Finite Group Databasebeta

← #29834492

Subgroups — #29834492

Order 15616.

738 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840197C211yesmaximal, characteristic, complemented
2C22×#14519717C211yesmaximal, characteristic, complemented
3C2×#16839998C211yesmaximal, characteristic, complemented
4#21447934C212yesmaximal, complemented
5#21447920C212yesmaximal, complemented
6#21447917C212yesmaximal, complemented
7#21447903C212yesmaximal, complemented
8#21447608C211yesmaximal, characteristic, complemented
9#21447534C211yesmaximal, characteristic, complemented
10#21447531C211yesmaximal, characteristic, complemented
11C61×#3050C211yesmaximal, characteristic, complemented
12C22×D488C2211yescharacteristic
13C2×#14519751—21no—
14#16840197—24no—
15C2×#14519720C2212yescomplemented
16C2×#14519718C2212yescomplemented
17C2×#14519717—24no—
18C2×#14519717C2211yescharacteristic
19C2×#14519717C2211yescharacteristic
20C2×#14519717C2212yescomplemented
21C22×#11758185C2212yescomplemented
22#16839998—21no—
23#16839998—21no—
24#16839998C2212yes—
25#16839998C2212yes—
26C2×#14519711C2211yescharacteristic
27C2×#14519699—21no—
28#16839589C2212yescomplemented
29#16839587C2212yescomplemented
30#16839566C2212yescomplemented
31#16839563C2212yescomplemented
32#16839562C2212yescomplemented
33#16839562C2212yescomplemented
34C2×#135×C61C2211yescharacteristic
35C23×C8×C61C2211yescharacteristic
36C61×#525—21no—
37C61×#467C2211yescharacteristic, complemented
38C61×#467C2211yescharacteristic, complemented
39C61×#466C2211yescharacteristic, complemented
40C61×#466C2211yescharacteristic, complemented
41C61×#465—21no—
42C61×#465—21no—
43C61×#427C2211yescharacteristic
44C61×#416—21no—
45C22×D244—21no—
46C22×C22⋊D61—41no—
47C2×D488—22no—
48C2×D488—22no—
49C2×D488—21no—
50C2×D488C2312yes—
51#14519751—22no—
52#14519751—22no—
53#14519751—41no—
54#14519751—41no—
55C2×C4⋊Dic61—21no—
56C2×#11758198—41no—
57#14519720—24no—
58#14519718—24no—
59#14519717—22no—
60#14519717—24no—
61#14519717D414yescomplemented
62C2×#11758185—24no—
63C2×#11758185C2312yes—
64C2×#11758185C2312yes—
65#14519711—21no—
66#14519711C2312yes—
67#14519699—22no—
68#14519699—22no—
69C2×#41×C61—21no—
70C23×C4×C61D411yescharacteristic
71#135×C61—22no—
72#135×C61—22no—
73C2×Q16×C61C2311yescharacteristic
74C2×SD16×C61—21no—
75C2×D8×C61C2311yescharacteristic
76C22×C8×C61—22no—
77C22×C8×C61—22no—
78C22×C8×C61D411yescharacteristic
79C22×C8×C61D411yescharacteristic
80C22×C8×C61D411yescharacteristic
81C22×C8×C61C2311yescharacteristic
82C22×C8×C61C2311yescharacteristic
83C22×C8×C61C2311yescharacteristic
84#123×C61—41no—
85C22⋊Q8×C61—21no—
86C22⋊Q8×C61—21no—
87#121×C61—21no—
88#121×C61—21no—
89#120×C61—41no—
90C4×D4×C61—41no—
91C4×D4×C61—41no—
92#117×C61—21no—
93#107×C61C2311yescharacteristic
94#107×C61C2311yescharacteristic
95#106×C61—21no—
96#103×C61—21no—
97#103×C61—21no—
98#102×C61—21no—
99#102×C61—21no—
100C24×C61D811yescharacteristic
101C23×D61—41no—
102#41×C61—42no—
103#41×C61—42no—
104C2×Q8×C61—21no—
105C2×D4×C61—21no—
106C2×D4×C61—41no—
107C2×D4×C61—41no—
108C2×D4×C61—41no—
109C22×C4×C61—22no—
110C22×C4×C61—22no—
111C22×C4×C61—41no—
112C22×C4×C61D811yescharacteristic
113C22×C4×C61D811yescharacteristic
114C22×C4×C61D811yescharacteristic
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2×D411yescharacteristic
118C2×C22⋊D61—42no—
119C2×C22⋊D61—42no—
120C2×C22⋊D61—84no—
121C22×Dic61—41no—
122C2×D244—22no—
123C2×D244—42no—
124C2×D244—42no—
125C2×D244—41no—
126Q16×C61—21no—
127Q16×C61—21no—
128SD16×C61—42no—
129D8×C61—21no—
130D8×C61—21no—
131C2×C8×C61—22no—
132C2×C8×C61—22no—
133C2×C8×C61—22no—
134C2×C8×C61—21no—
135C2×C8×C61—21no—
136C2×C8×C61—21no—
137C2×C8×C61—21no—
138C2×C8×C61C2×D411yescharacteristic
139C2×C8×C61C2×D411yescharacteristic
140C2×C8×C61C2×D411yescharacteristic
141C2×C8×C61C2×D412yes—
142C2×C8×C61#4112yes—
143C2×C8×C61C2411yescharacteristic
144C4⋊C4×C61—21no—
145C4⋊C4×C61—21no—
146C4⋊C4×C61—41no—
147C4⋊C4×C61—41no—
148C22⋊C4×C61—41no—
149C22⋊C4×C61—41no—
150C22⋊C4×C61—41no—
151C22⋊C4×C61—41no—
152C22⋊C4×C61—41no—
153C42×C61—41no—
154#11758198—42no—
155#11758198—42no—
156C4⋊Dic61—22no—
157C4⋊Dic61—41no—
158D488—22no—
159D488—22no—
160D488—42no—
161#11758185—22no—
162#11758185C2×D414yescomplemented
163C23×C61—22no—
164C23×C61—22no—
165C23×C61—41no—
166C23×C61—41no—
167C23×C61C2×D811yescharacteristic
168C23×C61C2×D811yescharacteristic
169C23×C61C2×D811yescharacteristic
170C22×D61—42no—
171C22×D61—82no—
172C22×D61—82no—
173C22×D61—82no—
174Q8×C61—41no—
175Q8×C61—41no—
176D4×C61—41no—
177D4×C61—41no—
178D4×C61—82no—
179D4×C61—82no—
180D4×C61—82no—
181C2×C4×C61—22no—
182C2×C4×C61—21no—
183C2×C4×C61—21no—
184C2×C4×C61—21no—
185C2×C4×C61—21no—
186C2×C4×C61—22no—
187C2×C4×C61—22no—
188C2×C4×C61—41no—
189C2×C4×C61—41no—
190C2×C4×C61—41no—
191C2×C4×C61—82no—
192C2×C4×C61#12112yes—
193C2×C4×C61C2×D811yescharacteristic
194C2×C4×C61C2×D811yescharacteristic
195C2×C4×C61C2×D811yescharacteristic
196C2×C4×C61#13512yes—
197C2×C4×C61C22×D411yescharacteristic
198C22⋊D61—84no—
199C22⋊D61—84no—
200C2×Dic61—42no—
201C2×Dic61—82no—
202D244—42no—
203D244—42no—
204D244—82no—
205C8×C61—21no—
206C8×C61—21no—
207C8×C61#12112yes—
208C8×C61C22×D411yescharacteristic
209C8×C61C2×#4111yescharacteristic
210#33605—611nomaximal, Sylow
211C22×C61—21no—
212C22×C61—21no—
213C22×C61—21no—
214C22×C61—21no—
215C22×C61—21no—
216C22×C61—21no—
217C22×C61—21no—
218C22×C61—44no—
219C22×C61—81no—
220C22×C61—81no—
221C22×C61—81no—
222C22×C61#46612yes—
223C22×C61#50912yescomplemented
224C22×C61C22×D811yescharacteristic
225C2×D61—82no—
226C2×D61—82no—
227C2×D61—82no—
228C2×D61—164no—
229C4×C61—21no—
230C4×C61—21no—
231C4×C61—81no—
232C4×C61—81no—
233C4×C61—81no—
234C4×C61#46612yes—
235C4×C61C2×#12111yescharacteristic
236C4×C61C2×#13511yescharacteristic
237Dic61—82no—
238C2×#509—611no—
239C22×M32—611no—
240#3050D6111yescharacteristic, complemented
241C2×#466—611no—
242#2220—612no—
243#2218—612no—
244#2217—612no—
245#2215—612no—
246#2159—611no—
247#2140—611no—
248#2139—611no—
249C2×C61—22no—
250C2×C61—22no—
251C2×C61—41no—
252C2×C61—161no—
253C2×C61C2×#46611yescharacteristic
254C2×C61C2×#50911yescharacteristic
255C2×C61#341411yescharacteristic
256D61—162no—
257C2×#135C2×D6111yescharacteristic
258C22×D8—611no—
259C23×C8C2×D6111yescharacteristic
260#525—21no—
261C2×#121—1221no—
262#509—1224no—
263C2×SD32—612no—
264C2×D16—612no—
265C2×M32—611no—
266C2×M32—611no—
267C2×M32—612no—
268C2×M32—1224no—
269C22×C16—612no—
270#467C2×D6111yescharacteristic, complemented
271#467C2×D6111yescharacteristic, complemented
272#466—612no—
273#466—612no—
274#466—1221no—
275#466—1221no—
276#466C2×D6111yescharacteristic, complemented
277#466C2×D6111yescharacteristic, complemented
278#465—21no—
279#465—21no—
280#427C2×D6111yescharacteristic
281C2×#107—611no—
282#416—21no—
283C2×#102—1221no—
284#367—612no—
285#366—612no—
286#358—612no—
287#357—612no—
288#357—612no—
289#357—612no—
290C61#3360511yescharacteristic, complemented, Sylow
291C2×#41—21no—
292C22×D4—1221no—
293C22×D4—2441no—
294C23×C4C22⋊D6111yescharacteristic
295#135—22no—
296#135—22no—
297C2×Q16C22×D6111yescharacteristic
298C2×SD16—21no—
299C2×D8—612no—
300C2×D8—1222no—
301C2×D8—1222no—
302C2×D8—1221no—
303C2×D8C22×D6111yescharacteristic
304C22×C8—22no—
305C22×C8—22no—
306C22×C8C22⋊D6111yescharacteristic
307C22×C8C22⋊D6111yescharacteristic
308C22×C8C22⋊D6111yescharacteristic
309C22×C8C22×D6111yescharacteristic
310C22×C8C22×D6111yescharacteristic
311C22×C8C22×D6111yescharacteristic
312#123—41no—
313C22⋊Q8—21no—
314C22⋊Q8—21no—
315#121—21no—
316#121—21no—
317#121—1222no—
318#121—1222no—
319#121—2441no—
320#121—2441no—
321#120—41no—
322C4×D4—41no—
323C4×D4—41no—
324#117—21no—
325C2×C4⋊C4—1221no—
326C2×C22⋊C4—2441no—
327SD32—1224no—
328D16—1224no—
329M32—614no—
330M32—1224no—
331M32—1222no—
332C2×C16—612no—
333C2×C16—612no—
334C2×C16—1224no—
335#107—612no—
336#107—1221no—
337#107C22×D6111yescharacteristic
338#107C22×D6111yescharacteristic
339#106—21no—
340#103—21no—
341#103—21no—
342#102—21no—
343#102—21no—
344#102—1222no—
345#102—1222no—
346C24—2441no—
347C24#1175819911yescharacteristic
348#41—42no—
349#41—42no—
350C2×Q8—21no—
351C2×D4—21no—
352C2×D4—41no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4—1222no—
356C2×D4—4884no—
357C2×D4—2442no—
358C2×D4—2442no—
359C2×D4—2442no—
360C2×D4—2441no—
361C2×D4—2442no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—41no—
365C22×C4—2441no—
366C22×C4#1175819911yescharacteristic
367C22×C4#1175819911yescharacteristic
368C22×C4#1175819911yescharacteristic
369C22×C4C2×C22⋊D6111yescharacteristic
370C22×C4C2×C22⋊D6111yescharacteristic
371C22×C4C2×C22⋊D6111yescharacteristic
372Q16—21no—
373Q16—21no—
374SD16—42no—
375D8—21no—
376D8—21no—
377D8—1222no—
378D8—1222no—
379D8—2442no—
380C2×C8—22no—
381C2×C8—21no—
382C2×C8—21no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8D4×D6112yes—
388C2×C8#1175822412yes—
389C2×C8C2×C22⋊D6111yescharacteristic
390C2×C8C2×C22⋊D6111yescharacteristic
391C2×C8C2×C22⋊D6111yescharacteristic
392C2×C8C23×D6111yescharacteristic
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—1222no—
398C4⋊C4—2441no—
399C22⋊C4—41no—
400C22⋊C4—41no—
401C22⋊C4—41no—
402C22⋊C4—41no—
403C22⋊C4—41no—
404C22⋊C4—2442no—
405C22⋊C4—2442no—
406C42—41no—
407C16—614no—
408C16—1222no—
409C23—22no—
410C23—22no—
411C23—41no—
412C23—41no—
413C23—4882no—
414C23—4882no—
415C23—4882no—
416C23—2442no—
417C23C2×#1175819911yescharacteristic
418C23C2×#1175819911yescharacteristic
419C23C2×#1175819911yescharacteristic
420Q8—41no—
421Q8—41no—
422D4—41no—
423D4—41no—
424D4—82no—
425D4—82no—
426D4—82no—
427D4—4884no—
428D4—4884no—
429D4—4882no—
430D4—2442no—
431D4—2442no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—82no—
443C2×C4—4882no—
444C2×C4—2442no—
445C2×C4#1451974912yes—
446C2×C4C2×#1175819911yescharacteristic
447C2×C4C2×#1175819911yescharacteristic
448C2×C4C2×#1175819911yescharacteristic
449C2×C4#1451979712yes—
450C2×C4C22×C22⋊D6111yescharacteristic
451C8—21no—
452C8—21no—
453C8#1451974912yes—
454C8C2×D4×D6111yescharacteristic
455C8C2×#1175822411yescharacteristic
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—44no—
464C22—81no—
465C22—81no—
466C22—81no—
467C22—9764no—
468C22—4882no—
469C22—4882no—
470C22—4882no—
471C22#1683999612yes—
472C22#1684019712yescomplemented
473C22C22×#1175819911yescharacteristic
474C4—21no—
475C4—21no—
476C4—81no—
477C4—81no—
478C4—81no—
479C4—4882no—
480C4#1683999612yes—
481C4C2×#1451974911yescharacteristic
482C4C2×#1451979711yescharacteristic
483C2—22no—
484C2—22no—
485C2—41no—
486C2—161no—
487C2—9762no—
488C2C2×#1683999611yescharacteristic
489C2C2×#1684019711yescharacteristic
490C2#2145895311yescharacteristic
subgroupG/H
centerC22C22×#11758199
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#3050C2
socleC22×C61C22×D8
hypercenter#3050D61
hypocenterC61#33605
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ #3050

Complements

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

NH ≅ G/Ntype
1. C2×#16840197C2semidirect
2. C22×#14519717C2semidirect
3. C2×#16839998C2semidirect
4. #21447934C2semidirect
5. #21447920C2semidirect
6. #21447917C2semidirect
7. #21447903C2semidirect
8. #21447608C2semidirect
9. #21447534C2semidirect
10. #21447531C2semidirect
11. C61×#3050C2semidirect
15. C2×#14519720C22semidirect
16. C2×#14519718C22semidirect
20. C2×#14519717C22semidirect
21. C22×#11758185C22semidirect
28. #16839589C22semidirect
29. #16839587C22semidirect
30. #16839566C22semidirect
31. #16839563C22semidirect
32. #16839562C22semidirect
33. #16839562C22semidirect
37. C61×#467C22semidirect
38. C61×#467C22semidirect
39. C61×#466C22semidirect
40. C61×#466C22semidirect
61. #14519717D4semidirect
162. #11758185C2×D4semidirect
223. C22×C61#509semidirect
240. #3050D61semidirect
270. #467C2×D61semidirect
271. #467C2×D61semidirect
276. #466C2×D61semidirect
277. #466C2×D61semidirect
290. C61#33605semidirect
472. C22#16840197semidirect

Maximal subgroups

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

subgroupindex
C2×#168401972
C22×#145197172
C2×#168399982
#214479342
#214479202
#214479172
#214479032
#214476082
#214475342
#214475312
C61×#30502
#3360561