Finite Group Databasebeta

← #29657608

Subgroups — #29657608

Order 15616.

926 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457161C211yesmaximal, characteristic, complemented
2#21456401C211yesmaximal, characteristic, complemented
3#21455616C211yesmaximal, characteristic
4#21445068C212yesmaximal, complemented
5#21445033C212yesmaximal, complemented
6#21444984C212yesmaximal, complemented
7#21444953C212yesmaximal, complemented
8#21444509C211yesmaximal, characteristic, complemented
9#21444274C211yesmaximal, characteristic, complemented
10C2×D4⋊#92418C211yesmaximal, characteristic
11C61×#3501C211yesmaximal, characteristic
12#16840319C2211yescharacteristic
13#16840165C2211yescharacteristic, complemented
14#16840141C2211yescharacteristic, complemented
15#16840114C2211yescharacteristic, complemented
16#16840112C2211yescharacteristic, complemented
17#16840062—22no—
18#16840057C2212yescomplemented
19#16840054—21no—
20#16840052C2212yes—
21#16840042C2212yes—
22#16840001—22no—
23#16840001—22no—
24#16839972C2212yescomplemented
25#16839969—21no—
26C4×#11758202—22no—
27#16839802—21no—
28#16839786C2211yescharacteristic
29C2×C4⋊#92418C2211yescharacteristic
30#16839767—21no—
31C2×C4×#92418C2211yescharacteristic
32#16839511C2212yescomplemented
33#16839508C2212yescomplemented
34Q8⋊#92418C2212yescomplemented
35Q8⋊#92418C2212yescomplemented
36D4⋊#92418C2212yes—
37D4⋊#92418C2212yes—
38C61×#554—21no—
39C61×#554—22no—
40C61×#554—22no—
41C61×#554C2212yes—
42D4×Q8×C61—21no—
43D4×Q8×C61C2212yes—
44C61×#544—21no—
45C61×#543—22no—
46C61×#533C2211yescharacteristic
47C61×#533C2211yescharacteristic
48C2×C22⋊Q8×C61—22no—
49C61×#518—21no—
50C2×C4×D4×C61C2211yescharacteristic
51C2×#11758202—22no—
52C2×#11758202—22no—
53C2×#11758194—21no—
54C22×#92418—21no—
55C22×#92418—21no—
56#14519777C2311yescharacteristic
57#14519777C2311yescharacteristic
58#14519769—21no—
59#14519756—21no—
60#14519756—21no—
61C4×Dic122—21no—
62C4×Dic122—21no—
63#14519736—21no—
64#14519710—22no—
65#14519710—22no—
66#14519710—22no—
67#14519708—22no—
68C4⋊#92418—21no—
69C4⋊#92418—21no—
70C4⋊#92418C2312yes—
71C4⋊#92418C2312yes—
72#14519701—22no—
73#14519701—22no—
74#14519700—22no—
75#14519700—22no—
76#14519698—22no—
77#14519698—22no—
78C4×#92418—21no—
79C4×#92418C2311yescharacteristic
80C4×#92418C2311yescharacteristic
81C22×Q8×C61—22no—
82C22×D4×C61D411yescharacteristic
83C23×C4×C61—21no—
84C4⋊Q8×C61—22no—
85C4⋊Q8×C61—22no—
86C4⋊Q8×C61D411yescharacteristic
87C4⋊Q8×C61D411yescharacteristic
88C4⋊Q8×C61C2311yescharacteristic
89C4⋊Q8×C61C2311yescharacteristic
90#125×C61—22no—
91#125×C61—22no—
92#125×C61—21no—
93#125×C61—21no—
94C22⋊Q8×C61—24no—
95C22⋊Q8×C61—24no—
96C22⋊Q8×C61—24no—
97C22⋊Q8×C61—24no—
98C22⋊Q8×C61—24no—
99C22⋊Q8×C61—22no—
100C22⋊Q8×C61—21no—
101C22⋊Q8×C61—21no—
102C22⋊Q8×C61—24no—
103C22⋊Q8×C61—22no—
104C22⋊Q8×C61D412yes—
105C22⋊Q8×C61D412yes—
106C4×Q8×C61—21no—
107C4×Q8×C61—21no—
108C4×Q8×C61D412yes—
109C4×Q8×C61C2312yes—
110C4×D4×C61—22no—
111C4×D4×C61—22no—
112C4×D4×C61—22no—
113C4×D4×C61—22no—
114C4×D4×C61—21no—
115C4×D4×C61—21no—
116C4×D4×C61D412yes—
117C4×D4×C61C2312yes—
118#117×C61—21no—
119#117×C61—21no—
120C2×C4⋊C4×C61—24no—
121C2×C4⋊C4×C61—22no—
122C2×C4⋊C4×C61D411yescharacteristic
123C2×C22⋊C4×C61—24no—
124C2×C22⋊C4×C61—21no—
125C2×C42×C61C2311yescharacteristic
126C24×C61—21no—
127C2×Q8×C61—24no—
128C2×Q8×C61—21no—
129C2×Q8×C61—21no—
130C2×Q8×C61—82no—
131C2×Q8×C61—82no—
132C2×Q8×C61C2×D412yes—
133C2×Q8×C61C2×D412yes—
134C2×D4×C61—21no—
135C2×D4×C61—22no—
136C2×D4×C61Q1614yescomplemented
137C2×D4×C61C2×D412yes—
138C22×C4×C61—24no—
139C22×C4×C61—24no—
140C22×C4×C61—21no—
141C22×C4×C61—22no—
142C22×C4×C61—22no—
143C22×C4×C61—81no—
144C22×C4×C61—81no—
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C2×Dic122—21no—
149C2×Dic122—21no—
150C4⋊C4×C61—24no—
151C4⋊C4×C61—24no—
152C4⋊C4×C61—24no—
153C4⋊C4×C61—22no—
154C4⋊C4×C61—22no—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—22no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—22no—
164C4⋊C4×C61C2×D412yes—
165C4⋊C4×C61C2×D412yes—
166C4⋊C4×C61C2×D412yes—
167C22⋊C4×C61—24no—
168C22⋊C4×C61—24no—
169C22⋊C4×C61—22no—
170C22⋊C4×C61—22no—
171C22⋊C4×C61—22no—
172C22⋊C4×C61—22no—
173C42×C61—22no—
174C42×C61—22no—
175C42×C61C2×D411yescharacteristic
176C42×C61C2×D411yescharacteristic
177C42×C61C2×D411yescharacteristic
178C42×C61C2411yescharacteristic
179C22⋊Dic61—41no—
180#11758202—44no—
181#11758202—82no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186#11758196—41no—
187#11758196—41no—
188C4×Dic61—41no—
189#11758194—42no—
190C2×#92418—21no—
191C2×#92418—21no—
192C2×#92418—22no—
193C2×#92418—22no—
194C2×#92418—42no—
195C2×#92418—81no—
196C23×C61—22no—
197C23×C61—24no—
198C23×C61—81no—
199C23×C61—81no—
200C23×C61#12011yescharacteristic
201Q8×C61—22no—
202Q8×C61—22no—
203Q8×C61—42no—
204Q8×C61—41no—
205Q8×C61—41no—
206Q8×C61—81no—
207Q8×C61—84no—
208D4×C61—22no—
209D4×C61—24no—
210D4×C61C2×Q1614yes—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—22no—
214C2×C4×C61—22no—
215C2×C4×C61—22no—
216C2×C4×C61—22no—
217C2×C4×C61—22no—
218C2×C4×C61—41no—
219C2×C4×C61—42no—
220C2×C4×C61—42no—
221C2×C4×C61—82no—
222C2×C4×C61—81no—
223C2×C4×C61—84no—
224C2×C4×C61—84no—
225C2×C4×C61#12011yescharacteristic
226C2×C4×C61#12011yescharacteristic
227C2×C4×C61#12011yescharacteristic
228C2×C4×C61C2×Q1612yes—
229C2×C4×C61#13712yes—
230C2×C4×C61C22×D411yescharacteristic
231C2×C4×C61C22×D411yescharacteristic
232C2×C4×C61C22×D411yescharacteristic
233C2×Dic61—41no—
234C2×Dic61—41no—
235Dic122—42no—
236Dic122—41no—
237Dic122—41no—
238#92418—22no—
239#92418—41no—
240#92418—42no—
241#92418—42no—
242#18665—611nomaximal, Sylow
243C22×C61—21no—
244C22×C61—21no—
245C22×C61—24no—
246C22×C61—42no—
247C22×C61—42no—
248C22×C61—84no—
249C22×C61#45112yes—
250C22×C61C2×#12011yescharacteristic
251C4×C61—21no—
252C4×C61—21no—
253C4×C61—21no—
254C4×C61—42no—
255C4×C61—42no—
256C4×C61—42no—
257C4×C61—42no—
258C4×C61—42no—
259C4×C61#45112yes—
260C4×C61C22×Q1611yescharacteristic
261C4×C61C2×#13711yescharacteristic
262C4×C61#57611yescharacteristic
263C4×C61#57711yescharacteristic
264Dic61—81no—
265Dic61—81no—
266#3501D6111yescharacteristic
267#3249—611no—
268#3170—611no—
269#3077—611no—
270#1649—612no—
271#1644—612no—
272#1637—612no—
273#1632—612no—
274#1550—611no—
275#1493—611no—
276C2×D4⋊C8—611no—
277C2×C61—22no—
278C2×C61—44no—
279C2×C61C2×#45111yescharacteristic
280C2×C61#301211yescharacteristic
281C2×C61#301811yescharacteristic
282#554—21no—
283#554—22no—
284#554—22no—
285#554C2×D6112yes—
286D4×Q8—21no—
287D4×Q8C2×D6112yes—
288#544—21no—
289#543—22no—
290#533—611no—
291#533C2×D6111yescharacteristic
292#533C2×D6111yescharacteristic
293C2×C22⋊Q8—22no—
294#518—21no—
295C2×C4×D4C2×D6111yescharacteristic
296#500—611no—
297#494—611no—
298#487—611no—
299#487—611no—
300#478—1222no—
301#477—612no—
302#477—1221no—
303#476—612no—
304#474—612no—
305#467—1222no—
306#467—1222no—
307#462—612no—
308#462—1221no—
309C4×Q16—1222no—
310#428—1221no—
311#423—611no—
312C2×C4⋊C8—611no—
313#418—1221no—
314C2×C4×C8—611no—
315#333—612no—
316#332—612no—
317Q8⋊C8—612no—
318Q8⋊C8—612no—
319D4⋊C8—612no—
320D4⋊C8—612no—
321C61#1866511yescharacteristic, complemented, Sylow
322C22×Q8—22no—
323C22×D4C22⋊D6111yescharacteristic
324C23×C4—21no—
325C2×Q16—1222no—
326C2×Q16—1222no—
327C2×M16—1221no—
328C22×C8—1221no—
329C22×C8—1221no—
330C4⋊Q8—22no—
331C4⋊Q8—22no—
332C4⋊Q8—611no—
333C4⋊Q8—611no—
334C4⋊Q8C22⋊D6111yescharacteristic
335C4⋊Q8C22⋊D6111yescharacteristic
336C4⋊Q8C22×D6111yescharacteristic
337C4⋊Q8C22×D6111yescharacteristic
338#125—22no—
339#125—22no—
340#125—21no—
341#125—21no—
342#125—1221no—
343C22⋊Q8—24no—
344C22⋊Q8—24no—
345C22⋊Q8—24no—
346C22⋊Q8—24no—
347C22⋊Q8—22no—
348C22⋊Q8—24no—
349C22⋊Q8—21no—
350C22⋊Q8—21no—
351C22⋊Q8—22no—
352C22⋊Q8—24no—
353C22⋊Q8—1221no—
354C22⋊Q8—1221no—
355C22⋊Q8C22⋊D6112yes—
356C22⋊Q8C22⋊D6112yes—
357C4×Q8—21no—
358C4×Q8—21no—
359C4×Q8—1221no—
360C4×Q8—1221no—
361C4×Q8C22⋊D6112yes—
362C4×Q8C22×D6112yes—
363C4×D4—22no—
364C4×D4—22no—
365C4×D4—22no—
366C4×D4—22no—
367C4×D4—21no—
368C4×D4—21no—
369C4×D4C22⋊D6112yes—
370C4×D4C22×D6112yes—
371#117—21no—
372#117—21no—
373#117—1221no—
374C2×C4⋊C4—24no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4C22⋊D6111yescharacteristic
377C2×C22⋊C4—24no—
378C2×C22⋊C4—21no—
379C2×C42C22×D6111yescharacteristic
380#107—1222no—
381#107—1222no—
382#107—1222no—
383#106—1222no—
384C4⋊C8—612no—
385C4⋊C8—612no—
386C4⋊C8—1221no—
387C4⋊C8—1221no—
388#103—1222no—
389#103—1222no—
390#103—1222no—
391#103—1222no—
392#102—1222no—
393#102—1222no—
394C4×C8—611no—
395C4×C8—611no—
396C4×C8—1221no—
397C24—21no—
398C2×Q8—24no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—82no—
402C2×Q8—82no—
403C2×Q8—1221no—
404C2×Q8—1221no—
405C2×Q8C2×C22⋊D6112yes—
406C2×Q8C2×C22⋊D6112yes—
407C2×D4—22no—
408C2×D4—21no—
409C2×D4#1175820214yescomplemented
410C2×D4C2×C22⋊D6112yes—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—81no—
417C22×C4—81no—
418C22×C4C2×C22⋊D6111yescharacteristic
419C22×C4C2×C22⋊D6111yescharacteristic
420C22×C4C2×C22⋊D6111yescharacteristic
421Q16—4882no—
422Q16—2444no—
423M16—2442no—
424C2×C8—1221no—
425C2×C8—1221no—
426C2×C8—1222no—
427C2×C8—1222no—
428C2×C8—4881no—
429C2×C8—2442no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—22no—
444C4⋊C4—1221no—
445C4⋊C4—1221no—
446C4⋊C4—1221no—
447C4⋊C4—1221no—
448C4⋊C4—2441no—
449C4⋊C4—2441no—
450C4⋊C4C2×C22⋊D6112yes—
451C4⋊C4C2×C22⋊D6112yes—
452C4⋊C4C2×C22⋊D6112yes—
453C22⋊C4—24no—
454C22⋊C4—24no—
455C22⋊C4—22no—
456C22⋊C4—22no—
457C22⋊C4—22no—
458C22⋊C4—22no—
459C22⋊C4—2441no—
460C42—22no—
461C42—22no—
462C42—2441no—
463C42C2×C22⋊D6111yescharacteristic
464C42C2×C22⋊D6111yescharacteristic
465C42C2×C22⋊D6111yescharacteristic
466C42C23×D6111yescharacteristic
467C23—22no—
468C23—24no—
469C23—81no—
470C23—81no—
471C23#1451974611yescharacteristic
472Q8—22no—
473Q8—22no—
474Q8—42no—
475Q8—41no—
476Q8—41no—
477Q8—81no—
478Q8—84no—
479Q8—2442no—
480Q8—2441no—
481Q8—2441no—
482D4—22no—
483D4—24no—
484D4C2×#1175820214yes—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—82no—
496C2×C4—81no—
497C2×C4—84no—
498C2×C4—84no—
499C2×C4—2441no—
500C2×C4—2441no—
501C2×C4#1451974611yescharacteristic
502C2×C4#1451974611yescharacteristic
503C2×C4#1451974611yescharacteristic
504C2×C4C2×#1175820212yes—
505C2×C4#1451980812yes—
506C2×C4C22×C22⋊D6111yescharacteristic
507C2×C4C22×C22⋊D6111yescharacteristic
508C2×C4C22×C22⋊D6111yescharacteristic
509C8—1222no—
510C8—2441no—
511C8—2442no—
512C8—2442no—
513C22—21no—
514C22—21no—
515C22—24no—
516C22—42no—
517C22—42no—
518C22—84no—
519C22#1683991812yes—
520C22C2×#1451974611yescharacteristic
521C4—21no—
522C4—21no—
523C4—21no—
524C4—42no—
525C4—42no—
526C4—42no—
527C4—42no—
528C4—42no—
529C4—4881no—
530C4—4881no—
531C4#1683991812yes—
532C4C22×#1175820211yescharacteristic
533C4C2×#1451980811yescharacteristic
534C4#1684056511yescharacteristic
535C4#1684057311yescharacteristic
536C2—22no—
537C2—44no—
538C2C2×#1683991811yescharacteristic
539C2#2145512311yescharacteristic
540C2#2145516811yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3501C2
socleC22×C61C2×#120
prime radicalC22×D4×C61D4
hypercenter#3501D61
hypocenterC61#18665
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3501

Complements

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

NH ≅ G/Ntype
1. #21457161C2semidirect
2. #21456401C2semidirect
4. #21445068C2semidirect
5. #21445033C2semidirect
6. #21444984C2semidirect
7. #21444953C2semidirect
8. #21444509C2semidirect
9. #21444274C2semidirect
13. #16840165C22semidirect
14. #16840141C22semidirect
15. #16840114C22semidirect
16. #16840112C22semidirect
18. #16840057C22semidirect
24. #16839972C22semidirect
32. #16839511C22semidirect
33. #16839508C22semidirect
34. Q8⋊#92418C22semidirect
35. Q8⋊#92418C22semidirect
136. C2×D4×C61Q16semidirect
321. C61#18665semidirect
409. C2×D4#11758202semidirect

Maximal subgroups

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

subgroupindex
#214571612
#214564012
#214556162
#214450682
#214450332
#214449842
#214449532
#214445092
#214442742
C2×D4⋊#924182
C61×#35012
#1866561