Finite Group Databasebeta

← #29833874

Subgroups — #29833874

Order 15616.

1150 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840567C211yesmaximal, characteristic, complemented
2C2×#16840197C211yesmaximal, characteristic, complemented
3C2×#16840192C211yesmaximal, characteristic, complemented
4#21447757C212yesmaximal, complemented
5#21447747C212yesmaximal, complemented
6#21447740C212yesmaximal, complemented
7#21447730C212yesmaximal, complemented
8#21447529C211yesmaximal, characteristic, complemented
9C2×#16839562C211yesmaximal, characteristic, complemented
10#21447433C211yesmaximal, characteristic, complemented
11C61×#3061C211yesmaximal, characteristic, complemented
12C2×#14519823—21no—
13#16840567—24no—
14#16840567—24no—
15C2×#14519805—21no—
16C2×#14519805—22no—
17C2×#14519799C2212yes—
18C2×#14519799C2211yescharacteristic
19C22×D488C2212yes—
20C22×D488C2211yescharacteristic
21#16840197—24no—
22#16840192—24no—
23C2×#14519723C2211yescharacteristic, complemented
24C2×#14519721C2211yescharacteristic, complemented
25C2×#14519720C2211yescharacteristic, complemented
26C2×#14519720C2212yescomplemented
27C2×#14519718C2211yescharacteristic, complemented
28C2×#14519718C2212yescomplemented
29C2×#14519717C2211yescharacteristic, complemented
30C22×#11758185C2211yescharacteristic, complemented
31#16839565C2212yescomplemented
32#16839562C2212yescomplemented
33#16839562C2212yescomplemented
34#16839562C2212yescomplemented
35C22⋊#11758185C2212yescomplemented
36C22⋊#11758185C2212yescomplemented
37C2×#135×C61C2211yescharacteristic, complemented
38C2×#131×C61C2211yescharacteristic
39C61×#537—21no—
40C61×#468—22no—
41C61×#467C2212yescomplemented
42C61×#466C2212yescomplemented
43C2×#107×C61C2211yescharacteristic, complemented
44C61×#417—21no—
45#14519823—28no—
46#14519823—44no—
47C2×#11758226—41no—
48C2×#11758222—22no—
49C2×#11758222—21no—
50C22×D244—22no—
51C22×D244—21no—
52C2×D4×D61—42no—
53C2×D4×D61—41no—
54#14519805—28no—
55#14519805—216no—
56#14519799—22no—
57#14519799—24no—
58#14519799—24no—
59#14519799D414yescomplemented
60C2×Dic244C2311yescharacteristic
61C2×#11758190—22no—
62C2×#11758190—21no—
63C2×D488—22no—
64C2×D488—21no—
65C2×D488—22no—
66C2×D488—24no—
67C2×D488—24no—
68C2×D488D414yescomplemented
69C2×D488C2311yescharacteristic
70C2×D488C2312yes—
71#14519723—22no—
72#14519721—22no—
73#14519720—22no—
74#14519720—24no—
75#14519718—22no—
76#14519718—24no—
77#14519717—22no—
78C2×#11758185—22no—
79C2×#11758185C2311yescharacteristic
80C2×#11758185C2311yescharacteristic
81C2×#11758185C2312yes—
82C2×#41×C61—21no—
83C2×#41×C61D411yescharacteristic
84#135×C61—22no—
85#135×C61—22no—
86C2×Q16×C61C2311yescharacteristic
87C2×SD16×C61—21no—
88C2×D8×C61C2311yescharacteristic
89#131×C61—22no—
90#131×C61—22no—
91C2×M16×C61D411yescharacteristic
92C2×M16×C61D412yes—
93C22×C8×C61C2312yes—
94C22×C8×C61C2311yescharacteristic
95C22⋊Q8×C61—22no—
96C22⋊Q8×C61—41no—
97#121×C61—22no—
98#121×C61—41no—
99#121×C61—42no—
100C2×C4⋊C4×C61—21no—
101#107×C61—21no—
102#107×C61C2312yes—
103#103×C61—22no—
104#102×C61—22no—
105C23×D61—42no—
106C23×D61—41no—
107#41×C61—22no—
108#41×C61—22no—
109#41×C61—42no—
110#41×C61—42no—
111C2×Q8×C61—21no—
112C2×Q8×C61D811yescharacteristic
113C2×D4×C61—21no—
114C2×D4×C61—41no—
115C2×D4×C61—42no—
116C2×D4×C61D812yes—
117C2×D4×C61D811yescharacteristic
118C22×C4×C61—41no—
119C22×C4×C61—41no—
120C22×C4×C61C2×D412yes—
121C22×C4×C61C2×D411yescharacteristic
122C2×C22⋊D61—42no—
123C2×C22⋊D61—41no—
124#11758226—48no—
125D4×D61—416no—
126D4×D61—48no—
127#11758222—28no—
128#11758222—24no—
129#11758222—44no—
130#11758222—42no—
131C2×D244—28no—
132C2×D244—24no—
133C2×D244—22no—
134C2×D244—21no—
135C2×D244—44no—
136C2×D244—42no—
137C2×D244—41no—
138C2×D244—42no—
139C2×C4×D61—42no—
140C2×C4×D61—41no—
141C2×Dic122—21no—
142Q16×C61—21no—
143Q16×C61—21no—
144SD16×C61—42no—
145D8×C61—21no—
146D8×C61—21no—
147M16×C61—22no—
148M16×C61—24no—
149C2×C8×C61—22no—
150C2×C8×C61—22no—
151C2×C8×C61C2×D412yes—
152C2×C8×C61C2×D411yescharacteristic
153C2×C8×C61C2×D412yes—
154C2×C8×C61C2×D412yes—
155C2×C8×C61C2411yescharacteristic
156C4⋊C4×C61—22no—
157C4⋊C4×C61—41no—
158C4⋊C4×C61—42no—
159C22⋊C4×C61—42no—
160C22⋊C4×C61—42no—
161Dic244—21no—
162Dic244C2×D412yes—
163D488—28no—
164D488—24no—
165D488—22no—
166D488—21no—
167D488C2×D412yes—
168D488C2×D414yes—
169#11758190—28no—
170#11758190—24no—
171#11758185—21no—
172#11758185—21no—
173#11758185—22no—
174C23×C61—41no—
175C23×C61C2×D812yes—
176C23×C61C2×D811yescharacteristic
177C22×D61—42no—
178C22×D61—41no—
179C22×D61—48no—
180C22×D61—44no—
181C22×D61—161no—
182C22×D61—162no—
183C22×D61—161no—
184C22×D61—162no—
185Q8×C61—22no—
186Q8×C61—41no—
187Q8×C61—41no—
188D4×C61—24no—
189D4×C61—22no—
190D4×C61—41no—
191D4×C61—41no—
192D4×C61—82no—
193D4×C61—162no—
194C2×C4×C61—22no—
195C2×C4×C61—22no—
196C2×C4×C61—42no—
197C2×C4×C61—41no—
198C2×C4×C61—82no—
199C2×C4×C61—161no—
200C2×C4×C61#12012yes—
201C2×C4×C61C2×D812yes—
202C2×C4×C61C2×D811yescharacteristic
203C2×C4×C61AGL(1,ℤ/8)12yes—
204C2×C4×C61C22×D411yescharacteristic
205C22⋊D61—48no—
206C22⋊D61—44no—
207C2×Dic61—41no—
208D244—24no—
209D244—22no—
210D244—48no—
211D244—41no—
212D244—42no—
213D244—44no—
214C4×D61—48no—
215C4×D61—44no—
216Dic122—22no—
217Dic122—41no—
218C8×C61—22no—
219C8×C61#12012yes—
220C8×C61C22×D411yescharacteristic
221C8×C61C22×D411yescharacteristic
222#33526—611nomaximal, Sylow
223C22×C61—21no—
224C22×C61—21no—
225C22×C61—22no—
226C22×C61—22no—
227C22×C61—22no—
228C22×C61—81no—
229C22×C61—81no—
230C22×C61—81no—
231C22×C61#44712yes—
232C22×C61C22×D811yescharacteristic
233C2×D61—44no—
234C2×D61—42no—
235C2×D61—82no—
236C2×D61—81no—
237C2×D61—82no—
238C2×D61—81no—
239C2×D61—164no—
240C2×D61—168no—
241C4×C61—22no—
242C4×C61—82no—
243C4×C61—81no—
244C4×C61#44712yes—
245C4×C61C2×#12011yescharacteristic
246C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
247Dic61—42no—
248C2×#576—611no—
249C2×#509—611no—
250C2×#508—611no—
251#3061D6111yescharacteristic, complemented
252#2189—612no—
253#2187—612no—
254#2186—612no—
255#2184—612no—
256#2139—611no—
257C2×#357—611no—
258#2113—611no—
259C2×C61—22no—
260C2×C61—42no—
261C2×C61—161no—
262C2×C61C2×#44711yescharacteristic
263C2×C61#341511yescharacteristic
264C2×C61#341611yescharacteristic
265D61—84no—
266D61—82no—
267C2×H(5,𝔽2)—1221no—
268#576—1224no—
269#576—1224no—
270C2×AGL(1,ℤ/8)—1221no—
271C2×AGL(1,ℤ/8)—1222no—
272C2×#135—611no—
273C2×#135—612no—
274C2×#135C2×D6111yescharacteristic, complemented
275C22×D8—611no—
276C22×D8—612no—
277C2×#131C2×D6111yescharacteristic
278#537—21no—
279#509—1224no—
280#508—1224no—
281C2×Q32—611no—
282C2×SD32—611no—
283C2×SD32—612no—
284C2×SD32—611no—
285C2×D16—611no—
286C2×D16—612no—
287C2×M32—611no—
288C22×C16—611no—
289#468—22no—
290#467C2×D6112yescomplemented
291#466C2×D6112yescomplemented
292C2×#107C2×D6111yescharacteristic, complemented
293#417—21no—
294#358—612no—
295#357—612no—
296#357—612no—
297#357—612no—
298C22⋊C16—612no—
299C22⋊C16—612no—
300C61#3352611yescharacteristic, complemented, Sylow
301H(5,𝔽2)—1228no—
302H(5,𝔽2)—2444no—
303C2×#41—21no—
304C2×#41—1222no—
305C2×#41—1221no—
306C2×#41—2441no—
307C2×#41C22⋊D6111yescharacteristic
308C22×D4—1222no—
309C22×D4—1221no—
310C22×D4—2442no—
311C22×D4—2441no—
312AGL(1,ℤ/8)—1228no—
313AGL(1,ℤ/8)—12216no—
314#135—22no—
315#135—22no—
316#135—614no—
317#135—1224no—
318#135—1224no—
319#135—1222no—
320C2×Q16—611no—
321C2×Q16C22×D6111yescharacteristic
322C2×SD16—21no—
323C2×SD16—1222no—
324C2×SD16—1221no—
325C2×D8—614no—
326C2×D8—612no—
327C2×D8—611no—
328C2×D8—1224no—
329C2×D8—1224no—
330C2×D8—1222no—
331C2×D8—1222no—
332C2×D8—1221no—
333C2×D8C22×D6111yescharacteristic
334#131—22no—
335#131—22no—
336C2×M16C22⋊D6111yescharacteristic
337C2×M16C22⋊D6112yes—
338C22×C8C22×D6112yes—
339C22×C8C22×D6111yescharacteristic
340C22⋊Q8—22no—
341C22⋊Q8—41no—
342#121—22no—
343#121—41no—
344#121—42no—
345C2×C4⋊C4—21no—
346Q32—1222no—
347SD32—1222no—
348SD32—1222no—
349SD32—1224no—
350D16—1222no—
351D16—1224no—
352M32—1222no—
353C2×C16—611no—
354C2×C16—611no—
355C2×C16—612no—
356C2×C16—1222no—
357#107—21no—
358#107C22×D6112yes—
359#103—22no—
360#102—22no—
361C24—2442no—
362C24—2441no—
363#41—22no—
364#41—22no—
365#41—42no—
366#41—42no—
367#41—1228no—
368#41—1224no—
369#41—2444no—
370#41—2448no—
371#41—2442no—
372C2×Q8—21no—
373C2×Q8—1221no—
374C2×Q8#1175819911yescharacteristic
375C2×D4—21no—
376C2×D4—41no—
377C2×D4—42no—
378C2×D4—1228no—
379C2×D4—1222no—
380C2×D4—1221no—
381C2×D4—1224no—
382C2×D4—2444no—
383C2×D4—24416no—
384C2×D4—2448no—
385C2×D4—2442no—
386C2×D4—2441no—
387C2×D4—2442no—
388C2×D4—2441no—
389C2×D4—2442no—
390C2×D4#1175819912yes—
391C2×D4#1175819911yescharacteristic
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—2442no—
395C22×C4—2441no—
396C22×C4C2×C22⋊D6112yes—
397C22×C4C2×C22⋊D6111yescharacteristic
398Q16—21no—
399Q16—21no—
400Q16—612no—
401Q16—1221no—
402SD16—42no—
403SD16—1228no—
404SD16—1224no—
405D8—21no—
406D8—21no—
407D8—614no—
408D8—612no—
409D8—1228no—
410D8—1221no—
411D8—1222no—
412D8—1224no—
413M16—22no—
414M16—24no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8D4×D6112yes—
418C2×C8D4×D6112yes—
419C2×C8C2×C22⋊D6112yes—
420C2×C8C2×C22⋊D6111yescharacteristic
421C2×C8C23×D6111yescharacteristic
422C4⋊C4—22no—
423C4⋊C4—41no—
424C4⋊C4—42no—
425C22⋊C4—42no—
426C22⋊C4—42no—
427C16—1221no—
428C16—1221no—
429C16—1222no—
430C23—41no—
431C23—9761no—
432C23—9761no—
433C23—9762no—
434C23—9762no—
435C23—2448no—
436C23—2444no—
437C23—2442no—
438C23—2441no—
439C23C2×#1175819912yes—
440C23C2×#1175819911yescharacteristic
441Q8—22no—
442Q8—41no—
443Q8—41no—
444Q8—1222no—
445Q8—2441no—
446D4—24no—
447D4—22no—
448D4—41no—
449D4—41no—
450D4—82no—
451D4—162no—
452D4—1224no—
453D4—1222no—
454D4—2448no—
455D4—2448no—
456D4—2444no—
457D4—2441no—
458D4—2442no—
459D4—2444no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—42no—
463C2×C4—41no—
464C2×C4—82no—
465C2×C4—161no—
466C2×C4—2448no—
467C2×C4—2441no—
468C2×C4—2444no—
469C2×C4#1451974512yes—
470C2×C4C2×#1175819912yes—
471C2×C4C2×#1175819911yescharacteristic
472C2×C4#1451980212yes—
473C2×C4C22×C22⋊D6111yescharacteristic
474C8—22no—
475C8#1451974512yes—
476C8C2×D4×D6111yescharacteristic
477C8C2×D4×D6111yescharacteristic
478C22—21no—
479C22—21no—
480C22—22no—
481C22—22no—
482C22—22no—
483C22—81no—
484C22—81no—
485C22—81no—
486C22—9764no—
487C22—9768no—
488C22—4882no—
489C22—4881no—
490C22—4882no—
491C22—4881no—
492C22—2444no—
493C22—2442no—
494C22#1683989512yes—
495C22C22×#1175819911yescharacteristic
496C4—22no—
497C4—82no—
498C4—81no—
499C4—2442no—
500C4#1683989512yes—
501C4C2×#1451974511yescharacteristic
502C4C2×#1451980211yescharacteristic
503C2—22no—
504C2—42no—
505C2—161no—
506C2—4884no—
507C2—4882no—
508C2C2×#1683989511yescharacteristic
509C2#2145895811yescharacteristic
510C2#2145896311yescharacteristic
subgroupG/H
centerC22C22×#11758199
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#3061C2
socleC22×C61C22×D8
hypercenter#3061D61
hypocenterC61#33526
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×#131 ◁ #3061

Complements

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

NH ≅ G/Ntype
1. C2×#16840567C2semidirect
2. C2×#16840197C2semidirect
3. C2×#16840192C2semidirect
4. #21447757C2semidirect
5. #21447747C2semidirect
6. #21447740C2semidirect
7. #21447730C2semidirect
8. #21447529C2semidirect
9. C2×#16839562C2semidirect
10. #21447433C2semidirect
11. C61×#3061C2semidirect
23. C2×#14519723C22semidirect
24. C2×#14519721C22semidirect
25. C2×#14519720C22semidirect
26. C2×#14519720C22semidirect
27. C2×#14519718C22semidirect
28. C2×#14519718C22semidirect
29. C2×#14519717C22semidirect
30. C22×#11758185C22semidirect
31. #16839565C22semidirect
32. #16839562C22semidirect
33. #16839562C22semidirect
34. #16839562C22semidirect
35. C22⋊#11758185C22semidirect
36. C22⋊#11758185C22semidirect
37. C2×#135×C61C22semidirect
41. C61×#467C22semidirect
42. C61×#466C22semidirect
43. C2×#107×C61C22semidirect
59. #14519799D4semidirect
68. C2×D488D4semidirect
251. #3061D61semidirect
274. C2×#135C2×D61semidirect
290. #467C2×D61semidirect
291. #466C2×D61semidirect
292. C2×#107C2×D61semidirect
300. C61#33526semidirect

Maximal subgroups

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

subgroupindex
C2×#168405672
C2×#168401972
C2×#168401922
#214477572
#214477472
#214477402
#214477302
#214475292
C2×#168395622
#214474332
C61×#30612
#3352661