Finite Group Databasebeta

← #27154388

Subgroups — #27154388

Order 12416.

350 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261457C211yesmaximal, characteristic, complemented
2#20261441C211yesmaximal, characteristic, complemented
3#20261092C211yesmaximal, characteristic, complemented
4#20261091C211yesmaximal, characteristic, complemented
5#20261084C211yesmaximal, characteristic, complemented
6#20261041C211yesmaximal, characteristic, complemented
7#20261024C211yesmaximal, characteristic, complemented
8#20260953C211yesmaximal, characteristic
9#20260951C211yesmaximal, characteristic
10#20260939C211yesmaximal, characteristic, complemented
11#20260859C211yesmaximal, characteristic, complemented
12C4×#13095498C211yesmaximal, characteristic, complemented
13#20260822C211yesmaximal, characteristic, complemented
14#20260817C211yesmaximal, characteristic
15#479×C97C211yesmaximal, characteristic, complemented
16C2×#13095500C2211yescharacteristic
17C2×#13095498C2211yescharacteristic
18C2×D776C2211yescharacteristic
19#15690809C2211yescharacteristic
20#15690808C2211yescharacteristic
21#15690805—21no—
22#15690804—21no—
23#15690801C2211yescharacteristic
24#15690799C2211yescharacteristic
25#15690798C2211yescharacteristic
26#15690790—21no—
27#15690788—21no—
28#15690788—21no—
29#15690788—21no—
30C4×Dic194C2211yescharacteristic
31#15690775C2211yescharacteristic
32C4×D388C2211yescharacteristic
33#15690772C2211yescharacteristic
34#15690772C2211yescharacteristic
35#15690772C2211yescharacteristic
36#15690767—21no—
37C4⋊C4×D97—21no—
38#15690741C2211yescharacteristic
39#15690739C2211yescharacteristic
40#15690739C2211yescharacteristic
41#15690735C2211yescharacteristic
42#15690734C2211yescharacteristic
43#15690733C2211yescharacteristic
44#15690733C2211yescharacteristic
45#15690732C2211yescharacteristic
46#15690730C2211yescharacteristic
47#15690730C2211yescharacteristic
48#15690730C2211yescharacteristic
49C4×#11715895C2211yescharacteristic
50C8×Dic97C2211yescharacteristic
51C8×Dic97C2211yescharacteristic
52#125×C97C2211yescharacteristic
53C4×Q8×C97C2211yescharacteristic
54#107×C97C2211yescharacteristic
55#106×C97C2211yescharacteristic
56C4⋊C8×C97C2211yescharacteristic
57#103×C97C2211yescharacteristic
58#103×C97C2211yescharacteristic
59C2×Dic194—21no—
60C2×Dic194D411yescharacteristic
61C2×Dic194C2311yescharacteristic
62C2×D388C2311yescharacteristic
63C2×D388C2311yescharacteristic
64C2×C4×D97—21no—
65C2×C4×D97—21no—
66C2×C4×D97—21no—
67C2×C4×D97—21no—
68#13095500—22no—
69#13095498—22no—
70D776—22no—
71C2×#11715895C2311yescharacteristic
72C2×#11715895C2311yescharacteristic
73C4⋊Dic97—21no—
74C4⋊Dic97D411yescharacteristic
75C4⋊Dic97C2311yescharacteristic
76#13095490—21no—
77#13095490—21no—
78#13095490—21no—
79#13095490—21no—
80#13095490—21no—
81#13095490D411yescharacteristic
82#13095490D411yescharacteristic
83#13095488—21no—
84#13095488—21no—
85#13095488—21no—
86#13095488—21no—
87#13095488—21no—
88#13095488—21no—
89#13095488—21no—
90#13095488—21no—
91C4×Dic97—21no—
92C4×Dic97C2311yescharacteristic
93C4×Dic97C2311yescharacteristic
94C2×Q8×C97C2311yescharacteristic
95C2×C8×C97C2311yescharacteristic
96C2×C8×C97C2311yescharacteristic
97C4⋊C4×C97—21no—
98C4⋊C4×C97—21no—
99C4⋊C4×C97—21no—
100C4⋊C4×C97C2311yescharacteristic
101C4⋊C4×C97C2311yescharacteristic
102C4⋊C4×C97C2311yescharacteristic
103C42×C97—21no—
104C42×C97C2311yescharacteristic
105C22×D97—21no—
106C22×D97—21no—
107Q8×C97—21no—
108Q8×C97—21no—
109C2×C4×C97—21no—
110C2×C4×C97—21no—
111C2×C4×C97—21no—
112C2×C4×C97—21no—
113C2×C4×C97C2×D411yescharacteristic
114C2×C4×C97C2×D411yescharacteristic
115C2×C4×C97C2411yescharacteristic
116C2×Dic97—21no—
117C2×Dic97—21no—
118C2×Dic97C2×D411yescharacteristic
119C2×Dic97C2×D411yescharacteristic
120C2×Dic97C2×D411yescharacteristic
121C2×Dic97C2×D411yescharacteristic
122D388—21no—
123D388—21no—
124D388—21no—
125D388#4112yes—
126C4×D97—42no—
127C4×D97—42no—
128C4×D97—81no—
129C4×D97—81no—
130Dic194—22no—
131Dic194—21no—
132Dic194—21no—
133Dic194—81no—
134C8×C97—21no—
135C8×C97—21no—
136#11715895—21no—
137#11715895—21no—
138C22×C97C22×D411yescharacteristic
139C2×D97—22no—
140C2×D97—41no—
141C2×D97—41no—
142C2×D97—41no—
143C2×D97—41no—
144C2×D97—41no—
145C4×C97—21no—
146C4×C97—21no—
147C4×C97—41no—
148C4×C97—41no—
149C4×C97—41no—
150C4×C97—41no—
151C4×C97C2×#4111yescharacteristic
152C4×C972−1+411yescharacteristic
153Dic97—21no—
154Dic97—21no—
155Dic97—21no—
156Dic97—41no—
157Dic97—41no—
158Dic97#13512yes—
159C2×C97#54811yescharacteristic
160C2×C97C2×#13511yescharacteristic
161C2×C97#57711yescharacteristic
162D97—42no—
163D97—81no—
164#3360—971nomaximal, Sylow
165C97#336011yescharacteristic, complemented, Sylow
166#556—971no—
167#554—971no—
168#488—971no—
169#488—971no—
170#487—971no—
171#479—971no—
172#479D9711yescharacteristic, complemented
173#476—971no—
174#463—971no—
175#463—971no—
176#461—971no—
177C8×Q8—971no—
178C4×SD16—971no—
179C4×SD16—971no—
180C4×D8—971no—
181C2×SD16—971no—
182C2×SD16—971no—
183C2×D8—971no—
184C4⋊Q8—971no—
185C4⋊Q8—971no—
186#126—1941no—
187#126—1941no—
188#125—971no—
189#125C2×D9711yescharacteristic
190#124—971no—
191#124—971no—
192C22⋊Q8—1941no—
193C22⋊Q8—1941no—
194C22⋊Q8—1941no—
195C22⋊Q8—1941no—
196C4×Q8—971no—
197C4×Q8—971no—
198C4×Q8C2×D9711yescharacteristic
199C4×D4—971no—
200C4×D4—971no—
201C4×D4—971no—
202C4×D4—971no—
203#117—1941no—
204C2×C4⋊C4—1941no—
205#107C2×D9711yescharacteristic
206#106—971no—
207#106C2×D9711yescharacteristic
208C4⋊C8—971no—
209C4⋊C8—971no—
210C4⋊C8C2×D9711yescharacteristic
211#103—971no—
212#103—971no—
213#103—971no—
214#103—971no—
215#103C2×D9711yescharacteristic
216#103C2×D9711yescharacteristic
217#102—971no—
218#102—971no—
219#102—971no—
220#102—971no—
221C4×C8—971no—
222C4×C8—971no—
223C4×C8—971no—
224C2×Q8—971no—
225C2×Q8—971no—
226C2×Q8—1941no—
227C2×Q8C22×D9711yescharacteristic
228C2×D4—971no—
229C2×D4—971no—
230C22×C4—1941no—
231C22×C4—1941no—
232C22×C4—1941no—
233C22×C4—1941no—
234SD16—1942no—
235SD16—1942no—
236D8—1942no—
237C2×C8—971no—
238C2×C8—971no—
239C2×C8C22×D9711yescharacteristic
240C2×C8C22×D9711yescharacteristic
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—971no—
245C4⋊C4—971no—
246C4⋊C4—971no—
247C4⋊C4—971no—
248C4⋊C4—1941no—
249C4⋊C4—1941no—
250C4⋊C4—1941no—
251C4⋊C4—1941no—
252C4⋊C4—1941no—
253C4⋊C4—1941no—
254C4⋊C4C22×D9711yescharacteristic
255C4⋊C4C22×D9711yescharacteristic
256C4⋊C4C22×D9711yescharacteristic
257C22⋊C4—1941no—
258C22⋊C4—1941no—
259C22⋊C4—1941no—
260C22⋊C4—1941no—
261C22⋊C4—1941no—
262C22⋊C4—1941no—
263C22⋊C4—1941no—
264C22⋊C4—1941no—
265C42—21no—
266C42—971no—
267C42—971no—
268C42—1941no—
269C42C22×D9711yescharacteristic
270C23—1941no—
271C23—1941no—
272Q8—21no—
273Q8—21no—
274Q8—7761no—
275Q8—1942no—
276Q8—1941no—
277Q8—1941no—
278D4—972no—
279D4—1941no—
280D4—1941no—
281D4—1941no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—7761no—
287C2×C4—7761no—
288C2×C4—971no—
289C2×C4—971no—
290C2×C4—971no—
291C2×C4—971no—
292C2×C4—3882no—
293C2×C4—3882no—
294C2×C4—1941no—
295C2×C4—1941no—
296C2×C4D4×D9711yescharacteristic
297C2×C4D4×D9711yescharacteristic
298C2×C4C23×D9711yescharacteristic
299C8—21no—
300C8—21no—
301C8—1941no—
302C8—1941no—
303C22—3881no—
304C22—3881no—
305C22—3881no—
306C22—3881no—
307C22—3881no—
308C22—1942no—
309C22C2×D4×D9711yescharacteristic
310C4—21no—
311C4—21no—
312C4—41no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—972no—
317C4—3881no—
318C4—3881no—
319C4—1941no—
320C4—1941no—
321C4—1941no—
322C4#41×D9711yescharacteristic
323C4#1569085811yescharacteristic
324C2—7761no—
325C2—3882no—
326C2#2026139511yescharacteristic
327C2#135×D9711yescharacteristic
328C2#2026154411yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#479×C97C2
socleC22×C97C22×D4
prime radicalC2×D776C22
hypercenter#479D97
hypocenterC97#3360
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #479

Complements

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

NH ≅ G/Ntype
1. #20261457C2semidirect
2. #20261441C2semidirect
3. #20261092C2semidirect
4. #20261091C2semidirect
5. #20261084C2semidirect
6. #20261041C2semidirect
7. #20261024C2semidirect
10. #20260939C2semidirect
11. #20260859C2semidirect
12. C4×#13095498C2semidirect
13. #20260822C2semidirect
15. #479×C97C2semidirect
165. C97#3360semidirect
172. #479D97semidirect

Maximal subgroups

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

subgroupindex
#202614572
#202614412
#202610922
#202610912
#202610842
#202610412
#202610242
#202609532
#202609512
#202609392
#202608592
C4×#130954982
#202608222
#202608172
#479×C972
#336097