Finite Group Databasebeta

← #27158661

Subgroups — #27158661

Order 12416.

398 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261787C212yesmaximal, complemented
2#20261767C212yesmaximal, complemented
3#20261766C212yesmaximal, complemented
4#20261762C212yesmaximal, complemented
5#20261757C211yesmaximal, characteristic, complemented
6#20261756C211yesmaximal, characteristic, complemented
7#20261751C211yesmaximal, characteristic, complemented
8#20261738C211yesmaximal, characteristic, complemented
9#20261736C211yesmaximal, characteristic, complemented
10C2×C4⋊#11715897C211yesmaximal, characteristic, complemented
11C4×#13095510C211yesmaximal, characteristic
12C2×#13095520C2212yes—
13C2×#13095519C2212yes—
14C22×#11715897C2212yes—
15#15690877C2212yes—
16#15690877C2212yes—
17C4⋊#11715897C2211yescharacteristic
18C4⋊#11715897C2211yescharacteristic
19C4⋊#11715897C2212yes—
20C4⋊#11715897C2211yescharacteristic
21C4⋊#11715897C2211yescharacteristic
22#15690867C2212yes—
23C22⋊#11715897C2212yes—
24C22⋊#11715897C2212yes—
25#15690865C2212yes—
26#15690865C2212yes—
27C4×#11715897C2212yes—
28C2×#13095510C411yescharacteristic
29C4×Dic194C411yescharacteristic
30#15690775C2211yescharacteristic
31D4×Dic97C2211yescharacteristic
32C4×C22⋊D97C411yescharacteristic
33C4×C22⋊D97C411yescharacteristic
34#15690770C2211yescharacteristic
35#15690770C2211yescharacteristic
36#15690768C411yescharacteristic
37#15690766C411yescharacteristic
38#15690766C411yescharacteristic
39C2×C4×Dic97C2211yescharacteristic
40C2×C4×Dic97C2211yescharacteristic
41C42×D97C2211yescharacteristic
42C4×D4×C97C411yescharacteristic
43#13095520—42no—
44#13095519—42no—
45C2×#11715897—42no—
46C2×#11715897C2312yes—
47C2×#11715897C2312yes—
48C2×#11715897C2312yes—
49C2×#11715897C2312yes—
50#13095510—24no—
51C2×Dic194C2×C411yescharacteristic
52C2×C22⋊D97C2×C411yescharacteristic
53C2×C22⋊D97C2×C411yescharacteristic
54C22×Dic97C2×C411yescharacteristic
55C22×Dic97C2×C411yescharacteristic
56C22×Dic97C2311yescharacteristic
57C22×Dic97C2311yescharacteristic
58C2×C4×D97C2×C411yescharacteristic
59C2×C4×D97C2×C411yescharacteristic
60C2×C4×D97C2311yescharacteristic
61C4⋊Dic97C2×C411yescharacteristic
62#13095490C2×C411yescharacteristic
63#13095490C2×C411yescharacteristic
64#13095490C2×C411yescharacteristic
65#13095490C2×C411yescharacteristic
66C22⋊Dic97C2×C411yescharacteristic
67C22⋊Dic97C2×C411yescharacteristic
68#13095488C2×C411yescharacteristic
69#13095488C2×C411yescharacteristic
70C4×Dic97C2×C411yescharacteristic
71C4×Dic97C2×C411yescharacteristic
72C4×Dic97C2×C411yescharacteristic
73C4×Dic97C2×C411yescharacteristic
74C4×Dic97C2×C411yescharacteristic
75C4×Dic97C2311yescharacteristic
76C4×Dic97C2311yescharacteristic
77C4×Dic97C2311yescharacteristic
78C4×Dic97C2311yescharacteristic
79C2×D4×C97C2×C411yescharacteristic
80C22×C4×C97C2×C411yescharacteristic
81C22×C4×C97C2×C411yescharacteristic
82C4⋊C4×C97C2×C411yescharacteristic
83C22⋊C4×C97C2×C411yescharacteristic
84C22⋊C4×C97C2×C411yescharacteristic
85C42×C97C2×C411yescharacteristic
86C23×C97C22×C411yescharacteristic
87C23×C97C22×C411yescharacteristic
88C22×D97C22×C411yescharacteristic
89D4×C97—22no—
90C2×C4×C97—22no—
91C2×C4×C97—22no—
92C2×C4×C97C22×C411yescharacteristic
93C2×C4×C97C22×C411yescharacteristic
94C2×C4×C97C22×C411yescharacteristic
95C2×C4×C97C22×C411yescharacteristic
96C2×C4×C97C22×C411yescharacteristic
97C22⋊D97—22no—
98C22⋊D97M1614yescomplemented
99C2×Dic97—22no—
100C2×Dic97—22no—
101C2×Dic97—21no—
102C2×Dic97—21no—
103C2×Dic97—21no—
104C2×Dic97—21no—
105C2×Dic97C22×C411yescharacteristic
106C2×Dic97C22×C411yescharacteristic
107C2×Dic97C22×C411yescharacteristic
108C2×Dic97C22×C411yescharacteristic
109C2×Dic97C22×C411yescharacteristic
110C2×Dic97C22×C411yescharacteristic
111C2×Dic97C2411yescharacteristic
112C4×D97—22no—
113C4×D97—22no—
114C4×D97—21no—
115C4×D97—21no—
116Dic194—22no—
117#11715897—22no—
118#11715897—22no—
119#11715897—22no—
120#11715897—22no—
121C22×C97—21no—
122C22×C97—21no—
123C22×C97—21no—
124C22×C97—21no—
125C22×C97—21no—
126C22×C97C2×M1612yes—
127C22×C97C23×C411yescharacteristic
128C2×D97—21no—
129C2×D97—21no—
130C2×D97C2×M1612yes—
131C4×C97—21no—
132C4×C97—21no—
133C4×C97—21no—
134C4×C97—21no—
135C4×C97#13112yes—
136Dic97—21no—
137Dic97—21no—
138Dic97—21no—
139Dic97—21no—
140Dic97—21no—
141Dic97C2×M1612yes—
142Dic97H(5,𝔽2)11yescharacteristic
143Dic972−1+411yescharacteristic
144C2×C97—22no—
145C2×C97—41no—
146C2×C97#52011yescharacteristic
147C2×C97C22×M1611yescharacteristic
148C2×C97C2×#13111yescharacteristic
149D97—22no—
150#2990—971nomaximal, Sylow
151C97#299011yescharacteristic, complemented, Sylow
152C4×#41—971no—
153#446—972no—
154#435—972no—
155#435—972no—
156C8×D4—972no—
157#432—971no—
158#432—971no—
159#431—971no—
160#423—971no—
161C2×C4⋊C8—971no—
162C2×C4⋊C8—971no—
163C2×#41—971no—
164C2×M16—972no—
165C22×C8—972no—
166C22×C8—972no—
167C4×Q8—971no—
168C4×Q8—971no—
169C4×D4—971no—
170C4×D4—971no—
171C4×D4—971no—
172C4×D4—971no—
173C4×D4—971no—
174C4×D4#8760411yescharacteristic
175#117—971no—
176#117—971no—
177#117—971no—
178C2×C42—971no—
179C2×C42—971no—
180C2×C42—971no—
181C4⋊C8—971no—
182C4⋊C8—971no—
183C4⋊C8—971no—
184C4⋊C8—972no—
185C4⋊C8—971no—
186C4⋊C8—972no—
187C4⋊C8—972no—
188C22⋊C8—972no—
189C22⋊C8—972no—
190C22⋊C8—972no—
191#97—972no—
192#97—972no—
193C4×C8—972no—
194#41—1944no—
195C2×Q8—971no—
196C2×D4—971no—
197C2×D4—971no—
198C2×D4C2×#8760411yescharacteristic
199C22×C4—971no—
200C22×C4—971no—
201C22×C4—971no—
202C22×C4—971no—
203C22×C4—971no—
204C22×C4—971no—
205C22×C4—971no—
206C22×C4C2×#8760411yescharacteristic
207C22×C4C2×#8760411yescharacteristic
208M16—3882no—
209C2×C8—972no—
210C2×C8—972no—
211C2×C8—972no—
212C2×C8—972no—
213C2×C8—3882no—
214C2×C8—3882no—
215C4⋊C4—971no—
216C4⋊C4—971no—
217C4⋊C4—971no—
218C4⋊C4—971no—
219C4⋊C4—971no—
220C4⋊C4C2×#8760411yescharacteristic
221C22⋊C4—971no—
222C22⋊C4—971no—
223C22⋊C4—971no—
224C22⋊C4—971no—
225C22⋊C4C2×#8760411yescharacteristic
226C22⋊C4C2×#8760411yescharacteristic
227C42—971no—
228C42—971no—
229C42—971no—
230C42—971no—
231C42—971no—
232C42—971no—
233C42—971no—
234C42—971no—
235C42—971no—
236C42C2×#8760411yescharacteristic
237C23—971no—
238C23C22×#8760411yescharacteristic
239C23C22×#8760411yescharacteristic
240Q8—1942no—
241D4—22no—
242D4—974no—
243D4—1942no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—971no—
247C2×C4—971no—
248C2×C4—971no—
249C2×C4—971no—
250C2×C4—971no—
251C2×C4—971no—
252C2×C4—971no—
253C2×C4—1941no—
254C2×C4—1941no—
255C2×C4—1942no—
256C2×C4—1941no—
257C2×C4—1941no—
258C2×C4—1942no—
259C2×C4—1941no—
260C2×C4—1941no—
261C2×C4—1942no—
262C2×C4—1942no—
263C2×C4C22×#8760411yescharacteristic
264C2×C4C22×#8760411yescharacteristic
265C2×C4C22×#8760411yescharacteristic
266C2×C4C22×#8760411yescharacteristic
267C2×C4C22×#8760411yescharacteristic
268C8—1942no—
269C8—1942no—
270C8—1942no—
271C8—1942no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—972no—
278C22—1941no—
279C22—1941no—
280C22D97⋊M1612yes—
281C22C23×#8760411yescharacteristic
282C4—21no—
283C4—21no—
284C4—21no—
285C4—21no—
286C4—971no—
287C4—971no—
288C4—972no—
289C4—1941no—
290C4—1941no—
291C4—1941no—
292C4—1941no—
293C4—1941no—
294C4#1569089712yes—
295C2—22no—
296C2—41no—
297C2—1942no—
298C2#2026180411yescharacteristic
299C2C2×D97⋊M1611yescharacteristic
300C2C2×#1569089711yescharacteristic
subgroupG/H
center, FrattiniC22C23×#87604
commutator, socleC22×C97C23×C4
FittingC4×D4×C97C4
prime radicalC2×#13095510C4
cosocleC2×Dic97C24
hypercenterC4×D4#87604
hypocenterC97#2990
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. #20261787C2semidirect
2. #20261767C2semidirect
3. #20261766C2semidirect
4. #20261762C2semidirect
5. #20261757C2semidirect
6. #20261756C2semidirect
7. #20261751C2semidirect
8. #20261738C2semidirect
9. #20261736C2semidirect
10. C2×C4⋊#11715897C2semidirect
98. C22⋊D97M16semidirect
151. C97#2990semidirect

Maximal subgroups

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

subgroupindex
#202617872
#202617672
#202617662
#202617622
#202617572
#202617562
#202617512
#202617382
#202617362
C2×C4⋊#117158972
C4×#130955102
#299097