Finite Group Databasebeta

← #27144503

Subgroups — #27144503

Order 12416.

596 conjugacy classes of subgroups; only the 217 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 217 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20260638C212yesmaximal, complemented
2#20260628C212yesmaximal, complemented
3#20260621C212yesmaximal, complemented
4#20260615C212yesmaximal, complemented
5#20260615C212yesmaximal, complemented
6#20260598C211yesmaximal, characteristic, complemented
7C22⋊C4×Dic97C212yesmaximal, complemented
8C4×C22⋊Dic97C211yesmaximal, characteristic, complemented
9C2×#121×C97C211yesmaximal, characteristic
10C23×Dic97C2212yes—
11C2×C4⋊Dic97C2212yes—
12C2×C22⋊Dic97C2212yes—
13C2×C22⋊Dic97C2212yes—
14C2×C22⋊Dic97C2212yes—
15C2×C22⋊Dic97C2211yescharacteristic
16C2×C22⋊Dic97C2212yes—
17C2×C22⋊Dic97C2211yescharacteristic
18C2×C4×Dic97C2212yes—
19C2×C4×Dic97C2211yescharacteristic
20C2×C4×Dic97C2211yescharacteristic
21#15690717C2212yes—
22#15690717C2212yes—
23#15690717C2212yes—
24#15690717C2212yes—
25#15690717C2212yes—
26C22×D4×C97C2211yescharacteristic
27C22×D4×C97C2212yes—
28C23×C4×C97C2211yescharacteristic
29#121×C97C418yescomplemented
30C2×C4⋊C4×C97C2211yescharacteristic
31C2×C22⋊C4×C97C2212yes—
32C22×Dic97—24no—
33C22×Dic97—42no—
34C22×Dic97—42no—
35C22×Dic97C2312yes—
36C22×Dic97C2312yes—
37C22×Dic97C2312yes—
38C22×Dic97C2312yes—
39C4⋊Dic97—42no—
40C22⋊Dic97—24no—
41C22⋊Dic97—24no—
42C22⋊Dic97—42no—
43C22⋊Dic97—41no—
44C22⋊Dic97—42no—
45C22⋊Dic97—41no—
46C4×Dic97—41no—
47C4×Dic97—42no—
48C4×Dic97—41no—
49C24×C97C2311yescharacteristic
50C24×C97C2312yes—
51C2×D4×C97—41no—
52C2×D4×C97—41no—
53C2×D4×C97—42no—
54C2×D4×C97—42no—
55C2×D4×C97C2×C414yes—
56C2×D4×C97C2×C418yescomplemented
57C22×C4×C97—41no—
58C22×C4×C97—41no—
59C22×C4×C97C2×C414yescomplemented
60C22×C4×C97C2311yescharacteristic
61C22×C4×C97C2312yes—
62C22×C4×C97C2311yescharacteristic
63C4⋊C4×C97C2×C414yescomplemented
64C22⋊C4×C97C2×C418yescomplemented
65C23×C97—22no—
66C23×C97—22no—
67C23×C97—22no—
68C23×C97—21no—
69C23×C97—21no—
70C23×C97—22no—
71C23×C97—22no—
72C23×C97—22no—
73C23×C97—22no—
74C23×C97—22no—
75C23×C97C22×C414yes—
76C23×C97C22×C412yes—
77C23×C97C2411yescharacteristic
78D4×C97—44no—
79D4×C97—48no—
80C2×C4×C97—21no—
81C2×C4×C97—21no—
82C2×C4×C97—21no—
83C2×C4×C97—21no—
84C2×C4×C97—22no—
85C2×C4×C97—22no—
86C2×C4×C97—44no—
87C2×C4×C97C22×C412yes—
88C2×C4×C97C22×C414yes—
89C2×C4×C97C22×C412yes—
90C2×Dic97—22no—
91C2×Dic97—22no—
92C2×Dic97—22no—
93C2×Dic97—22no—
94C2×Dic97—22no—
95C2×Dic97—22no—
96C2×Dic97—22no—
97C2×Dic97—22no—
98C2×Dic97—22no—
99C2×Dic97—22no—
100C2×Dic97—22no—
101C2×Dic97—48no—
102C2×Dic97#4114yes—
103C22×C97—22no—
104C22×C97—24no—
105C22×C97—22no—
106C22×C97—22no—
107C22×C97—24no—
108C22×C97—24no—
109C22×C97—42no—
110C22×C97—42no—
111C22×C97—42no—
112C22×C97—42no—
113C22×C97—42no—
114C22×C97—42no—
115C22×C97C23×C411yescharacteristic
116C22×C97C2×#4112yes—
117C22×C97H(5,𝔽2)11yescharacteristic
118C22×C97H(5,𝔽2)12yes—
119C22×C97H(5,𝔽2)11yescharacteristic
120C4×C97—22no—
121C4×C97—22no—
122C4×C97—24no—
123Dic97—24no—
124Dic97—42no—
125Dic97—42no—
126Dic97—42no—
127C2×C97—42no—
128C2×C97—44no—
129C2×C97C4×#4111yescharacteristic
130C2×C97#51811yescharacteristic
131C2×C97#51811yescharacteristic
132C2×C97#53812yes—
133C2×C97#54012yes—
134#2333—971nomaximal, Sylow
135C97#233311yescharacteristic, complemented, Sylow
136C2×#121D9711yescharacteristic
137#390—972no—
138#388—972no—
139#387—972no—
140#386—972no—
141#386—972no—
142#381—971no—
143C4×C22⋊C4—971no—
144C4×C22⋊C4—972no—
145C22×D4C2×D9711yescharacteristic
146C22×D4C2×D9712yes—
147C23×C4—972no—
148C23×C4C2×D9711yescharacteristic
149#121Dic9718yescomplemented
150C2×C4⋊C4—972no—
151C2×C4⋊C4C2×D9711yescharacteristic
152C2×C22⋊C4—971no—
153C2×C22⋊C4—971no—
154C2×C22⋊C4—972no—
155C2×C22⋊C4—972no—
156C2×C22⋊C4—972no—
157C2×C22⋊C4—972no—
158C2×C22⋊C4C2×D9712yes—
159C2×C42—971no—
160C2×C42—972no—
161C2×C42—971no—
162#95—972no—
163#95—972no—
164#95—972no—
165#95—972no—
166#95—972no—
167C24C22×D9711yescharacteristic
168C24C22×D9712yes—
169C2×D4—42no—
170C2×D4—42no—
171C2×D4—41no—
172C2×D4—41no—
173C2×D4C2×Dic9714yes—
174C2×D4C2×Dic9718yescomplemented
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—972no—
178C22×C4—972no—
179C22×C4—972no—
180C22×C4—972no—
181C22×C4—3882no—
182C22×C4—3882no—
183C22×C4—1944no—
184C22×C4C2×Dic9714yescomplemented
185C22×C4C22×D9711yescharacteristic
186C22×C4C22×D9711yescharacteristic
187C22×C4C22×D9712yes—
188C4⋊C4—3882no—
189C4⋊C4C2×Dic9714yescomplemented
190C22⋊C4—3882no—
191C22⋊C4—3881no—
192C22⋊C4—3882no—
193C22⋊C4—3881no—
194C22⋊C4—1944no—
195C22⋊C4—1944no—
196C22⋊C4C2×Dic9718yescomplemented
197C42—3881no—
198C42—3881no—
199C42—3882no—
200C23—22no—
201C23—22no—
202C23—22no—
203C23—21no—
204C23—21no—
205C23—22no—
206C23—22no—
207C23—22no—
208C23—22no—
209C23—22no—
210C23C22×Dic9714yes—
211C23C22×Dic9712yes—
212C23C23×D9711yescharacteristic
213D4—44no—
214D4—48no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—44no—
222C2×C4—974no—
223C2×C4—3888no—
224C2×C4—1942no—
225C2×C4—1942no—
226C2×C4—1942no—
227C2×C4—1942no—
228C2×C4—1942no—
229C2×C4—1942no—
230C2×C4—1942no—
231C2×C4—1942no—
232C2×C4—1942no—
233C2×C4—1942no—
234C2×C4—1942no—
235C2×C4C22×Dic9712yes—
236C2×C4C22×Dic9714yes—
237C2×C4C22×Dic9712yes—
238C22—22no—
239C22—24no—
240C22—22no—
241C22—22no—
242C22—24no—
243C22—24no—
244C22—42no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22—42no—
249C22—42no—
250C22C23×Dic9711yescharacteristic
251C22#41×D9712yes—
252C22#1569085511yescharacteristic
253C22#1569085512yes—
254C22#1569085511yescharacteristic
255C4—22no—
256C4—22no—
257C4—24no—
258C4—3882no—
259C4—3882no—
260C4—3882no—
261C4—1944no—
262C2—42no—
263C2—44no—
264C2#41×Dic9711yescharacteristic
265C2#2026121011yescharacteristic
266C2#2026121011yescharacteristic
267C2#2026131712yes—
268C2#2026133512yes—
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
Fitting, prime radicalC2×#121×C97C2
socle, cosocleC23×C97C24
hypercenterC2×#121D97
hypocenterC97#2333
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. #20260638C2semidirect
2. #20260628C2semidirect
3. #20260621C2semidirect
4. #20260615C2semidirect
5. #20260615C2semidirect
6. #20260598C2semidirect
7. C22⋊C4×Dic97C2semidirect
8. C4×C22⋊Dic97C2semidirect
29. #121×C97C4semidirect
56. C2×D4×C97C2×C4semidirect
59. C22×C4×C97C2×C4semidirect
63. C4⋊C4×C97C2×C4semidirect
64. C22⋊C4×C97C2×C4semidirect
135. C97#2333semidirect
149. #121Dic97semidirect
174. C2×D4C2×Dic97semidirect
184. C22×C4C2×Dic97semidirect
189. C4⋊C4C2×Dic97semidirect
196. C22⋊C4C2×Dic97semidirect

Maximal subgroups

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

subgroupindex
#202606382
#202606282
#202606212
#202606152
#202605982
C22⋊C4×Dic972
C4×C22⋊Dic972
C2×#121×C972
#233397