Finite Group Databasebeta

← #27154422

Subgroups — #27154422

Order 12416.

336 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261461C211yesmaximal, characteristic, complemented
2#20261457C211yesmaximal, characteristic
3#20261091C211yesmaximal, characteristic
4#20261090C211yesmaximal, characteristic
5#20261084C211yesmaximal, characteristic, complemented
6#20261042C211yesmaximal, characteristic, complemented
7#20261041C211yesmaximal, characteristic, complemented
8#20261014C211yesmaximal, characteristic
9#20260959C211yesmaximal, characteristic
10#20260957C211yesmaximal, characteristic
11#20260943C211yesmaximal, characteristic, complemented
12C8×Dic194C211yesmaximal, characteristic, complemented
13#20260827C211yesmaximal, characteristic, complemented
14#20260820C211yesmaximal, characteristic
15C4×Q16×C97C211yesmaximal, characteristic, complemented
16C2×#13095501C2211yescharacteristic
17C2×#13095498C2211yescharacteristic
18#15690809—21no—
19#15690809C2211yescharacteristic
20#15690808C2211yescharacteristic
21#15690805—21no—
22#15690804—21no—
23#15690802—21no—
24#15690802C2211yescharacteristic
25#15690801—21no—
26#15690801C2211yescharacteristic
27#15690799C2211yescharacteristic
28#15690798C2211yescharacteristic
29#15690788—21no—
30Q8×Dic97C2211yescharacteristic
31Q8×Dic97C2211yescharacteristic
32C4×Dic194—21no—
33C4×Dic194C2211yescharacteristic
34#15690775C2211yescharacteristic
35#15690772C2211yescharacteristic
36#15690767—21no—
37#15690743C2211yescharacteristic
38#15690741C2211yescharacteristic
39C4⋊#11715895C2211yescharacteristic
40#15690739C2211yescharacteristic
41#15690739C2211yescharacteristic
42#15690735C2211yescharacteristic
43#15690735C2211yescharacteristic
44#15690734C2211yescharacteristic
45#15690734C2211yescharacteristic
46#15690733C2211yescharacteristic
47#15690732C2211yescharacteristic
48#15690732C2211yescharacteristic
49#15690730C2211yescharacteristic
50C8×Dic97C2211yescharacteristic
51C8×Dic97C2211yescharacteristic
52C2×Q16×C97C2211yescharacteristic
53C4×Q8×C97C2211yescharacteristic
54C4×Q8×C97C2211yescharacteristic
55#107×C97C2211yescharacteristic
56#103×C97C2211yescharacteristic
57#103×C97C2211yescharacteristic
58C4×C8×C97C2211yescharacteristic
59C2×Dic194—21no—
60C2×Dic194D411yescharacteristic
61C2×Dic194C2311yescharacteristic
62C2×D388C2311yescharacteristic
63C2×C4×D97—21no—
64#13095501—22no—
65#13095498—22no—
66C2×#11715895C2311yescharacteristic
67C2×#11715895C2311yescharacteristic
68C4⋊Dic97—21no—
69C4⋊Dic97—21no—
70C4⋊Dic97—21no—
71C4⋊Dic97D411yescharacteristic
72C4⋊Dic97C2311yescharacteristic
73C4⋊Dic97C2311yescharacteristic
74#13095490—21no—
75#13095490—21no—
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—
88C4×Dic97—21no—
89C4×Dic97—21no—
90C4×Dic97—21no—
91C4×Dic97C2311yescharacteristic
92C4×Dic97C2311yescharacteristic
93C2×Q8×C97C2311yescharacteristic
94C2×Q8×C97C2311yescharacteristic
95Q16×C97—22no—
96C2×C8×C97C2311yescharacteristic
97C2×C8×C97C2311yescharacteristic
98C4⋊C4×C97—21no—
99C4⋊C4×C97—21no—
100C4⋊C4×C97C2311yescharacteristic
101C4⋊C4×C97C2311yescharacteristic
102C42×C97—21no—
103C42×C97—21no—
104C42×C97C2311yescharacteristic
105C22×D97—21no—
106Q8×C97—21no—
107Q8×C97—21no—
108Q8×C97—21no—
109Q8×C97#4112yes—
110C2×C4×C97—21no—
111C2×C4×C97—21no—
112C2×C4×C97—21no—
113C2×C4×C97—21no—
114C2×C4×C97C2×D411yescharacteristic
115C2×C4×C97C2×D411yescharacteristic
116C2×C4×C97C2411yescharacteristic
117C2×Dic97—21no—
118C2×Dic97—21no—
119C2×Dic97—21no—
120C2×Dic97C2×D411yescharacteristic
121C2×Dic97C2×D411yescharacteristic
122C2×Dic97C2×D411yescharacteristic
123C2×Dic97C2×D411yescharacteristic
124D388—21no—
125D388—21no—
126C4×D97—42no—
127Dic194—22no—
128Dic194—21no—
129Dic194—21no—
130Dic194—42no—
131C8×C97—21no—
132C8×C97—21no—
133#11715895—21no—
134#11715895—21no—
135C22×C97C22×D411yescharacteristic
136C2×D97—41no—
137C2×D97—41no—
138C2×D97—41no—
139C4×C97—22no—
140C4×C97—21no—
141C4×C97—21no—
142C4×C97—41no—
143C4×C97—41no—
144C4×C97—41no—
145C4×C97C2×#4111yescharacteristic
146C4×C972−1+411yescharacteristic
147Dic97—21no—
148Dic97—21no—
149Dic97—21no—
150Dic97—41no—
151Dic97—41no—
152Dic97—41no—
153Dic97#13512yes—
154C2×C97#54811yescharacteristic
155C2×C97C2×#13511yescharacteristic
156C2×C97#57811yescharacteristic
157D97—81no—
158#3364—971nomaximal, Sylow
159C97#336411yescharacteristic, complemented, Sylow
160#557—971no—
161#556—971no—
162#488—971no—
163#488—971no—
164#487—971no—
165#479—971no—
166#479—971no—
167#474—971no—
168#464—971no—
169#464—971no—
170#462—971no—
171C8×Q8—971no—
172C4×Q16—971no—
173C4×Q16D9711yescharacteristic, complemented
174C4×SD16—971no—
175C2×Q16—971no—
176C2×Q16C2×D9711yescharacteristic
177C2×SD16—971no—
178C4⋊Q8—971no—
179C4⋊Q8—971no—
180C4⋊Q8—1941no—
181#126—1941no—
182#126—1941no—
183#125—971no—
184#125—971no—
185#125—1941no—
186#125—1941no—
187#124—971no—
188#124—971no—
189C22⋊Q8—1941no—
190C4×Q8—971no—
191C4×Q8—971no—
192C4×Q8—971no—
193C4×Q8—971no—
194C4×Q8—1941no—
195C4×Q8C2×D9711yescharacteristic
196C4×Q8C2×D9711yescharacteristic
197C4×D4—971no—
198#117—1941no—
199#107—971no—
200#107C2×D9711yescharacteristic
201#106—971no—
202C4⋊C8—971no—
203C4⋊C8—971no—
204C4⋊C8—971no—
205#103—971no—
206#103—971no—
207#103—971no—
208#103—971no—
209#103—971no—
210#103C2×D9711yescharacteristic
211#103C2×D9711yescharacteristic
212#102—971no—
213#102—971no—
214#102—971no—
215C4×C8—971no—
216C4×C8—971no—
217C4×C8C2×D9711yescharacteristic
218C2×Q8—971no—
219C2×Q8—971no—
220C2×Q8—1941no—
221C2×Q8C22×D9711yescharacteristic
222C2×Q8C22×D9711yescharacteristic
223C2×D4—971no—
224C22×C4—1941no—
225Q16—22no—
226Q16—1942no—
227SD16—1942no—
228C2×C8—971no—
229C2×C8—971no—
230C2×C8C22×D9711yescharacteristic
231C2×C8C22×D9711yescharacteristic
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—971no—
235C4⋊C4—971no—
236C4⋊C4—971no—
237C4⋊C4—971no—
238C4⋊C4—971no—
239C4⋊C4—1941no—
240C4⋊C4—1941no—
241C4⋊C4—1941no—
242C4⋊C4—1941no—
243C4⋊C4—1941no—
244C4⋊C4—1941no—
245C4⋊C4—1941no—
246C4⋊C4—1941no—
247C4⋊C4—1941no—
248C4⋊C4—1941no—
249C4⋊C4C22×D9711yescharacteristic
250C4⋊C4C22×D9711yescharacteristic
251C22⋊C4—1941no—
252C22⋊C4—1941no—
253C22⋊C4—1941no—
254C22⋊C4—1941no—
255C22⋊C4—1941no—
256C42—21no—
257C42—21no—
258C42—971no—
259C42—971no—
260C42—1941no—
261C42—1941no—
262C42—1941no—
263C42C22×D9711yescharacteristic
264C23—1941no—
265Q8—21no—
266Q8—21no—
267Q8—21no—
268Q8—3882no—
269Q8—1942no—
270Q8—1941no—
271Q8—1941no—
272Q8#1309551112yes—
273D4—1941no—
274D4—1941no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—971no—
280C2×C4—971no—
281C2×C4—971no—
282C2×C4—971no—
283C2×C4—3882no—
284C2×C4—1941no—
285C2×C4—1941no—
286C2×C4—1941no—
287C2×C4D4×D9711yescharacteristic
288C2×C4D4×D9711yescharacteristic
289C2×C4C23×D9711yescharacteristic
290C8—21no—
291C8—21no—
292C8—1941no—
293C8—1941no—
294C22—3881no—
295C22—3881no—
296C22—3881no—
297C22C2×D4×D9711yescharacteristic
298C4—22no—
299C4—21no—
300C4—21no—
301C4—41no—
302C4—41no—
303C4—41no—
304C4—972no—
305C4—3881no—
306C4—3881no—
307C4—3881no—
308C4—1941no—
309C4—1941no—
310C4—1941no—
311C4C2×#1309551111yescharacteristic
312C4#1569085711yescharacteristic
313C2—7761no—
314C2#2026139411yescharacteristic
315C2#135×D9711yescharacteristic
316C2#2026154811yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC4×Q16×C97C2
socleC22×C97C22×D4
prime radicalC2×D388C23
hypercenterC4×Q16D97
hypocenterC97#3364
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q16

Complements

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

NH ≅ G/Ntype
1. #20261461C2semidirect
5. #20261084C2semidirect
6. #20261042C2semidirect
7. #20261041C2semidirect
11. #20260943C2semidirect
12. C8×Dic194C2semidirect
13. #20260827C2semidirect
15. C4×Q16×C97C2semidirect
159. C97#3364semidirect
173. C4×Q16D97semidirect

Maximal subgroups

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

subgroupindex
#202614612
#202614572
#202610912
#202610902
#202610842
#202610422
#202610412
#202610142
#202609592
#202609572
#202609432
C8×Dic1942
#202608272
#202608202
C4×Q16×C972
#336497