Finite Group Databasebeta

← #30612697

Subgroups — #30612697

Order 15616.

596 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#21462764C211yesmaximal, characteristic
2#21462712C211yesmaximal, characteristic, complemented
3#21461446C211yesmaximal, characteristic, complemented
4#21461444C211yesmaximal, characteristic, complemented
5#21461432C211yesmaximal, characteristic, complemented
6#21461426C211yesmaximal, characteristic, complemented
7#21461412C211yesmaximal, characteristic, complemented
8#21461409C211yesmaximal, characteristic, complemented
9Q16×#92420C211yesmaximal, characteristic, complemented
10#21461403C211yesmaximal, characteristic, complemented
11#21461395C211yesmaximal, characteristic, complemented
12#21461147C211yesmaximal, characteristic, complemented
13#21461132C211yesmaximal, characteristic
14#21461040C211yesmaximal, characteristic, complemented
15#21455900C211yesmaximal, characteristic
16#16840813C2211yescharacteristic
17Q8×#92420—21no—
18Q8×#92420—21no—
19Q8×#92420C2211yescharacteristic
20Q8×#92420C2211yescharacteristic
21Q8×#92420C2211yescharacteristic
22Q8×#92420C2211yescharacteristic
23Q8×#92420C2211yescharacteristic
24#16840796C2211yescharacteristic
25#16840795—21no—
26#16840794—21no—
27#16840790C2211yescharacteristic
28#16840787—21no—
29#16840779C2211yescharacteristic
30#16840778—21no—
31#16840778—21no—
32#16840778C2211yescharacteristic
33#16840778C2211yescharacteristic
34#16840769—21no—
35#16840765C2211yescharacteristic
36#16840626C2211yescharacteristic
37#16840625C2211yescharacteristic
38#16840624C2211yescharacteristic
39#16840623C2211yescharacteristic
40Q8⋊#92420C2211yescharacteristic
41Q8⋊#92420C2211yescharacteristic
42Q8⋊#92420C2211yescharacteristic
43Q8⋊#92420C2211yescharacteristic
44Q8⋊#92420C2211yescharacteristic
45Q8⋊#92420C2211yescharacteristic
46#16840610C2211yescharacteristic
47#16840610C2211yescharacteristic
48#92420⋊C8C2211yescharacteristic
49#92420⋊C8C2211yescharacteristic
50#16840603C2211yescharacteristic
51C8×#92420C2211yescharacteristic
52#16840298C2211yescharacteristic
53C2×Q8×Dic61C2211yescharacteristic
54#16840090C411yescharacteristic
55#16840076C411yescharacteristic
56#16840071C411yescharacteristic
57#16840071C411yescharacteristic
58#16839986C411yescharacteristic
59#16839986C411yescharacteristic
60#16839901C411yescharacteristic
61Q16×Dic61C2211yescharacteristic
62Q16×Dic61C2211yescharacteristic
63#16839847C2211yescharacteristic
64#16839847C2211yescharacteristic
65#16839812C2211yescharacteristic
66C61×#451C411yescharacteristic
67C2×#11758213—21no—
68C2×#11758217—21no—
69C2×D61⋊C8—21no—
70#14519844—21no—
71#14519844—21no—
72#14519844—21no—
73C4⋊#92420—21no—
74C4⋊#92420—21no—
75C4⋊#92420—21no—
76C4⋊#92420—21no—
77C4⋊#92420—21no—
78C4⋊#92420D411yescharacteristic
79C4⋊#92420D411yescharacteristic
80C4⋊#92420C2311yescharacteristic
81C4⋊#92420C2311yescharacteristic
82C4⋊#92420C2311yescharacteristic
83C4⋊#92420C2311yescharacteristic
84#14519834—21no—
85#14519834—21no—
86C22⋊#92420—21no—
87C22⋊#92420—21no—
88C22⋊#92420D411yescharacteristic
89C22⋊#92420D411yescharacteristic
90#14519832—21no—
91C4×#92420—21no—
92C4×#92420—21no—
93C4×#92420—21no—
94C4×#92420—21no—
95C4×#92420—21no—
96C4×#92420—21no—
97C4×#92420C2311yescharacteristic
98C4×#92420C2311yescharacteristic
99C4×#92420C2311yescharacteristic
100C4×#92420C2311yescharacteristic
101C2×#11758201C2×C411yescharacteristic
102C2×#11758201C2×C411yescharacteristic
103#14519772—21no—
104#14519768C2×C411yescharacteristic
105#14519766C2×C411yescharacteristic
106#14519761—21no—
107#14519751C2×C411yescharacteristic
108Q8×Dic61—21no—
109Q8×Dic61—21no—
110Q8×Dic61C2×C411yescharacteristic
111Q8×Dic61C2×C411yescharacteristic
112Q8×Dic61C2311yescharacteristic
113Q8×Dic61C2311yescharacteristic
114Q8×Dic61C2311yescharacteristic
115#14519739C2311yescharacteristic
116#14519737—21no—
117#14519735—21no—
118C2×C4⋊Dic61—21no—
119C2×C4⋊Dic61C2×C411yescharacteristic
120C2×C4×Dic61—21no—
121C2×C4×Dic61C2311yescharacteristic
122#14519711C2×C411yescharacteristic
123#14519711C2×C411yescharacteristic
124#14519708C2×C411yescharacteristic
125#14519708C2×C411yescharacteristic
126#14519706C2×C411yescharacteristic
127#14519706C2×C411yescharacteristic
128#14519701C2×C411yescharacteristic
129#14519701C2×C411yescharacteristic
130#14519701C2×C411yescharacteristic
131#14519701C2×C411yescharacteristic
132#14519699C2×C411yescharacteristic
133#14519699C2×C411yescharacteristic
134C22⋊#92418C2×C411yescharacteristic
135C8×Dic61C2311yescharacteristic
136C8×Dic61C2311yescharacteristic
137C22×Q8×C61C2×C411yescharacteristic
138C2×Q16×C61C2×C411yescharacteristic
139C2×Q16×C61C2×C411yescharacteristic
140C22⋊Q8×C61C2×C411yescharacteristic
141#103×C61C2×C411yescharacteristic
142#103×C61C2×C411yescharacteristic
143C22⋊C8×C61C2×C411yescharacteristic
144C2×Q8×C61—22no—
145C2×Q8×C61—22no—
146C2×Q8×C61—21no—
147C2×Q8×C61C22×C411yescharacteristic
148C2×Q8×C61C22×C411yescharacteristic
149C2×Q8×C61C22×C411yescharacteristic
150C22×C4×C61—21no—
151C22×C4×C61C22×C411yescharacteristic
152C2×C22⋊D61—21no—
153C22×Dic61—21no—
154C22×Dic61C2×D411yescharacteristic
155C22×Dic61#4111yescharacteristic
156C2×D244C22×C411yescharacteristic
157C2×C4×D61—21no—
158#11758217—81no—
159C2×#92420—21no—
160C2×#92420—21no—
161C2×#92420—21no—
162C2×#92420—21no—
163C2×#92420—21no—
164C2×#92420—21no—
165C2×#92420C2×D411yescharacteristic
166C2×#92420C2×D411yescharacteristic
167C2×#92420C2×D411yescharacteristic
168C2×#92420C2×D411yescharacteristic
169#11758213—81no—
170D61⋊C8—81no—
171Q16×C61—22no—
172Q16×C61—22no—
173C2×C8×C61C22×C411yescharacteristic
174C2×C8×C61C22×C411yescharacteristic
175C4⋊C4×C61—21no—
176C4⋊C4×C61C22×C411yescharacteristic
177C22⋊C4×C61—21no—
178#11758201—22no—
179#11758201—22no—
180#11758198—21no—
181#11758198—21no—
182#11758198—21no—
183#11758198—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61C22×C411yescharacteristic
192C4⋊Dic61C22×C411yescharacteristic
193C4⋊Dic61C22×C411yescharacteristic
194#11758196—21no—
195C4×Dic61—21no—
196C4×Dic61—21no—
197C4×Dic61—21no—
198C4×Dic61—21no—
199C4×Dic61—21no—
200C4×Dic61C22×C411yescharacteristic
201C4×Dic61C2×D411yescharacteristic
202C4×Dic61#4111yescharacteristic
203C4×Dic61C2411yescharacteristic
204C2×#92418C22×C411yescharacteristic
205C2×#92418C22×C411yescharacteristic
206C23×C61C4×D411yescharacteristic
207C22×D61—21no—
208Q8×C61—24no—
209Q8×C61—21no—
210Q8×C61—21no—
211Q8×C61—21no—
212Q8×C61—21no—
213Q8×C61#13112yes—
214Q8×C61#13112yes—
215C2×C4×C61—24no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61C4×D411yescharacteristic
223C2×C4×C61C23×C411yescharacteristic
224C22⋊D61—81no—
225C2×Dic61—21no—
226C2×Dic61—21no—
227C2×Dic61—21no—
228C2×Dic61—21no—
229C2×Dic61—21no—
230C2×Dic61—41no—
231C2×Dic61—81no—
232C2×Dic61C4×D411yescharacteristic
233C2×Dic61C4×D411yescharacteristic
234C2×Dic61C22×D411yescharacteristic
235C2×Dic61C2×#4111yescharacteristic
236D244—21no—
237D244—21no—
238C4×D61—41no—
239C4×D61—41no—
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#17919—611nomaximal, Sylow
255C22×C61—21no—
256C22×C61—21no—
257C22×C61—21no—
258C22×C61C2×C4×D411yescharacteristic
259C2×D61—41no—
260C2×D61—41no—
261C2×D61—41no—
262C4×C61—22no—
263C4×C61—22no—
264C4×C61—21no—
265C4×C61—41no—
266C4×C61—41no—
267C4×C61C2×#13111yescharacteristic
268C4×C61C2×#13111yescharacteristic
269Dic61—21no—
270Dic61—21no—
271Dic61—21no—
272Dic61—41no—
273Dic61—41no—
274Dic61—41no—
275Dic61#57711yescharacteristic
276Dic61#57811yescharacteristic
277#3108—611no—
278#2993—611no—
279#2982—611no—
280#1588—611no—
281#1587—611no—
282#1584—611no—
283#1583—611no—
284#1579—611no—
285SD16⋊C8—611no—
286C8×Q16—611no—
287C8×SD16—611no—
288#1574—611no—
289#1487—611no—
290#1482—611no—
291#1454—611no—
292C2×C61—41no—
293C2×C61#292911yescharacteristic
294C2×C61#294711yescharacteristic
295C2×C61#295711yescharacteristic
296D61—81no—
297#529—611no—
298C2×C4×Q8—611no—
299#483—611no—
300#480—611no—
301#479—611no—
302#479—611no—
303#464—611no—
304#464—611no—
305#451#692311yescharacteristic
306#448—611no—
307#446—611no—
308C8×Q8—611no—
309C8×Q8—611no—
310C8×Q8—611no—
311C8×Q8—611no—
312C8×Q8—611no—
313C8×Q8—1221no—
314C8×Q8—1221no—
315C4×Q16—611no—
316C4×Q16—611no—
317C4×SD16—611no—
318C4×SD16—611no—
319#436—611no—
320#436—1221no—
321#435—1221no—
322C8×D4—611no—
323#433—1221no—
324#431—611no—
325#431—611no—
326#431—611no—
327#431—611no—
328#431—1221no—
329#431—1221no—
330#424—1221no—
331#423—611no—
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#1791911yescharacteristic, complemented, Sylow
349C22×Q8C2×#692311yescharacteristic
350C2×Q16C2×#692311yescharacteristic
351C2×Q16C2×#692311yescharacteristic
352C2×SD16—611no—
353C2×SD16—611no—
354C2×M16—1221no—
355C2×M16—1221no—
356C22×C8—1221no—
357#126—1221no—
358#125—611no—
359#124—611no—
360#123—1221no—
361C22⋊Q8C2×#692311yescharacteristic
362#121—611no—
363C4×Q8—611no—
364C4×Q8—611no—
365C4×Q8—611no—
366C4×Q8—611no—
367C4×Q8—611no—
368C4×Q8—1221no—
369C4×Q8—1221no—
370C4×D4—611no—
371C4×D4—1221no—
372#117—1221no—
373C2×C4⋊C4—611no—
374C2×C4⋊C4—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—611no—
389C4⋊C8—1221no—
390C4⋊C8—1221no—
391C4⋊C8—1221no—
392C4⋊C8—1221no—
393C4⋊C8—1221no—
394C4⋊C8—1221no—
395C4⋊C8—1221no—
396C4⋊C8—1221no—
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—1221no—
409C22⋊C8—1221no—
410C22⋊C8—1221no—
411C22⋊C8—1221no—
412C22⋊C8C2×#692311yescharacteristic
413#97—1221no—
414C4×C8—611no—
415C4×C8—611no—
416C4×C8—611no—
417C4×C8—611no—
418C4×C8—611no—
419C4×C8—611no—
420C4×C8—1221no—
421C4×C8—1221no—
422C4×C8—1221no—
423C4×C8—1221no—
424C4×C8—1221no—
425C4×C8—1221no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—21no—
429C2×Q8C22×#692311yescharacteristic
430C2×Q8C22×#692311yescharacteristic
431C2×Q8C22×#692311yescharacteristic
432C2×D4—611no—
433C2×D4—1221no—
434C22×C4—21no—
435C22×C4—611no—
436C22×C4—611no—
437C22×C4—1221no—
438C22×C4—1221no—
439C22×C4C22×#692311yescharacteristic
440Q16—22no—
441Q16—22no—
442SD16—1222no—
443SD16—1222no—
444M16—4881no—
445M16—4881no—
446C2×C8—611no—
447C2×C8—611no—
448C2×C8—611no—
449C2×C8—611no—
450C2×C8—611no—
451C2×C8—611no—
452C2×C8—1221no—
453C2×C8—1221no—
454C2×C8—1221no—
455C2×C8—1221no—
456C2×C8—1221no—
457C2×C8—1221no—
458C2×C8—4881no—
459C2×C8C22×#692311yescharacteristic
460C2×C8C22×#692311yescharacteristic
461C4⋊C4—21no—
462C4⋊C4—611no—
463C4⋊C4—611no—
464C4⋊C4—611no—
465C4⋊C4—1221no—
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—611no—
483C42—1221no—
484C42—1221no—
485C42—1221no—
486C42—1221no—
487C42—1221no—
488C23—1221no—
489C23D4×#692311yescharacteristic
490Q8—24no—
491Q8—21no—
492Q8—21no—
493Q8—21no—
494Q8—21no—
495Q8#1451986412yes—
496Q8#1451986412yes—
497D4—1221no—
498D4—1221no—
499D4—4881no—
500C2×C4—24no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—611no—
508C2×C4—611no—
509C2×C4—611no—
510C2×C4—611no—
511C2×C4—1221no—
512C2×C4—1221no—
513C2×C4—1221no—
514C2×C4—1221no—
515C2×C4—1221no—
516C2×C4—4881no—
517C2×C4—2441no—
518C2×C4—2441no—
519C2×C4—2441no—
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—21no—
537C22—21no—
538C22—21no—
539C22—2441no—
540C22—2441no—
541C22—2441no—
542C22C2×D4×#692311yescharacteristic
543C4—22no—
544C4—22no—
545C4—21no—
546C4—41no—
547C4—41no—
548C4—611no—
549C4—611no—
550C4—1221no—
551C4—1221no—
552C4—1221no—
553C4—2441no—
554C4—2441no—
555C4—2441no—
556C4C2×#1451986411yescharacteristic
557C4C2×#1451986411yescharacteristic
558C2—41no—
559C2—4881no—
560C2#2146253411yescharacteristic
561C2#2146257711yescharacteristic
562C2#2146259911yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#451C4
socleC22×C61C2×C4×D4
prime radical#14519751C2×C4
cosocleC4×Dic61C24
hypercenter#451#6923
hypocenterC61#17919
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
2. #21462712C2semidirect
3. #21461446C2semidirect
4. #21461444C2semidirect
5. #21461432C2semidirect
6. #21461426C2semidirect
7. #21461412C2semidirect
8. #21461409C2semidirect
9. Q16×#92420C2semidirect
10. #21461403C2semidirect
11. #21461395C2semidirect
12. #21461147C2semidirect
14. #21461040C2semidirect
348. C61#17919semidirect

Maximal subgroups

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

subgroupindex
#214627642
#214627122
#214614462
#214614442
#214614322
#214614262
#214614122
#214614092
Q16×#924202
#214614032
#214613952
#214611472
#214611322
#214610402
#214559002
#1791961