Finite Group Databasebeta

← #29657182

Subgroups — #29657182

Order 15616.

1056 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456967C211yesmaximal, characteristic, complemented
2C2×#16840134C211yesmaximal, characteristic, complemented
3#21455634C211yesmaximal, characteristic, complemented
4#21444946C214yesmaximal, complemented
5#21444940C214yesmaximal, complemented
6#21444487C211yesmaximal, characteristic, complemented
7#21444274C211yesmaximal, characteristic, complemented
8#21444201C211yesmaximal, characteristic, complemented
9C61×#3508C211yesmaximal, characteristic
10C22×#11758200—22no—
11C2×#14519777C2211yescharacteristic
12#16840165—21no—
13#16840134—22no—
14#16840134C2214yes—
15#16840112—21no—
16#16840095—22no—
17#16840057C2212yescomplemented
18#16840057C2214yescomplemented
19#16840052C2212yescomplemented
20#16840047—21no—
21#16840042—21no—
22#16839815C2211yescharacteristic
23#16839805—21no—
24#16839786C2211yescharacteristic
25#16839763—21no—
26C2×C4×#92418C2211yescharacteristic
27#16839506C2214yes—
28Q8⋊#92418C2212yescomplemented
29Q8⋊#92418C2214yescomplemented
30D4⋊#92418C2212yescomplemented
31C61×#559—21no—
32C61×#559C2212yes—
33C61×#550—21no—
34C61×#550C2212yes—
35C61×#546—22no—
36C61×#546—22no—
37C61×#540—22no—
38C61×#535—21no—
39C61×#533C2211yescharacteristic
40C61×#532—21no—
41C61×#532—21no—
42C2×#127×C61C2211yescharacteristic
43C61×#518—21no—
44C4×#41×C61C2211yescharacteristic
45C22×Dic122—21no—
46C2×#11758200—28no—
47C2×#11758200—48no—
48C2×#11758194—21no—
49C22×#92418—21no—
50#14519777—21no—
51#14519777—21no—
52#14519777—21no—
53#14519777C2312yes—
54C2×C4⋊Dic61—21no—
55C2×C4⋊Dic61—21no—
56#14519710—22no—
57#14519710—24no—
58#14519708—22no—
59C4⋊#92418—21no—
60C4⋊#92418C2312yes—
61C4⋊#92418C2312yes—
62#14519701—22no—
63#14519701—24no—
64#14519698—22no—
65C22⋊#92418—21no—
66C4×#92418—21no—
67C4×#92418—21no—
68C4×#92418C2312yes—
69C2×#41×C61—22no—
70C2×#41×C61—22no—
71C2×#41×C61D411yescharacteristic
72C22×D4×C61—21no—
73C22×D4×C61—21no—
74C22×D4×C61D412yes—
75C4⋊Q8×C61D411yescharacteristic
76C4⋊Q8×C61D411yescharacteristic
77#127×C61—22no—
78#127×C61—21no—
79#127×C61—21no—
80#127×C61D412yes—
81#127×C61C2312yes—
82#125×C61—21no—
83#124×C61—22no—
84#124×C61—22no—
85#124×C61—22no—
86#123×C61—24no—
87#123×C61—24no—
88C22⋊Q8×C61—21no—
89C22⋊Q8×C61D412yes—
90#121×C61—22no—
91#121×C61—22no—
92#121×C61—22no—
93#121×C61—24no—
94#121×C61—24no—
95#121×C61—22no—
96#120×C61—24no—
97C4×Q8×C61—21no—
98C4×Q8×C61D411yescharacteristic
99C4×Q8×C61C2312yes—
100C4×Q8×C61C2311yescharacteristic
101C4×D4×C61—22no—
102C4×D4×C61—22no—
103C4×D4×C61—22no—
104C4×D4×C61—21no—
105C4×D4×C61—21no—
106C4×D4×C61—22no—
107C4×D4×C61—22no—
108C4×D4×C61—22no—
109C4×D4×C61D411yescharacteristic
110C4×D4×C61C2311yescharacteristic
111#117×C61—21no—
112#117×C61—21no—
113#117×C61D411yescharacteristic
114C2×C22⋊C4×C61—24no—
115C2×C42×C61—21no—
116C2×C42×C61C2311yescharacteristic
117C24×C61—22no—
118#41×C61—22no—
119#41×C61—22no—
120#41×C61—84no—
121#41×C61—84no—
122C2×Q8×C61—21no—
123C2×Q8×C61C2×D411yescharacteristic
124C2×Q8×C61C2×D412yes—
125C2×D4×C61—21no—
126C2×D4×C61—24no—
127C2×D4×C61—24no—
128C2×D4×C61—22no—
129C2×D4×C61—22no—
130C2×D4×C61—22no—
131C2×D4×C61—22no—
132C2×D4×C61—22no—
133C2×D4×C61—44no—
134C2×D4×C61—42no—
135C2×D4×C61—42no—
136C2×D4×C61—82no—
137C2×D4×C61C2×D411yescharacteristic
138C2×D4×C61C2×D414yes—
139C22×C4×C61—24no—
140C22×C4×C61—24no—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×Dic61—41no—
148C2×Dic122—22no—
149C2×Dic122—44no—
150C2×Dic122—41no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—22no—
158C4⋊C4×C61C2×D411yescharacteristic
159C4⋊C4×C61C2×D411yescharacteristic
160C4⋊C4×C61C2×D412yes—
161C22⋊C4×C61—24no—
162C22⋊C4×C61—24no—
163C22⋊C4×C61—24no—
164C22⋊C4×C61—24no—
165C22⋊C4×C61—21no—
166C22⋊C4×C61—22no—
167C22⋊C4×C61—21no—
168C22⋊C4×C61—21no—
169C42×C61—21no—
170C42×C61—21no—
171C42×C61—21no—
172C42×C61—22no—
173C42×C61C2×D411yescharacteristic
174C42×C61C2×D411yescharacteristic
175C42×C61C2×D411yescharacteristic
176C42×C61C2411yescharacteristic
177#11758200—416no—
178C4⋊Dic61—22no—
179C4⋊Dic61—22no—
180C4⋊Dic61—41no—
181C4⋊Dic61—41no—
182#11758194—42no—
183C2×#92418—24no—
184C2×#92418—22no—
185C2×#92418—22no—
186C2×#92418—22no—
187C23×C61—24no—
188C23×C61—21no—
189C23×C61—44no—
190C23×C61—82no—
191C23×C61—82no—
192C23×C61#12011yescharacteristic
193Q8×C61—21no—
194Q8×C61—21no—
195Q8×C61—24no—
196Q8×C61—42no—
197D4×C61—21no—
198D4×C61—21no—
199D4×C61—44no—
200D4×C61—42no—
201D4×C61—42no—
202D4×C61—42no—
203D4×C61—44no—
204D4×C61—44no—
205D4×C61—44no—
206D4×C61—82no—
207D4×C61—84no—
208D4×C61—84no—
209C2×C4×C61—22no—
210C2×C4×C61—22no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—22no—
215C2×C4×C61—22no—
216C2×C4×C61—42no—
217C2×C4×C61—42no—
218C2×C4×C61—42no—
219C2×C4×C61—84no—
220C2×C4×C61—84no—
221C2×C4×C61#12011yescharacteristic
222C2×C4×C61#12011yescharacteristic
223C2×C4×C61#12011yescharacteristic
224C2×C4×C61#13512yes—
225C2×C4×C61#13712yes—
226C2×C4×C61C22×D411yescharacteristic
227C2×C4×C61C22×D411yescharacteristic
228C2×C4×C61C22×D411yescharacteristic
229C2×Dic61—42no—
230C2×Dic61—82no—
231Dic122—44no—
232Dic122—82no—
233#92418—24no—
234#92418—42no—
235#92418—42no—
236#18625—611nomaximal, Sylow
237C22×C61—21no—
238C22×C61—21no—
239C22×C61—41no—
240C22×C61—44no—
241C22×C61—41no—
242C22×C61—44no—
243C22×C61—41no—
244C22×C61—44no—
245C22×C61—164no—
246C22×C61#45212yes—
247C22×C61C2×#12011yescharacteristic
248C4×C61—21no—
249C4×C61—21no—
250C4×C61—41no—
251C4×C61—41no—
252C4×C61—42no—
253C4×C61—41no—
254C4×C61—42no—
255C4×C61#45212yes—
256C4×C61#45312yes—
257C4×C61C2×#13511yescharacteristic
258C4×C61C2×#13711yescharacteristic
259C4×C61#57712yes—
260Dic61—82no—
261#3508D6111yescharacteristic
262#3230—611no—
263C2×#492—611no—
264#3079—611no—
265#1631—614no—
266#1630—614no—
267#1545—611no—
268#1493—611no—
269#1476—611no—
270C2×C61—22no—
271C2×C61—81no—
272C2×C61—84no—
273C2×C61C2×#45211yescharacteristic
274C2×C61#300911yescharacteristic
275C2×C61C2×#45311yescharacteristic
276C22×SD16—1222no—
277#559—21no—
278#559C2×D6112yes—
279#550—21no—
280#550C2×D6112yes—
281#546—22no—
282#546—22no—
283#540—22no—
284#535—21no—
285#533C2×D6111yescharacteristic
286#532—21no—
287#532—21no—
288C2×C4⋊Q8—611no—
289C2×#127C2×D6111yescharacteristic
290#518—21no—
291C4×#41C2×D6111yescharacteristic
292#500—1221no—
293#492—614no—
294#492—1222no—
295#487—1221no—
296#484—1222no—
297#477—612no—
298#477—614no—
299#476—612no—
300#475—1221no—
301#474—1221no—
302#431—611no—
303#428—1221no—
304#423—611no—
305#417—1221no—
306C2×C4×C8—611no—
307#331—614no—
308Q8⋊C8—612no—
309Q8⋊C8—614no—
310D4⋊C8—612no—
311C61#1862511yescharacteristic, complemented, Sylow
312C2×#41—22no—
313C2×#41—22no—
314C2×#41C22⋊D6111yescharacteristic
315C22×Q8—1221no—
316C22×D4—21no—
317C22×D4—21no—
318C22×D4C22⋊D6112yes—
319C2×SD16—1228no—
320C2×SD16—2448no—
321C2×M16—1221no—
322C22×C8—1221no—
323C4⋊Q8—612no—
324C4⋊Q8—1221no—
325C4⋊Q8—1221no—
326C4⋊Q8—1221no—
327C4⋊Q8C22⋊D6111yescharacteristic
328C4⋊Q8C22⋊D6111yescharacteristic
329#127—22no—
330#127—21no—
331#127—21no—
332#127C22⋊D6112yes—
333#127C22×D6112yes—
334#125—21no—
335#124—22no—
336#124—22no—
337#124—22no—
338#123—24no—
339#123—24no—
340C22⋊Q8—21no—
341C22⋊Q8C22⋊D6112yes—
342#121—22no—
343#121—22no—
344#121—22no—
345#121—24no—
346#121—24no—
347#121—22no—
348#120—24no—
349C4×Q8—21no—
350C4×Q8C22⋊D6111yescharacteristic
351C4×Q8C22×D6111yescharacteristic
352C4×Q8C22×D6112yes—
353C4×D4—22no—
354C4×D4—22no—
355C4×D4—22no—
356C4×D4—21no—
357C4×D4—21no—
358C4×D4—22no—
359C4×D4—22no—
360C4×D4—22no—
361C4×D4C22⋊D6111yescharacteristic
362C4×D4C22×D6111yescharacteristic
363#117—21no—
364#117—21no—
365#117C22⋊D6111yescharacteristic
366C2×C4⋊C4—1221no—
367C2×C4⋊C4—1221no—
368C2×C22⋊C4—24no—
369C2×C42—21no—
370C2×C42C22×D6111yescharacteristic
371#107—1222no—
372#107—1224no—
373#106—1222no—
374C4⋊C8—612no—
375C4⋊C8—612no—
376C4⋊C8—1221no—
377#103—1222no—
378#103—1224no—
379#102—1222no—
380C22⋊C8—1221no—
381C4×C8—612no—
382C4×C8—1221no—
383C4×C8—1221no—
384C24—22no—
385#41—22no—
386#41—22no—
387#41—84no—
388#41—84no—
389C2×Q8—21no—
390C2×Q8—1222no—
391C2×Q8—2444no—
392C2×Q8—2441no—
393C2×Q8C2×C22⋊D6111yescharacteristic
394C2×Q8C2×C22⋊D6112yes—
395C2×D4—21no—
396C2×D4—24no—
397C2×D4—24no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—44no—
404C2×D4—42no—
405C2×D4—42no—
406C2×D4—82no—
407C2×D4C2×C22⋊D6111yescharacteristic
408C2×D4C2×C22⋊D6114yes—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—2441no—
415C22×C4C2×C22⋊D6111yescharacteristic
416C22×C4C2×C22⋊D6111yescharacteristic
417C22×C4C2×C22⋊D6111yescharacteristic
418SD16—24416no—
419M16—2442no—
420C2×C8—1224no—
421C2×C8—1222no—
422C2×C8—1222no—
423C2×C8—1222no—
424C4⋊C4—21no—
425C4⋊C4—22no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—22no—
431C4⋊C4—1222no—
432C4⋊C4—1222no—
433C4⋊C4—2441no—
434C4⋊C4—2441no—
435C4⋊C4C2×C22⋊D6111yescharacteristic
436C4⋊C4C2×C22⋊D6111yescharacteristic
437C4⋊C4C2×C22⋊D6112yes—
438C22⋊C4—24no—
439C22⋊C4—24no—
440C22⋊C4—24no—
441C22⋊C4—24no—
442C22⋊C4—21no—
443C22⋊C4—22no—
444C22⋊C4—21no—
445C22⋊C4—21no—
446C42—21no—
447C42—22no—
448C42—21no—
449C42—21no—
450C42C2×C22⋊D6111yescharacteristic
451C42C2×C22⋊D6111yescharacteristic
452C42C2×C22⋊D6111yescharacteristic
453C42C23×D6111yescharacteristic
454C23—24no—
455C23—21no—
456C23—44no—
457C23—82no—
458C23—82no—
459C23#1451974611yescharacteristic
460Q8—21no—
461Q8—21no—
462Q8—24no—
463Q8—42no—
464Q8—4882no—
465Q8—2444no—
466D4—21no—
467D4—21no—
468D4—44no—
469D4—42no—
470D4—42no—
471D4—42no—
472D4—44no—
473D4—44no—
474D4—44no—
475D4—82no—
476D4—84no—
477D4—84no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—84no—
489C2×C4—84no—
490C2×C4—4882no—
491C2×C4—2442no—
492C2×C4#1451974611yescharacteristic
493C2×C4#1451974611yescharacteristic
494C2×C4#1451974611yescharacteristic
495C2×C4#1451979712yes—
496C2×C4#1451980712yes—
497C2×C4C22×C22⋊D6111yescharacteristic
498C2×C4C22×C22⋊D6111yescharacteristic
499C2×C4C22×C22⋊D6111yescharacteristic
500C8—1224no—
501C8—2442no—
502C8—2442no—
503C22—21no—
504C22—21no—
505C22—41no—
506C22—44no—
507C22—41no—
508C22—44no—
509C22—41no—
510C22—44no—
511C22—164no—
512C22#1683992412yes—
513C22C2×#1451974611yescharacteristic
514C4—21no—
515C4—21no—
516C4—41no—
517C4—42no—
518C4—41no—
519C4—41no—
520C4—42no—
521C4—4882no—
522C4#1683992412yes—
523C4#1683993012yes—
524C4C2×#1451979711yescharacteristic
525C4C2×#1451980711yescharacteristic
526C4#1684057312yes—
527C2—22no—
528C2—84no—
529C2—81no—
530C2C2×#1683992411yescharacteristic
531C2#2145509611yescharacteristic
532C2C2×#1683993011yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3508C2
socleC22×C61C2×#120
hypercenter#3508D61
hypocenterC61#18625
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3508

Complements

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

NH ≅ G/Ntype
1. #21456967C2semidirect
2. C2×#16840134C2semidirect
3. #21455634C2semidirect
4. #21444946C2semidirect
5. #21444940C2semidirect
6. #21444487C2semidirect
7. #21444274C2semidirect
8. #21444201C2semidirect
17. #16840057C22semidirect
18. #16840057C22semidirect
19. #16840052C22semidirect
28. Q8⋊#92418C22semidirect
29. Q8⋊#92418C22semidirect
30. D4⋊#92418C22semidirect
311. C61#18625semidirect

Maximal subgroups

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

subgroupindex
#214569672
C2×#168401342
#214556342
#214449462
#214449402
#214444872
#214442742
#214442012
C61×#35082
#1862561