Finite Group Databasebeta

← #29834363

Subgroups — #29834363

Order 15616.

726 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840200C211yesmaximal, characteristic, complemented
2C2×#16840192C211yesmaximal, characteristic, complemented
3#21456306C211yesmaximal, characteristic, complemented
4#21454693C211yesmaximal, characteristic, complemented
5#21447884C212yesmaximal, complemented
6#21447877C212yesmaximal, complemented
7#21447858C212yesmaximal, complemented
8#21447853C212yesmaximal, complemented
9#21447579C211yesmaximal, characteristic, complemented
10C2×#16839562C211yesmaximal, characteristic, complemented
11C61×#2139C211yesmaximal, characteristic, complemented
12C2×#14519802—21no—
13C2×#14519799C2211yescharacteristic
14C22×D488C2211yescharacteristic
15#16840326—21no—
16#16840243—21no—
17#16840200—24no—
18#16840192—24no—
19C2×#14519723C2211yescharacteristic, complemented
20C2×#14519722C2212yes—
21C2×#14519721C2211yescharacteristic, complemented
22C2×#14519720C2211yescharacteristic, complemented
23C2×D976C2212yes—
24C2×#14519718C2211yescharacteristic, complemented
25C22×#11758185C2211yescharacteristic
26#16840148—22no—
27#16840147C2212yes—
28#16840137C2212yes—
29#16839861—22no—
30#16839856C2212yescomplemented
31#16839844C2212yescomplemented
32#16839761—21no—
33#16839717C2211yescharacteristic
34#16839580C2212yescomplemented
35#16839580C2212yescomplemented
36#16839562C2212yes—
37#16839562C2212yes—
38C2×#135×C61C2211yescharacteristic
39C2×M32×C61C2211yescharacteristic
40C2×#107×C61C2211yescharacteristic
41C61×#358C2212yescomplemented
42C61×#357C2212yescomplemented
43C2×#11758222—21no—
44C22×D244—21no—
45#14519802—44no—
46#14519799—22no—
47#14519799D414yescomplemented
48C2×Dic244C2311yescharacteristic
49C2×#11758190—21no—
50C2×#11758201—22no—
51C2×D488—22no—
52C2×D488—22no—
53C2×D488—21no—
54C2×D488C2312yes—
55C2×D488C2311yescharacteristic
56C2×#11758199—22no—
57C2×#11758194D411yescharacteristic
58#14519773—22no—
59#14519766—22no—
60#14519751—41no—
61#14519751—41no—
62#14519745—42no—
63#14519742—22no—
64#14519739—22no—
65C4×C22⋊D61—42no—
66#14519736D411yescharacteristic
67#14519735—41no—
68C4⋊C4×D61—41no—
69#14519723—22no—
70#14519722—24no—
71#14519721—22no—
72#14519720—22no—
73D976—24no—
74#14519718—22no—
75C2×#11758185—22no—
76C2×#11758185C2311yescharacteristic
77C2×#11758185C2311yescharacteristic
78#14519700—22no—
79#14519697—22no—
80#14519687D412yes—
81C8×Dic61C2312yes—
82C2×#41×C61—21no—
83#135×C61—22no—
84#135×C61—22no—
85C2×Q16×C61C2311yescharacteristic
86C2×SD16×C61—21no—
87C2×D8×C61C2311yescharacteristic
88C22×C8×C61C2311yescharacteristic
89C2×C4⋊C4×C61—21no—
90M32×C61—22no—
91C2×C16×C61C2312yes—
92#107×C61—21no—
93#107×C61C2312yes—
94C23×D61—41no—
95#41×C61—42no—
96#41×C61—42no—
97C2×Q8×C61—21no—
98C2×D4×C61—21no—
99C2×D4×C61—41no—
100C22×C4×C61—41no—
101C22×C4×C61—41no—
102C22×C4×C61C2×D411yescharacteristic
103C2×C22⋊D61—41no—
104C2×C22⋊D61—42no—
105C2×C22⋊D61—42no—
106#11758222—24no—
107#11758222—42no—
108C2×D244—21no—
109C2×D244—22no—
110C2×D244—42no—
111C2×D244—42no—
112C2×D244—41no—
113C2×C4×D61—41no—
114C2×C4×D61—42no—
115C2×Dic122—21no—
116Q16×C61—21no—
117Q16×C61—21no—
118SD16×C61—42no—
119D8×C61—21no—
120D8×C61—21no—
121C2×C8×C61C2×D411yescharacteristic
122C2×C8×C61C2×D412yes—
123C2×C8×C61#4112yes—
124C2×C8×C61C2411yescharacteristic
125C4⋊C4×C61—22no—
126C4⋊C4×C61—41no—
127C22⋊Dic61D812yes—
128#11758201—44no—
129#11758199—44no—
130#11758198—42no—
131#11758198—42no—
132#11758198—42no—
133C4⋊Dic61D811yescharacteristic
134C4⋊Dic61D811yescharacteristic
135#11758196—42no—
136C4×Dic61—42no—
137C4×Dic61C2×D412yes—
138#11758194—22no—
139C2×#92418C2×D412yes—
140Dic244—21no—
141Dic244C2×D412yes—
142D488—22no—
143D488—21no—
144D488—22no—
145D488—42no—
146D488C2×D412yes—
147#11758190—24no—
148C16×C61—22no—
149#11758185—21no—
150#11758185—21no—
151C23×C61—41no—
152C23×C61C2×D811yescharacteristic
153C22×D61—41no—
154C22×D61—42no—
155C22×D61—82no—
156C22×D61—161no—
157C22×D61—161no—
158Q8×C61—41no—
159Q8×C61—41no—
160D4×C61—41no—
161D4×C61—41no—
162D4×C61—82no—
163C2×C4×C61—42no—
164C2×C4×C61—41no—
165C2×C4×C61—82no—
166C2×C4×C61—161no—
167C2×C4×C61#12112yes—
168C2×C4×C61C2×D811yescharacteristic
169C2×C4×C61AGL(1,ℤ/8)12yes—
170C2×C4×C61C22×D411yescharacteristic
171C22⋊D61—44no—
172C22⋊D61—162no—
173C22⋊D61—162no—
174C2×Dic61—41no—
175C2×Dic61C2×D812yes—
176C2×Dic61C2×D812yes—
177D244—22no—
178D244—42no—
179D244—41no—
180D244—42no—
181D244—82no—
182C4×D61—44no—
183C4×D61—162no—
184Dic122—22no—
185Dic122—41no—
186C8×C61#12112yes—
187C8×C61C22×D411yescharacteristic
188C8×C61C2×#4111yescharacteristic
189#92418—22no—
190#33589—611nomaximal, Sylow
191C22×C61—21no—
192C22×C61—21no—
193C22×C61—81no—
194C22×C61—81no—
195C22×C61—81no—
196C22×C61#45912yes—
197C22×C61C22×D811yescharacteristic
198C2×D61—42no—
199C2×D61—81no—
200C2×D61—81no—
201C2×D61—82no—
202C2×D61—82no—
203C2×D61—82no—
204C2×D61—164no—
205C4×C61—82no—
206C4×C61—81no—
207C4×C61#45912yes—
208C4×C61C2×#12111yescharacteristic
209C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
210Dic61—22no—
211Dic61—22no—
212Dic61—42no—
213C2×#509—611no—
214C2×#508—611no—
215#3155—611no—
216#2951—611no—
217#2211—612no—
218#2210—612no—
219#2207—612no—
220#2206—612no—
221#2151—611no—
222#2139D6111yescharacteristic, complemented
223C2×#357—611no—
224C2×C61—22no—
225C2×C61—161no—
226C2×C61C2×#45911yescharacteristic
227C2×C61#341511yescharacteristic
228C2×C61#341611yescharacteristic
229D61—82no—
230D61—162no—
231C2×AGL(1,ℤ/8)—1221no—
232C2×#135—611no—
233C2×#135C2×D6111yescharacteristic
234C22×D8—611no—
235#535—1221no—
236#520—1221no—
237#509—1224no—
238#508—1224no—
239C2×Q32—611no—
240C2×SD32—611no—
241C2×SD32—611no—
242C2×SD32—612no—
243C2×D16—611no—
244C2×D16—612no—
245C2×M32C2×D6111yescharacteristic
246C22×C16—611no—
247#496—1222no—
248#495—612no—
249#493—612no—
250#440—1222no—
251C4×Q16—612no—
252C4×D8—612no—
253C2×#107C2×D6111yescharacteristic
254#417—1221no—
255#405—611no—
256C4⋊C16—612no—
257C4⋊C16—612no—
258#358C2×D6112yescomplemented
259#357—612no—
260#357—612no—
261#357C2×D6112yescomplemented
262C61#3358911yescharacteristic, complemented, Sylow
263C2×#41—21no—
264C2×#41—1221no—
265C22×D4—1221no—
266AGL(1,ℤ/8)—2444no—
267#135—22no—
268#135—22no—
269#135—614no—
270#135—1222no—
271C2×Q16—611no—
272C2×Q16C22×D6111yescharacteristic
273C2×SD16—21no—
274C2×SD16—1222no—
275C2×SD16—1221no—
276C2×D8—611no—
277C2×D8—612no—
278C2×D8—1222no—
279C2×D8—1222no—
280C2×D8—1222no—
281C2×D8—1221no—
282C2×D8C22×D6111yescharacteristic
283C2×M16—611no—
284C22×C8C22×D6111yescharacteristic
285#127—1222no—
286#124—1222no—
287#121—2441no—
288#121—2441no—
289#120—2442no—
290C4×Q8—1222no—
291C4×D4—1222no—
292C4×D4—2442no—
293#117—611no—
294#117—2441no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—2441no—
297Q32—1222no—
298SD32—1224no—
299SD32—1222no—
300SD32—1222no—
301D16—1224no—
302D16—1222no—
303M32—22no—
304C2×C16—611no—
305C2×C16—611no—
306C2×C16—1222no—
307C2×C16C22×D6112yes—
308#107—21no—
309#107C22×D6112yes—
310#103—1222no—
311#102—1222no—
312#97—612no—
313C4×C8—612no—
314C24—2441no—
315#41—42no—
316#41—42no—
317#41—1224no—
318#41—2442no—
319C2×Q8—21no—
320C2×Q8—1221no—
321C2×D4—21no—
322C2×D4—41no—
323C2×D4—1221no—
324C2×D4—1222no—
325C2×D4—2442no—
326C2×D4—2442no—
327C2×D4—2441no—
328C2×D4—2442no—
329C2×D4—2441no—
330C2×D4—2442no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—2442no—
334C22×C4—2441no—
335C22×C4D4×D6111yescharacteristic
336Q16—21no—
337Q16—21no—
338Q16—612no—
339Q16—1221no—
340SD16—42no—
341SD16—1224no—
342SD16—2444no—
343D8—21no—
344D8—21no—
345D8—612no—
346D8—1222no—
347D8—1221no—
348D8—1222no—
349D8—2444no—
350D8—2442no—
351M16—1222no—
352C2×C8—612no—
353C2×C8#1175822212yes—
354C2×C8D4×D6112yes—
355C2×C8D4×D6111yescharacteristic
356C2×C8C23×D6111yescharacteristic
357C4⋊C4—22no—
358C4⋊C4—41no—
359C4⋊C4—611no—
360C4⋊C4—611no—
361C4⋊C4—2442no—
362C22⋊C4—612no—
363C22⋊C4—2442no—
364C22⋊C4—2442no—
365C22⋊C4—2442no—
366C42—612no—
367C42—2442no—
368C16—22no—
369C16—1221no—
370C16—1221no—
371C23—41no—
372C23—9761no—
373C23—9761no—
374C23—4882no—
375C23—2442no—
376C23—2441no—
377C23D8×D6111yescharacteristic
378Q8—41no—
379Q8—41no—
380Q8—1222no—
381Q8—2441no—
382D4—41no—
383D4—41no—
384D4—82no—
385D4—1222no—
386D4—9762no—
387D4—9762no—
388D4—4882no—
389D4—2444no—
390D4—2442no—
391D4—2442no—
392D4—2441no—
393C2×C4—42no—
394C2×C4—41no—
395C2×C4—82no—
396C2×C4—161no—
397C2×C4—612no—
398C2×C4—612no—
399C2×C4—9762no—
400C2×C4—2441no—
401C2×C4—2444no—
402C2×C4#1451975012yes—
403C2×C4D8×D6111yescharacteristic
404C2×C4#1451980412yes—
405C2×C4C2×D4×D6111yescharacteristic
406C8—1222no—
407C8#1451975012yes—
408C8C2×D4×D6111yescharacteristic
409C8C2×#1175822211yescharacteristic
410C22—21no—
411C22—21no—
412C22—81no—
413C22—81no—
414C22—81no—
415C22—9764no—
416C22—4882no—
417C22—4882no—
418C22—4882no—
419C22—4881no—
420C22—4881no—
421C22—2442no—
422C22#1683995612yes—
423C22C2×D8×D6111yescharacteristic
424C4—82no—
425C4—81no—
426C4—1222no—
427C4—1222no—
428C4—2442no—
429C4#1683995612yes—
430C4C2×#1451975011yescharacteristic
431C4C2×#1451980411yescharacteristic
432C2—22no—
433C2—161no—
434C2—9762no—
435C2—4882no—
436C2C2×#1683995611yescharacteristic
437C2#2145895911yescharacteristic
438C2#2145896411yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2139C2
socleC22×C61C22×D8
hypercenter#2139D61
hypocenterC61#33589
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2139

Complements

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

NH ≅ G/Ntype
1. C2×#16840200C2semidirect
2. C2×#16840192C2semidirect
3. #21456306C2semidirect
4. #21454693C2semidirect
5. #21447884C2semidirect
6. #21447877C2semidirect
7. #21447858C2semidirect
8. #21447853C2semidirect
9. #21447579C2semidirect
10. C2×#16839562C2semidirect
11. C61×#2139C2semidirect
19. C2×#14519723C22semidirect
21. C2×#14519721C22semidirect
22. C2×#14519720C22semidirect
24. C2×#14519718C22semidirect
30. #16839856C22semidirect
31. #16839844C22semidirect
34. #16839580C22semidirect
35. #16839580C22semidirect
41. C61×#358C22semidirect
42. C61×#357C22semidirect
47. #14519799D4semidirect
222. #2139D61semidirect
258. #358C2×D61semidirect
261. #357C2×D61semidirect
262. C61#33589semidirect

Maximal subgroups

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

subgroupindex
C2×#168402002
C2×#168401922
#214563062
#214546932
#214478842
#214478772
#214478582
#214478532
#214475792
C2×#168395622
C61×#21392
#3358961