Finite Group Databasebeta

← #27150681

Subgroups — #27150681

Order 12416.

356 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261289C211yesmaximal, characteristic
2#20260866C211yesmaximal, characteristic, complemented
3#20260864C211yesmaximal, characteristic, complemented
4#20260864C211yesmaximal, characteristic, complemented
5#20260863C211yesmaximal, characteristic, complemented
6#20260863C211yesmaximal, characteristic, complemented
7C8×Dic194C211yesmaximal, characteristic, complemented
8#20260859C211yesmaximal, characteristic, complemented
9#20260760C211yesmaximal, characteristic
10#20260760C211yesmaximal, characteristic
11C4⋊C8×D97C211yesmaximal, characteristic
12#20260692C211yesmaximal, characteristic
13#20260689C211yesmaximal, characteristic
14#20260687C211yesmaximal, characteristic
15#446×C97C211yesmaximal, characteristic, complemented
16C2×#13095494C2211yescharacteristic
17C2×#13095494C2211yescharacteristic
18C2×C8×D97C2211yescharacteristic
19C2×C8×D97C2211yescharacteristic
20#15690809C411yescharacteristic
21#15690808C411yescharacteristic
22#15690800C411yescharacteristic
23#15690800C411yescharacteristic
24#15690790C411yescharacteristic
25#15690788C411yescharacteristic
26#15690788C411yescharacteristic
27#15690786C411yescharacteristic
28C4×Dic194C2211yescharacteristic
29C4×Dic194C2211yescharacteristic
30C4×Dic194C2211yescharacteristic
31#15690767C2211yescharacteristic
32#15690767C2211yescharacteristic
33C42×D97C2211yescharacteristic
34C4⋊#11715895C2211yescharacteristic
35C4⋊#11715895C2211yescharacteristic
36C4⋊#11715895C2211yescharacteristic
37#15690739C2211yescharacteristic
38#15690739C2211yescharacteristic
39#15690739C2211yescharacteristic
40#15690739C2211yescharacteristic
41#15690739C2211yescharacteristic
42#15690739C2211yescharacteristic
43#15690721C2211yescharacteristic
44#15690721C2211yescharacteristic
45#15690720C2211yescharacteristic
46#15690720C2211yescharacteristic
47C4×#11715895C2211yescharacteristic
48C8×Dic97C2211yescharacteristic
49C8×Dic97C2211yescharacteristic
50C8×Dic97C2211yescharacteristic
51C8×Dic97C2211yescharacteristic
52C4×Q8×C97C2211yescharacteristic
53C4⋊C8×C97C2211yescharacteristic
54C4⋊C8×C97C2211yescharacteristic
55C4⋊C8×C97C2211yescharacteristic
56#97×C97C2211yescharacteristic
57#97×C97C2211yescharacteristic
58C4×C8×C97C2211yescharacteristic
59C2×Dic194C2×C411yescharacteristic
60C2×Dic194C2×C411yescharacteristic
61C2×Dic194C2×C411yescharacteristic
62C2×C4×D97C2×C411yescharacteristic
63C2×C4×D97C2×C411yescharacteristic
64C2×C4×D97C2311yescharacteristic
65#13095494—22no—
66#13095494Q814yescomplemented
67C2×#11715895C2311yescharacteristic
68C2×#11715895C2311yescharacteristic
69C2×#11715895C2311yescharacteristic
70C2×#11715895C2311yescharacteristic
71C8×D97—22no—
72C8×D97—22no—
73C4⋊Dic97C2×C411yescharacteristic
74C4⋊Dic97C2×C411yescharacteristic
75C4⋊Dic97C2×C411yescharacteristic
76#13095490C2×C411yescharacteristic
77#13095490C2×C411yescharacteristic
78#13095490C2×C411yescharacteristic
79#13095490C2×C411yescharacteristic
80#13095490C2×C411yescharacteristic
81#13095490C2×C411yescharacteristic
82#13095490C2×C411yescharacteristic
83#13095490C2×C411yescharacteristic
84#13095490C2×C411yescharacteristic
85#13095490C2×C411yescharacteristic
86#13095488C2×C411yescharacteristic
87#13095488C2×C411yescharacteristic
88#13095488C2×C411yescharacteristic
89#13095488C2×C411yescharacteristic
90C4×Dic97C2×C411yescharacteristic
91C4×Dic97C2×C411yescharacteristic
92C4×Dic97C2311yescharacteristic
93C4×Dic97C2311yescharacteristic
94C4×Dic97C2311yescharacteristic
95C2×Q8×C97C2×C411yescharacteristic
96C2×C8×C97C2311yescharacteristic
97C2×C8×C97C2311yescharacteristic
98C2×C8×C97C2311yescharacteristic
99C2×C8×C97C2311yescharacteristic
100C4⋊C4×C97C2×C411yescharacteristic
101C4⋊C4×C97C2×C411yescharacteristic
102C4⋊C4×C97C2×C411yescharacteristic
103C42×C97C2311yescharacteristic
104C42×C97C2311yescharacteristic
105C42×C97C2311yescharacteristic
106C22×D97C22×C411yescharacteristic
107Q8×C97—41no—
108C2×C4×C97C22×C411yescharacteristic
109C2×C4×C97C22×C411yescharacteristic
110C2×C4×C97C22×C411yescharacteristic
111C2×C4×C97C22×C411yescharacteristic
112C2×C4×C97C22×C411yescharacteristic
113C2×C4×C97C22×C411yescharacteristic
114C2×C4×C97C2411yescharacteristic
115C2×Dic97C22×C411yescharacteristic
116C2×Dic97C22×C411yescharacteristic
117C2×Dic97C22×C411yescharacteristic
118C2×Dic97C22×C411yescharacteristic
119C2×Dic97C22×C411yescharacteristic
120C2×Dic97C22×C411yescharacteristic
121C2×Dic97C22×C411yescharacteristic
122C4×D97—22no—
123C4×D97—41no—
124C4×D97C2×Q812yes—
125C4×D97#4112yes—
126Dic194—22no—
127Dic194—22no—
128Dic194—41no—
129C8×C97—21no—
130C8×C97—21no—
131C8×C97—21no—
132C8×C97C2×Q812yes—
133#11715895—21no—
134#11715895—21no—
135#11715895—21no—
136#11715895C2×Q812yes—
137C22×C97C23×C411yescharacteristic
138C2×D97—21no—
139C2×D97—21no—
140C2×D97C4×Q812yes—
141C4×C97—21no—
142C4×C97—21no—
143C4×C97—21no—
144C4×C97—21no—
145C4×C97—21no—
146C4×C97—21no—
147C4×C97C22×Q811yescharacteristic
148C4×C97C2×#4111yescharacteristic
149Dic97—21no—
150Dic97—21no—
151Dic97—21no—
152Dic97—21no—
153Dic97—21no—
154Dic97C4×Q812yes—
155Dic97#13112yes—
156C2×C97C2×C4×Q811yescharacteristic
157C2×C97C2×#13111yescharacteristic
158C2×C97#56811yescharacteristic
159D97—22no—
160#2962—971nomaximal, Sylow
161C97#296211yescharacteristic, complemented, Sylow
162#533—971no—
163#446—971no—
164#446—971no—
165#446—971no—
166#446—971no—
167#446—971no—
168#446D9711yescharacteristic, complemented
169C8×Q8—971no—
170C8×Q8—971no—
171#424—971no—
172#424—971no—
173C2×C4⋊C8—971no—
174#405—971no—
175#405—971no—
176C4×M16—971no—
177C2×M16—971no—
178C2×M16—971no—
179C22×C8—971no—
180C22×C8—971no—
181C4⋊Q8—971no—
182C4⋊Q8—971no—
183#125—971no—
184#125—971no—
185C22⋊Q8—971no—
186C22⋊Q8—971no—
187C22⋊Q8—971no—
188C22⋊Q8—971no—
189C4×Q8—971no—
190C4×Q8—971no—
191C4×Q8—971no—
192C4×Q8C2×D9711yescharacteristic
193#117—971no—
194#117—971no—
195C2×C42—971no—
196C4⋊C8—971no—
197C4⋊C8—971no—
198C4⋊C8—971no—
199C4⋊C8—971no—
200C4⋊C8—971no—
201C4⋊C8—971no—
202C4⋊C8—971no—
203C4⋊C8—971no—
204C4⋊C8—971no—
205C4⋊C8C2×D9711yescharacteristic
206C4⋊C8C2×D9711yescharacteristic
207C4⋊C8C2×D9711yescharacteristic
208#97—971no—
209#97—971no—
210#97—971no—
211#97—971no—
212#97C2×D9711yescharacteristic
213#97C2×D9711yescharacteristic
214C4×C8—971no—
215C4×C8—971no—
216C4×C8—971no—
217C4×C8—971no—
218C4×C8—971no—
219C4×C8C2×D9711yescharacteristic
220C2×Q8—971no—
221C2×Q8—971no—
222C2×Q8—971no—
223C2×Q8C4×D9711yescharacteristic
224C22×C4—971no—
225C22×C4—971no—
226C22×C4—971no—
227M16—974no—
228M16—1942no—
229C2×C8—971no—
230C2×C8—971no—
231C2×C8—971no—
232C2×C8—971no—
233C2×C8—1942no—
234C2×C8—1942no—
235C2×C8C22×D9711yescharacteristic
236C2×C8C22×D9711yescharacteristic
237C2×C8C22×D9711yescharacteristic
238C2×C8C22×D9711yescharacteristic
239C4⋊C4—971no—
240C4⋊C4—971no—
241C4⋊C4—971no—
242C4⋊C4—971no—
243C4⋊C4—971no—
244C4⋊C4—971no—
245C4⋊C4—971no—
246C4⋊C4—971no—
247C4⋊C4—971no—
248C4⋊C4—971no—
249C4⋊C4—971no—
250C4⋊C4—971no—
251C4⋊C4—971no—
252C4⋊C4C4×D9711yescharacteristic
253C4⋊C4C4×D9711yescharacteristic
254C4⋊C4C4×D9711yescharacteristic
255C22⋊C4—971no—
256C22⋊C4—971no—
257C22⋊C4—971no—
258C22⋊C4—971no—
259C42—971no—
260C42—971no—
261C42—971no—
262C42—971no—
263C42—971no—
264C42C22×D9711yescharacteristic
265C42C22×D9711yescharacteristic
266C42C22×D9711yescharacteristic
267C23—971no—
268Q8—41no—
269Q8—3881no—
270Q8—1942no—
271Q8—1942no—
272C2×C4—971no—
273C2×C4—971no—
274C2×C4—972no—
275C2×C4—972no—
276C2×C4—971no—
277C2×C4—971no—
278C2×C4—971no—
279C2×C4—971no—
280C2×C4—971no—
281C2×C4—3881no—
282C2×C4—1942no—
283C2×C4C2×C4×D9711yescharacteristic
284C2×C4C2×C4×D9711yescharacteristic
285C2×C4C2×C4×D9711yescharacteristic
286C2×C4C2×C4×D9711yescharacteristic
287C2×C4C2×C4×D9711yescharacteristic
288C2×C4C2×C4×D9711yescharacteristic
289C2×C4C23×D9711yescharacteristic
290C8—21no—
291C8—21no—
292C8—21no—
293C8—972no—
294C8—1941no—
295C8—1941no—
296C8—1941no—
297C8Q8×D9712yes—
298C22—972no—
299C22—1941no—
300C22—1941no—
301C22C22×C4×D9711yescharacteristic
302C4—21no—
303C4—21no—
304C4—21no—
305C4—21no—
306C4—21no—
307C4—21no—
308C4—972no—
309C4—972no—
310C4—1941no—
311C4—1941no—
312C4—1941no—
313C4—1941no—
314C4—1941no—
315C4C2×Q8×D9711yescharacteristic
316C4#41×D9711yescharacteristic
317C2—1942no—
318C2C4×Q8×D9711yescharacteristic
319C2#131×D9711yescharacteristic
320C2#2026150111yescharacteristic
subgroupG/H
center, FrattiniC2×C4C23×D97
commutator, socleC22×C97C23×C4
Fitting#446×C97C2
prime radicalC22×D97C22×C4
cosocleC2×C4×C97C24
hypercenter#446D97
hypocenterC97#2962
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C2×C4 ◁ #446

Complements

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

NH ≅ G/Ntype
2. #20260866C2semidirect
3. #20260864C2semidirect
4. #20260864C2semidirect
5. #20260863C2semidirect
6. #20260863C2semidirect
7. C8×Dic194C2semidirect
8. #20260859C2semidirect
15. #446×C97C2semidirect
66. #13095494Q8semidirect
161. C97#2962semidirect
168. #446D97semidirect

Maximal subgroups

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

subgroupindex
#202612892
#202608662
#202608642
#202608632
C8×Dic1942
#202608592
#202607602
C4⋊C8×D972
#202606922
#202606892
#202606872
#446×C972
#296297