61C2×C4×D4
prime radicalC2×#11758224C2×C4
cosocleC4×Dic61C24
hypercenter#461#6923
hypocenterC61#17921
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
2. #21462724C2semidirect
3. #21461445C2semidirect
4. #21461436C2semidirect
5. #21461433C2semidirect
6. #21461424C2semidirect
7. #21461413C2semidirect
8. SD16⋊#92420C2semidirect
9. #21461406C2semidirect
10. SD16×#92420C2semidirect
11. #21461393C2semidirect
13. #21461128C2semidirect
14. #21461041C2semidirect
348. C61#17921semidirect

Maximal subgroups

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

subgroupindex
#214627352
#214627242
#214614452
#214614362
#214614332
#214614242
#214614132
SD16⋊#924202
#214614062
SD16×#924202
#214613932
#214611512
#214611282
#214610412
#214558872
#1792161