Finite Group Databasebeta

← #27150707

Subgroups — #27150707

Order 12416.

380 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261263C211yesmaximal, characteristic
2#20260863C211yesmaximal, characteristic, complemented
3C8×Dic194C211yesmaximal, characteristic, complemented
4#20260811C211yesmaximal, characteristic, complemented
5#20260808C211yesmaximal, characteristic, complemented
6#20260804C211yesmaximal, characteristic, complemented
7C8×D388C211yesmaximal, characteristic, complemented
8C8×C22⋊D97C211yesmaximal, characteristic, complemented
9C8×C22⋊D97C211yesmaximal, characteristic, complemented
10#20260797C211yesmaximal, characteristic, complemented
11#20260796C211yesmaximal, characteristic, complemented
12#20260790C211yesmaximal, characteristic, complemented
13#20260689C211yesmaximal, characteristic, complemented
14C4×C8×D97C211yesmaximal, characteristic, complemented
15#405×C97C211yesmaximal, characteristic, complemented
16C2×#13095494C2211yescharacteristic
17C2×C8×D97C2211yescharacteristic
18C2×C8×D97C2211yescharacteristic
19C2×C8×D97C2211yescharacteristic
20#15690805C411yescharacteristic
21#15690803C411yescharacteristic
22#15690800C411yescharacteristic
23#15690796C411yescharacteristic
24#15690795C411yescharacteristic
25#15690792C411yescharacteristic
26#15690788C411yescharacteristic
27#15690783C411yescharacteristic
28C4×Dic194C2211yescharacteristic
29C4×D388C2211yescharacteristic
30C4×C22⋊D97C2211yescharacteristic
31C4×C22⋊D97C2211yescharacteristic
32#15690767C2211yescharacteristic
33C42×D97C2211yescharacteristic
34C4⋊#11715895C2211yescharacteristic
35C4⋊#11715895C2211yescharacteristic
36#15690739C2211yescharacteristic
37#15690739C2211yescharacteristic
38#15690739C2211yescharacteristic
39#15690739C2211yescharacteristic
40C22⋊#11715895C2211yescharacteristic
41C22⋊#11715895C2211yescharacteristic
42#15690722C2211yescharacteristic
43#15690722C2211yescharacteristic
44#15690722C2211yescharacteristic
45#15690722C2211yescharacteristic
46#15690721C2211yescharacteristic
47#15690720C2211yescharacteristic
48C4×#11715895C2211yescharacteristic
49C8×Dic97C2211yescharacteristic
50C8×Dic97C2211yescharacteristic
51C8×Dic97C2211yescharacteristic
52C2×M16×C97C2211yescharacteristic
53C22×C8×C97C2211yescharacteristic
54#117×C97C2211yescharacteristic
55#97×C97C2211yescharacteristic
56#97×C97C2211yescharacteristic
57C4×C8×C97C2211yescharacteristic
58C4×C8×C97C2211yescharacteristic
59C2×Dic194C2×C411yescharacteristic
60C2×D388C2×C411yescharacteristic
61C2×C22⋊D97C2×C411yescharacteristic
62C2×C22⋊D97C2×C411yescharacteristic
63C2×C4×D97C2×C411yescharacteristic
64C2×C4×D97C2×C411yescharacteristic
65C2×C4×D97C2311yescharacteristic
66C2×C4×D97C2311yescharacteristic
67#13095494—22no—
68C2×#11715895C2311yescharacteristic
69C2×#11715895C2311yescharacteristic
70C2×#11715895C2311yescharacteristic
71C2×#11715895C2311yescharacteristic
72C8×D97—22no—
73C8×D97—22no—
74C8×D97—22no—
75C4⋊Dic97C2×C411yescharacteristic
76C4⋊Dic97C2×C411yescharacteristic
77#13095490C2×C411yescharacteristic
78#13095490C2×C411yescharacteristic
79#13095490C2×C411yescharacteristic
80#13095490C2×C411yescharacteristic
81#13095490C2×C411yescharacteristic
82#13095490C2×C411yescharacteristic
83C22⋊Dic97C2×C411yescharacteristic
84C22⋊Dic97C2×C411yescharacteristic
85#13095488C2×C411yescharacteristic
86#13095488C2×C411yescharacteristic
87#13095488C2×C411yescharacteristic
88#13095488C2×C411yescharacteristic
89#13095488C2×C411yescharacteristic
90#13095488C2×C411yescharacteristic
91C4×Dic97C2×C411yescharacteristic
92C4×Dic97C2×C411yescharacteristic
93C4×Dic97C2311yescharacteristic
94C4×Dic97C2311yescharacteristic
95C22×C4×C97C2311yescharacteristic
96M16×C97—22no—
97C2×C8×C97—22no—
98C2×C8×C97C2311yescharacteristic
99C2×C8×C97C2311yescharacteristic
100C2×C8×C97C2311yescharacteristic
101C2×C8×C97C2311yescharacteristic
102C4⋊C4×C97C2×C411yescharacteristic
103C4⋊C4×C97C2×C411yescharacteristic
104C22⋊C4×C97C2×C411yescharacteristic
105C22⋊C4×C97C2×C411yescharacteristic
106C42×C97C2311yescharacteristic
107C42×C97C2311yescharacteristic
108C23×C97C22×C411yescharacteristic
109C22×D97C22×C411yescharacteristic
110C22×D97C22×C411yescharacteristic
111C2×C4×C97—21no—
112C2×C4×C97—21no—
113C2×C4×C97C22×C411yescharacteristic
114C2×C4×C97C22×C411yescharacteristic
115C2×C4×C97C22×C411yescharacteristic
116C2×C4×C97C22×C411yescharacteristic
117C2×C4×C97C22×C411yescharacteristic
118C2×C4×C97C2411yescharacteristic
119C22⋊D97—22no—
120C22⋊D97—22no—
121C2×Dic97C22×C411yescharacteristic
122C2×Dic97C22×C411yescharacteristic
123C2×Dic97C22×C411yescharacteristic
124C2×Dic97C22×C411yescharacteristic
125C2×Dic97C22×C411yescharacteristic
126C2×Dic97C22×C411yescharacteristic
127D388—22no—
128C4×D97—22no—
129C4×D97—21no—
130C4×D97—21no—
131C4×D97—21no—
132C4×D97—21no—
133C4×D97—22no—
134Dic194—22no—
135C8×C97—21no—
136C8×C97—21no—
137C8×C97#4112yes—
138C8×C97#4112yes—
139#11715895—21no—
140#11715895—21no—
141#11715895—21no—
142#11715895—21no—
143C22×C97—21no—
144C22×C97—21no—
145C22×C97—21no—
146C22×C97C23×C411yescharacteristic
147C2×D97—21no—
148C2×D97—21no—
149C2×D97—21no—
150C2×D97—21no—
151C2×D97—21no—
152C2×D97#13112yes—
153C4×C97—21no—
154C4×C97—21no—
155C4×C97—21no—
156C4×C97—21no—
157C4×C97—21no—
158C4×C97C2×#4111yescharacteristic
159C4×C97C2×#4111yescharacteristic
160Dic97—21no—
161Dic97—21no—
162Dic97—21no—
163Dic97—21no—
164Dic97—21no—
165Dic97#13112yes—
166C2×C97—41no—
167C2×C97C4×#4111yescharacteristic
168C2×C97C2×#13111yescharacteristic
169C2×C97C2×#13111yescharacteristic
170D97—22no—
171D97—41no—
172#2966—971nomaximal, Sylow
173C97#296611yescharacteristic, complemented, Sylow
174#529—971no—
175#446—971no—
176C8×Q8—971no—
177#436—971no—
178#435—971no—
179#435—971no—
180C8×D4—971no—
181C8×D4—971no—
182C8×D4—971no—
183#433—971no—
184#433—971no—
185#432—971no—
186#405—971no—
187#405D9711yescharacteristic, complemented
188C2×C4×C8—971no—
189C2×M16—971no—
190C2×M16C2×D9711yescharacteristic
191C22×C8—971no—
192C22×C8—971no—
193C22×C8—971no—
194C22×C8C2×D9711yescharacteristic
195#126—971no—
196#126—971no—
197#125—971no—
198#124—971no—
199#123—971no—
200#123—971no—
201C22⋊Q8—971no—
202#121—971no—
203C4×Q8—971no—
204C4×D4—971no—
205C4×D4—971no—
206C4×D4—971no—
207#117—971no—
208#117C2×D9711yescharacteristic
209C2×C42—971no—
210C4⋊C8—971no—
211C4⋊C8—971no—
212C4⋊C8—971no—
213C4⋊C8—971no—
214C4⋊C8—971no—
215C4⋊C8—971no—
216C22⋊C8—971no—
217C22⋊C8—971no—
218C22⋊C8—971no—
219C22⋊C8—971no—
220C22⋊C8—971no—
221C22⋊C8—971no—
222#97—971no—
223#97—971no—
224#97C2×D9711yescharacteristic
225#97C2×D9711yescharacteristic
226C4×C8—971no—
227C4×C8—971no—
228C4×C8—971no—
229C4×C8—971no—
230C4×C8C2×D9711yescharacteristic
231C4×C8C2×D9711yescharacteristic
232C2×Q8—971no—
233C2×D4—971no—
234C2×D4—971no—
235C2×D4—971no—
236C22×C4—971no—
237C22×C4—971no—
238C22×C4—971no—
239C22×C4—971no—
240C22×C4C22×D9711yescharacteristic
241M16—22no—
242M16—1942no—
243C2×C8—22no—
244C2×C8—971no—
245C2×C8—971no—
246C2×C8—971no—
247C2×C8—971no—
248C2×C8—1942no—
249C2×C8—1942no—
250C2×C8—1942no—
251C2×C8C22×D9711yescharacteristic
252C2×C8C22×D9711yescharacteristic
253C2×C8C22×D9711yescharacteristic
254C2×C8C22×D9711yescharacteristic
255C4⋊C4—971no—
256C4⋊C4—971no—
257C4⋊C4—971no—
258C4⋊C4—971no—
259C4⋊C4—971no—
260C4⋊C4—971no—
261C4⋊C4—971no—
262C4⋊C4—971no—
263C4⋊C4C4×D9711yescharacteristic
264C4⋊C4C4×D9711yescharacteristic
265C22⋊C4—971no—
266C22⋊C4—971no—
267C22⋊C4—971no—
268C22⋊C4—971no—
269C22⋊C4—971no—
270C22⋊C4—971no—
271C22⋊C4—971no—
272C22⋊C4—971no—
273C22⋊C4C4×D9711yescharacteristic
274C22⋊C4C4×D9711yescharacteristic
275C42—971no—
276C42—971no—
277C42—971no—
278C42—971no—
279C42C22×D9711yescharacteristic
280C42C22×D9711yescharacteristic
281C23—971no—
282C23—971no—
283C23C2×C4×D9711yescharacteristic
284Q8—1942no—
285D4—1942no—
286D4—1942no—
287D4—1942no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—971no—
291C2×C4—971no—
292C2×C4—971no—
293C2×C4—971no—
294C2×C4—971no—
295C2×C4—971no—
296C2×C4—1942no—
297C2×C4—1942no—
298C2×C4—1941no—
299C2×C4—1941no—
300C2×C4—1941no—
301C2×C4—1941no—
302C2×C4C2×C4×D9711yescharacteristic
303C2×C4C2×C4×D9711yescharacteristic
304C2×C4C2×C4×D9711yescharacteristic
305C2×C4C2×C4×D9711yescharacteristic
306C2×C4C2×C4×D9711yescharacteristic
307C2×C4C23×D9711yescharacteristic
308C8—21no—
309C8—21no—
310C8—1941no—
311C8—1941no—
312C8—1941no—
313C8—1941no—
314C8#1309551112yes—
315C8#1309551112yes—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—972no—
320C22—1941no—
321C22—1941no—
322C22—1941no—
323C22—1941no—
324C22—1941no—
325C22C22×C4×D9711yescharacteristic
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—972no—
332C4—1941no—
333C4—1941no—
334C4—1941no—
335C4—1941no—
336C4—1941no—
337C4C2×#1309551111yescharacteristic
338C4C2×#1309551111yescharacteristic
339C2—41no—
340C2—3881no—
341C2—1942no—
342C2C4×#1309551111yescharacteristic
343C2#131×D9711yescharacteristic
344C2#131×D9711yescharacteristic
subgroupG/H
centerC2×C8C22×D97
commutator, socleC22×C97C23×C4
FrattiniC2×C4C23×D97
Fitting#405×C97C2
prime radical#15690783C4
cosocleC2×C4×C97C24
hypercenter#405D97
hypocenterC97#2966
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C2×C8 ◁ #405

Complements

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

NH ≅ G/Ntype
2. #20260863C2semidirect
3. C8×Dic194C2semidirect
4. #20260811C2semidirect
5. #20260808C2semidirect
6. #20260804C2semidirect
7. C8×D388C2semidirect
8. C8×C22⋊D97C2semidirect
9. C8×C22⋊D97C2semidirect
10. #20260797C2semidirect
11. #20260796C2semidirect
12. #20260790C2semidirect
13. #20260689C2semidirect
14. C4×C8×D97C2semidirect
15. #405×C97C2semidirect
173. C97#2966semidirect
187. #405D97semidirect

Maximal subgroups

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

subgroupindex
#202612632
#202608632
C8×Dic1942
#202608112
#202608082
#202608042
C8×D3882
C8×C22⋊D972
#202607972
#202607962
#202607902
#202606892
C4×C8×D972
#405×C972
#296697