Finite Group Databasebeta

← #27153251

Subgroups — #27153251

Order 12416.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690843C211yesmaximal, characteristic, complemented
2C22×#13095500C211yesmaximal, characteristic, complemented
3D4×Dic194C211yesmaximal, characteristic, complemented
4#20261381C211yesmaximal, characteristic
5#20261129C211yesmaximal, characteristic, complemented
6#20260981C211yesmaximal, characteristic, complemented
7#20260976C211yesmaximal, characteristic, complemented
8#20260958C211yesmaximal, characteristic, complemented
9#20260941C211yesmaximal, characteristic, complemented
10#20260899C211yesmaximal, characteristic, complemented
11#20260893C211yesmaximal, characteristic, complemented
12#20260888C211yesmaximal, characteristic, complemented
13#20260876C211yesmaximal, characteristic, complemented
14#20260837C211yesmaximal, characteristic, complemented
15#435×C97C211yesmaximal, characteristic, complemented
16C2×#13095511C2211yescharacteristic
17C22×Dic194C2211yescharacteristic
18C22×Dic194C2211yescharacteristic, complemented
19C22×D388C2211yescharacteristic
20#15690843—24no—
21C2×Dic388C2211yescharacteristic, complemented
22C2×Dic388C2211yescharacteristic, complemented
23C2×Dic388C2211yescharacteristic, complemented
24C2×#13095500—24no—
25C2×#13095500C2211yescharacteristic
26C2×#13095500C2211yescharacteristic
27C2×#13095500C2211yescharacteristic
28C2×#13095500C2211yescharacteristic
29C2×#13095500C2211yescharacteristic
30C2×#13095500C2211yescharacteristic
31#15690810C2211yescharacteristic, complemented
32#15690809—21no—
33#15690798C2211yescharacteristic
34#15690795—21no—
35#15690789C2211yescharacteristic
36#15690789C2211yescharacteristic
37#15690789C2211yescharacteristic, complemented
38#15690787—21no—
39#15690787—21no—
40#15690784C2211yescharacteristic
41#15690783—21no—
42#15690778—21no—
43C4×Dic194C2211yescharacteristic, complemented
44C4×D388C2211yescharacteristic
45D4×Dic97—21no—
46C22⋊C4×D97—21no—
47#15690744C2211yescharacteristic, complemented
48#15690735C2211yescharacteristic, complemented
49#15690735C2211yescharacteristic, complemented
50#15690735C2211yescharacteristic, complemented
51#15690735C2211yescharacteristic, complemented
52#15690732C2211yescharacteristic
53#15690732C2211yescharacteristic
54C2×M16×C97C2211yescharacteristic, complemented
55C22×C8×C97C2211yescharacteristic
56C4×D4×C97C2211yescharacteristic
57C4⋊C8×C97C2211yescharacteristic, complemented
58C22⋊C8×C97C2211yescharacteristic
59C22⋊C8×C97C2211yescharacteristic, complemented
60#97×C97C2211yescharacteristic, complemented
61C23×D97—21no—
62#13095511—22no—
63#13095511—41no—
64C2×Dic194—22no—
65C2×Dic194—22no—
66C2×Dic194—21no—
67C2×Dic194—21no—
68C2×Dic194—41no—
69C2×Dic194—41no—
70C2×Dic194C2311yescharacteristic
71C2×Dic194C2311yescharacteristic
72C2×Dic194C2311yescharacteristic
73C2×Dic194C2311yescharacteristic
74C2×D388—22no—
75C2×D388—21no—
76C2×D388—41no—
77C2×D388C2311yescharacteristic
78C2×D388C2311yescharacteristic
79C2×C22⋊D97—21no—
80C2×C22⋊D97—21no—
81C22×Dic97—21no—
82C22×Dic97—21no—
83C2×C4×D97—21no—
84C2×C4×D97—21no—
85Dic388—22no—
86Dic388—22no—
87Dic388—41no—
88#13095500—22no—
89#13095500—22no—
90#13095500—22no—
91#13095500—22no—
92#13095500—41no—
93#13095500D414yescomplemented
94C4⋊Dic97—21no—
95C4⋊Dic97—21no—
96C4⋊Dic97C2311yescharacteristic
97C4⋊Dic97C2311yescharacteristic
98#13095490—21no—
99#13095490—21no—
100#13095490—21no—
101#13095490—21no—
102C22⋊Dic97—21no—
103C22⋊Dic97—21no—
104C22⋊Dic97—21no—
105#13095488—21no—
106#13095488—21no—
107#13095488—21no—
108#13095488—21no—
109C4×Dic97—21no—
110C2×D4×C97D411yescharacteristic
111C22×C4×C97C2311yescharacteristic
112C22×C4×C97C2311yescharacteristic
113M16×C97—22no—
114C2×C8×C97—22no—
115C2×C8×C97C2311yescharacteristic
116C2×C8×C97C2311yescharacteristic
117C2×C8×C97C2311yescharacteristic
118C2×C8×C97C2311yescharacteristic
119C4⋊C4×C97D411yescharacteristic
120C22⋊C4×C97D411yescharacteristic
121C22⋊C4×C97D411yescharacteristic
122C42×C97C2311yescharacteristic
123C23×C97C2×D411yescharacteristic
124C23×C97C2×D411yescharacteristic
125C22×D97—21no—
126C22×D97—21no—
127C22×D97—42no—
128C22×D97—42no—
129C22×D97—81no—
130D4×C97—22no—
131C2×C4×C97—21no—
132C2×C4×C97—21no—
133C2×C4×C97—21no—
134C2×C4×C97—21no—
135C2×C4×C97C2×D411yescharacteristic
136C2×C4×C97C2×D411yescharacteristic
137C2×C4×C97C2×D411yescharacteristic
138C2×C4×C97C2×D411yescharacteristic
139C2×C4×C97C2411yescharacteristic
140C22⋊D97—42no—
141C22⋊D97—81no—
142C2×Dic97—21no—
143C2×Dic97—21no—
144C2×Dic97—21no—
145C2×Dic97—21no—
146C2×Dic97—21no—
147C2×Dic97—21no—
148C2×Dic97—42no—
149C2×Dic97—42no—
150D388—21no—
151D388—21no—
152D388—21no—
153D388—42no—
154D388C2×D412yes—
155C4×D97—42no—
156C4×D97—81no—
157Dic194—21no—
158Dic194—21no—
159Dic194—21no—
160Dic194—21no—
161Dic194—21no—
162Dic194—21no—
163Dic194—21no—
164Dic194—42no—
165Dic194—42no—
166Dic194C2×D412yes—
167C8×C97—21no—
168C8×C97—21no—
169C8×C97—21no—
170C8×C97C2×D412yes—
171C22×C97—21no—
172C22×C97—21no—
173C22×C97—21no—
174C22×C97—21no—
175C22×C97—21no—
176C22×C97#13712yescomplemented
177C22×C97C22×D411yescharacteristic
178C2×D97—22no—
179C2×D97—41no—
180C2×D97—41no—
181C2×D97—41no—
182C2×D97—41no—
183C2×D97—41no—
184C2×D97—84no—
185C4×C97—21no—
186C4×C97—21no—
187C4×C97—21no—
188C4×C97—21no—
189C4×C97C22×D411yescharacteristic
190C4×C97H(5,𝔽2)11yescharacteristic
191Dic97—22no—
192Dic97—41no—
193Dic97—41no—
194Dic97—41no—
195Dic97—41no—
196Dic97—41no—
197C2×C97—22no—
198C2×C97—41no—
199C2×C97D4211yescharacteristic
200C2×C97C2×#13711yescharacteristic
201C2×C97#57711yescharacteristic
202D97—42no—
203D97—81no—
204#3266—971nomaximal, Sylow
205C97#326611yescharacteristic, complemented, Sylow
206C2×#137—971no—
207C22×SD16—971no—
208D4×Q8—971no—
209#546—971no—
210#497—971no—
211#468—971no—
212#467—971no—
213#464—971no—
214#461—971no—
215#452—971no—
216#451—971no—
217#450—971no—
218#448—971no—
219#440—971no—
220#435D9711yescharacteristic, complemented
221C2×#41—971no—
222C22×Q8—971no—
223C22×Q8—971no—
224C22×D4—971no—
225#137—1944no—
226C2×Q16—971no—
227C2×Q16—971no—
228C2×Q16—971no—
229C2×SD16—971no—
230C2×SD16—971no—
231C2×SD16—971no—
232C2×SD16—971no—
233C2×SD16—971no—
234C2×SD16—971no—
235C2×SD16—1944no—
236C2×M16C2×D9711yescharacteristic, complemented
237C22×C8C2×D9711yescharacteristic
238C4⋊Q8—971no—
239C4⋊Q8—1941no—
240#124—971no—
241#123—1941no—
242C22⋊Q8—971no—
243C22⋊Q8—971no—
244C22⋊Q8—971no—
245C22⋊Q8—1941no—
246C22⋊Q8—1941no—
247#121—971no—
248#121—1941no—
249#120—1941no—
250C4×Q8—971no—
251C4×D4—971no—
252C4×D4—1941no—
253C4×D4C2×D9711yescharacteristic
254C2×C22⋊C4—1941no—
255#107—971no—
256C4⋊C8C2×D9711yescharacteristic, complemented
257#103—971no—
258#103—971no—
259#103—971no—
260#103—971no—
261#102—971no—
262#102—971no—
263C22⋊C8C2×D9711yescharacteristic
264C22⋊C8C2×D9711yescharacteristic, complemented
265#97C2×D9711yescharacteristic, complemented
266C24—1941no—
267#41—3881no—
268#41—1942no—
269C2×Q8—971no—
270C2×Q8—971no—
271C2×Q8—971no—
272C2×Q8—971no—
273C2×Q8—3881no—
274C2×Q8—3881no—
275C2×Q8—1942no—
276C2×Q8—1942no—
277C2×Q8—1941no—
278C2×Q8—1941no—
279C2×D4—971no—
280C2×D4—971no—
281C2×D4—3881no—
282C2×D4—1942no—
283C2×D4—1941no—
284C2×D4—1941no—
285C2×D4—1941no—
286C2×D4D38811yescharacteristic
287C22×C4—1941no—
288C22×C4—1941no—
289C22×C4—1941no—
290C22×C4—1941no—
291C22×C4C22×D9711yescharacteristic
292C22×C4C22×D9711yescharacteristic
293Q16—3881no—
294Q16—1942no—
295Q16—1942no—
296SD16—974no—
297SD16—3881no—
298SD16—1942no—
299SD16—1942no—
300SD16—1942no—
301SD16—1942no—
302M16—22no—
303C2×C8—22no—
304C2×C8C22×D9711yescharacteristic
305C2×C8C22×D9711yescharacteristic
306C2×C8C22×D9711yescharacteristic
307C2×C8C22×D9711yescharacteristic
308C4⋊C4—971no—
309C4⋊C4—971no—
310C4⋊C4—1941no—
311C4⋊C4—1941no—
312C4⋊C4—1941no—
313C4⋊C4—1941no—
314C4⋊C4—1941no—
315C4⋊C4—1941no—
316C4⋊C4D38811yescharacteristic
317C22⋊C4—1941no—
318C22⋊C4—1941no—
319C22⋊C4—1941no—
320C22⋊C4—1941no—
321C22⋊C4—1941no—
322C22⋊C4—1941no—
323C22⋊C4—1941no—
324C22⋊C4D38811yescharacteristic
325C22⋊C4D38811yescharacteristic
326C42—1941no—
327C42C22×D9711yescharacteristic
328C23—7761no—
329C23—3882no—
330C23—3882no—
331C23—1941no—
332C23—1941no—
333C23C2×D38811yescharacteristic
334C23C2×D38811yescharacteristic
335Q8—972no—
336Q8—3882no—
337Q8—3882no—
338Q8—1941no—
339Q8—1941no—
340Q8—1941no—
341Q8—1941no—
342Q8—1941no—
343Q8—1941no—
344Q8—1941no—
345D4—22no—
346D4—7761no—
347D4—972no—
348D4—3882no—
349D4—3882no—
350D4—1941no—
351D4—1941no—
352D4—1941no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—7761no—
358C2×C4—3882no—
359C2×C4—3882no—
360C2×C4—3882no—
361C2×C4—1941no—
362C2×C4—1941no—
363C2×C4—1941no—
364C2×C4—1941no—
365C2×C4—1941no—
366C2×C4—1941no—
367C2×C4C2×D38811yescharacteristic
368C2×C4C2×D38811yescharacteristic
369C2×C4C2×D38811yescharacteristic
370C2×C4C2×D38811yescharacteristic
371C2×C4C23×D9711yescharacteristic
372C8—21no—
373C8—21no—
374C8—21no—
375C8D4×D9712yes—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—7764no—
382C22—3881no—
383C22—3881no—
384C22—3881no—
385C22—3881no—
386C22—3881no—
387C22—1942no—
388C22#1569084312yescomplemented
389C22C22×D38811yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—3881no—
395C4—3881no—
396C4—3881no—
397C4—3881no—
398C4—3881no—
399C4—1942no—
400C4C2×D4×D9711yescharacteristic
401C4#1569085611yescharacteristic
402C2—22no—
403C2—41no—
404C2—7761no—
405C2—3882no—
406C2D4×D38811yescharacteristic
407C2C2×#1569084311yescharacteristic
408C2#2026154711yescharacteristic
subgroupG/H
centerC22C22×D388
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#435×C97C2
socleC22×C97C22×D4
prime radical#20261381C2
hypercenter#435D97
hypocenterC97#3266
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. C2×#15690843C2semidirect
2. C22×#13095500C2semidirect
3. D4×Dic194C2semidirect
5. #20261129C2semidirect
6. #20260981C2semidirect
7. #20260976C2semidirect
8. #20260958C2semidirect
9. #20260941C2semidirect
10. #20260899C2semidirect
11. #20260893C2semidirect
12. #20260888C2semidirect
13. #20260876C2semidirect
14. #20260837C2semidirect
15. #435×C97C2semidirect
18. C22×Dic194C22semidirect
21. C2×Dic388C22semidirect
22. C2×Dic388C22semidirect
23. C2×Dic388C22semidirect
31. #15690810C22semidirect
37. #15690789C22semidirect
43. C4×Dic194C22semidirect
47. #15690744C22semidirect
48. #15690735C22semidirect
49. #15690735C22semidirect
50. #15690735C22semidirect
51. #15690735C22semidirect
54. C2×M16×C97C22semidirect
57. C4⋊C8×C97C22semidirect
59. C22⋊C8×C97C22semidirect
60. #97×C97C22semidirect
93. #13095500D4semidirect
176. C22×C97#137semidirect
205. C97#3266semidirect
220. #435D97semidirect
236. C2×M16C2×D97semidirect
256. C4⋊C8C2×D97semidirect
264. C22⋊C8C2×D97semidirect
265. #97C2×D97semidirect
388. C22#15690843semidirect

Maximal subgroups

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

subgroupindex
C2×#156908432
C22×#130955002
D4×Dic1942
#202613812
#202611292
#202609812
#202609762
#202609582
#202609412
#202608992
#202608932
#202608882
#202608762
#202608372
#435×C972
#326697