Finite Group Databasebeta

← #27158665

Subgroups — #27158665

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#20261765C212yesmaximal, complemented
3#20261764C212yesmaximal, complemented
4#20261762C212yesmaximal, complemented
5#20261758C211yesmaximal, characteristic, complemented
6#20261755C211yesmaximal, characteristic, complemented
7#20261753C211yesmaximal, characteristic, complemented
8#20261740C211yesmaximal, characteristic, complemented
9C2×#15690877C211yesmaximal, characteristic, complemented
10#20261736C211yesmaximal, characteristic, complemented
11#20261203C211yesmaximal, characteristic
12C2×#13095521C2212yes—
13C2×#13095519C2212yes—
14C22×#11715897C2212yes—
15#15690877C2211yescharacteristic
16#15690877C2211yescharacteristic
17#15690877C2211yescharacteristic
18#15690877C2212yes—
19#15690877C2212yes—
20#15690877C2211yescharacteristic
21C4⋊#11715897C2212yes—
22#15690867C2212yes—
23#15690867C2212yes—
24C22⋊#11715897C2212yes—
25#15690865C2212yes—
26#15690865C2212yes—
27C4×#11715897C2212yes—
28C2×#13095511C411yescharacteristic
29#15690775C411yescharacteristic
30#15690775C2211yescharacteristic
31#15690772C411yescharacteristic
32#15690772C2211yescharacteristic
33#15690770C411yescharacteristic
34#15690770C411yescharacteristic
35#15690770C2211yescharacteristic
36#15690770C2211yescharacteristic
37#15690767C411yescharacteristic
38#15690767C411yescharacteristic
39C2×C4×Dic97C2211yescharacteristic
40C42×D97C2211yescharacteristic
41C42×D97C2211yescharacteristic
42#117×C97C411yescharacteristic
43#13095521—42no—
44#13095519—42no—
45C2×#11715897—42no—
46C2×#11715897C2312yes—
47C2×#11715897C2312yes—
48C2×#11715897C2312yes—
49C2×#11715897C2312yes—
50#13095511—24no—
51C2×Dic194C2×C411yescharacteristic
52C2×D388C2×C411yescharacteristic
53C2×C22⋊D97C2×C411yescharacteristic
54C2×C22⋊D97C2×C411yescharacteristic
55C22×Dic97C2×C411yescharacteristic
56C22×Dic97C2311yescharacteristic
57C2×C4×D97C2×C411yescharacteristic
58C2×C4×D97C2×C411yescharacteristic
59C2×C4×D97C2×C411yescharacteristic
60C2×C4×D97C2×C411yescharacteristic
61C2×C4×D97C2311yescharacteristic
62C2×C4×D97C2311yescharacteristic
63#13095490C2×C411yescharacteristic
64#13095490C2×C411yescharacteristic
65#13095490C2×C411yescharacteristic
66#13095490C2×C411yescharacteristic
67#13095488C2×C411yescharacteristic
68#13095488C2×C411yescharacteristic
69#13095488C2×C411yescharacteristic
70#13095488C2×C411yescharacteristic
71C4×Dic97C2×C411yescharacteristic
72C4×Dic97C2×C411yescharacteristic
73C4×Dic97C2×C411yescharacteristic
74C4×Dic97C2×C411yescharacteristic
75C4×Dic97C2311yescharacteristic
76C4×Dic97C2311yescharacteristic
77C4×Dic97C2311yescharacteristic
78C4×Dic97C2311yescharacteristic
79C22×C4×C97C2×C411yescharacteristic
80C4⋊C4×C97C2×C411yescharacteristic
81C4⋊C4×C97C2×C411yescharacteristic
82C22⋊C4×C97C2×C411yescharacteristic
83C22⋊C4×C97C2×C411yescharacteristic
84C42×C97C2×C411yescharacteristic
85C42×C97C2×C411yescharacteristic
86C23×C97C22×C411yescharacteristic
87C22×D97C22×C411yescharacteristic
88C22×D97C22×C411yescharacteristic
89C2×C4×C97—22no—
90C2×C4×C97C22×C411yescharacteristic
91C2×C4×C97C22×C411yescharacteristic
92C2×C4×C97C22×C411yescharacteristic
93C2×C4×C97C22×C411yescharacteristic
94C2×C4×C97C22×C411yescharacteristic
95C2×C4×C97C22×C411yescharacteristic
96C22⋊D97—22no—
97C22⋊D97M1614yescomplemented
98C2×Dic97—22no—
99C2×Dic97—21no—
100C2×Dic97—21no—
101C2×Dic97C22×C411yescharacteristic
102C2×Dic97C22×C411yescharacteristic
103C2×Dic97C22×C411yescharacteristic
104C2×Dic97C22×C411yescharacteristic
105C2×Dic97C22×C411yescharacteristic
106C2×Dic97C2411yescharacteristic
107D388—22no—
108C4×D97—22no—
109C4×D97—22no—
110C4×D97—22no—
111C4×D97—22no—
112C4×D97—21no—
113C4×D97—21no—
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×C97C2×M1612yes—
124C22×C97C23×C411yescharacteristic
125C2×D97—21no—
126C2×D97—21no—
127C2×D97—21no—
128C2×D97—21no—
129C2×D97—21no—
130C2×D97C2×M1612yes—
131C4×C97—21no—
132C4×C97—21no—
133C4×C97—21no—
134C4×C97—21no—
135C4×C97—21no—
136C4×C97—21no—
137Dic97—21no—
138Dic97—21no—
139Dic97—21no—
140Dic97C2×M1612yes—
141Dic97#13112yes—
142Dic97H(5,𝔽2)11yescharacteristic
143Dic972−1+411yescharacteristic
144C2×C97—22no—
145C2×C97#52011yescharacteristic
146C2×C97C22×M1611yescharacteristic
147C2×C97C2×#13111yescharacteristic
148D97—22no—
149D97—41no—
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—971no—
175#117—971no—
176#117—971no—
177#117#8760411yescharacteristic
178C2×C42—971no—
179C2×C42—971no—
180C2×C42—971no—
181C4⋊C8—972no—
182C4⋊C8—971no—
183C4⋊C8—971no—
184C4⋊C8—971no—
185C4⋊C8—972no—
186C4⋊C8—972no—
187C4⋊C8—971no—
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×D4—971no—
199C22×C4—971no—
200C22×C4—971no—
201C22×C4—971no—
202C22×C4—971no—
203C22×C4—971no—
204C22×C4—971no—
205C22×C4—971no—
206C22×C4—971no—
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⋊C4C2×#8760411yescharacteristic
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—
235C42C2×#8760411yescharacteristic
236C42C2×#8760411yescharacteristic
237C23—971no—
238C23—971no—
239C23C22×#8760411yescharacteristic
240Q8—1942no—
241D4—974no—
242D4—1942no—
243D4—1942no—
244C2×C4—22no—
245C2×C4—971no—
246C2×C4—971no—
247C2×C4—971no—
248C2×C4—971no—
249C2×C4—971no—
250C2×C4—971no—
251C2×C4—1941no—
252C2×C4—1941no—
253C2×C4—1942no—
254C2×C4—1941no—
255C2×C4—1941no—
256C2×C4—1942no—
257C2×C4—1942no—
258C2×C4—1942no—
259C2×C4—1941no—
260C2×C4—1941no—
261C2×C4—1942no—
262C2×C4C22×#8760411yescharacteristic
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—972no—
275C22—1941no—
276C22—1941no—
277C22—1941no—
278C22—1941no—
279C22—1941no—
280C22D97⋊M1612yes—
281C22C23×#8760411yescharacteristic
282C4—21no—
283C4—21no—
284C4—21no—
285C4—21no—
286C4—21no—
287C4—21no—
288C4—971no—
289C4—971no—
290C4—972no—
291C4—972no—
292C4—1941no—
293C4—1941no—
294C4—1941no—
295C2—22no—
296C2—3881no—
297C2—1942no—
298C2#2026180411yescharacteristic
299C2C2×D97⋊M1611yescharacteristic
300C2D97⋊#13111yescharacteristic
subgroupG/H
center, FrattiniC22C23×#87604
commutator, socleC22×C97C23×C4
Fitting#117×C97C4
prime radicalC2×#13095511C4
cosocleC2×Dic97C24
hypercenter#117#87604
hypocenterC97#2990
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ #117

Complements

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

NH ≅ G/Ntype
1. #20261787C2semidirect
2. #20261765C2semidirect
3. #20261764C2semidirect
4. #20261762C2semidirect
5. #20261758C2semidirect
6. #20261755C2semidirect
7. #20261753C2semidirect
8. #20261740C2semidirect
9. C2×#15690877C2semidirect
10. #20261736C2semidirect
97. 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
#202617652
#202617642
#202617622
#202617582
#202617552
#202617532
#202617402
C2×#156908772
#202617362
#202612032
#299097