Finite Group Databasebeta

← #27150057

Subgroups — #27150057

Order 12416.

582 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 153 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
1C2×#15690791C211yesmaximal, characteristic
2C22×C4×Dic97C211yesmaximal, characteristic, complemented
3#20260629C212yesmaximal, complemented
4#20260628C212yesmaximal, complemented
5#20260618C212yesmaximal, complemented
6#20260617C212yesmaximal, complemented
7#20260615C211yesmaximal, characteristic
8#20260610C211yesmaximal, characteristic, complemented
9#20260608C211yesmaximal, characteristic, complemented
10#20260596C211yesmaximal, characteristic, complemented
11#386×C97C211yesmaximal, characteristic
12C23×Dic97C2211yescharacteristic
13C23×Dic97C2211yescharacteristic
14#15690791—24no—
15C2×C4⋊Dic97C2212yes—
16C2×#13095490C2212yes—
17C2×#13095490C2211yescharacteristic
18C2×#13095490C2211yescharacteristic
19C2×C22⋊Dic97C2211yescharacteristic
20C2×C22⋊Dic97C2211yescharacteristic
21C2×C22⋊Dic97C2212yes—
22C2×C22⋊Dic97C2211yescharacteristic
23C2×C22⋊Dic97C2211yescharacteristic
24C2×C4×Dic97—24no—
25C2×C4×Dic97C2212yes—
26C2×C4×Dic97C2212yes—
27#15690717C2212yes—
28#15690717C2212yes—
29#15690717C2212yes—
30#15690717C2212yes—
31#15690717C2212yes—
32C22×D4×C97C2211yescharacteristic
33C23×C4×C97C2211yescharacteristic
34C2×C22⋊C4×C97C2211yescharacteristic
35C2×C22⋊C4×C97C2212yes—
36#95×C97C2212yes—
37C22×Dic97—22no—
38C22×Dic97—22no—
39C22×Dic97—22no—
40C22×Dic97—24no—
41C22×Dic97D414yes—
42C22×Dic97C2311yescharacteristic
43C22×Dic97C2311yescharacteristic
44C22×Dic97C2312yes—
45C22×Dic97C2311yescharacteristic
46C22×Dic97C2311yescharacteristic
47C22×Dic97C2312yes—
48C4⋊Dic97—24no—
49#13095490—24no—
50#13095490—22no—
51#13095490—22no—
52C22⋊Dic97—22no—
53C22⋊Dic97—22no—
54C22⋊Dic97—22no—
55C22⋊Dic97—22no—
56C22⋊Dic97—24no—
57C4×Dic97—24no—
58C4×Dic97—24no—
59C24×C97C2311yescharacteristic
60C24×C97C2311yescharacteristic
61C2×D4×C97—22no—
62C2×D4×C97—22no—
63C2×D4×C97—22no—
64C22×C4×C97—22no—
65C22×C4×C97—24no—
66C22×C4×C97C2311yescharacteristic
67C22×C4×C97C2312yes—
68C22×C4×C97C2312yes—
69C22⋊C4×C97—22no—
70C22⋊C4×C97—24no—
71C23×C97—22no—
72C23×C97—22no—
73C23×C97—21no—
74C23×C97—22no—
75C23×C97—21no—
76C23×C97—21no—
77C23×C97—22no—
78C23×C97C2×D412yes—
79C23×C97#4112yes—
80C23×C97#4112yes—
81C23×C97C2411yescharacteristic
82D4×C97—44no—
83C2×C4×C97—28no—
84C2×C4×C97—22no—
85C2×C4×C97—22no—
86C2×C4×C97—21no—
87C2×C4×C97—22no—
88C2×C4×C97—22no—
89C2×C4×C97—21no—
90C2×C4×C97—21no—
91C2×C4×C97—22no—
92C2×C4×C97#4114yes—
93C2×Dic97—28no—
94C2×Dic97—28no—
95C2×Dic97—22no—
96C2×Dic97—22no—
97C2×Dic97—21no—
98C2×Dic97—21no—
99C2×Dic97—21no—
100C2×Dic97—21no—
101C2×Dic97—21no—
102C2×Dic97—21no—
103C2×Dic97—22no—
104C2×Dic97—22no—
105C2×Dic97—22no—
106C2×Dic97—22no—
107C2×Dic97—22no—
108C2×Dic97C2×D412yes—
109C2×Dic97C2×D412yes—
110C2×Dic97#4114yes—
111C22×C97—22no—
112C22×C97—22no—
113C22×C97—24no—
114C22×C97—24no—
115C22×C97—41no—
116C22×C97—41no—
117C22×C97—41no—
118C22×C97—42no—
119C22×C97—42no—
120C22×C97#12314yes—
121C22×C97C22×D411yescharacteristic
122C22×C97C2×#4112yes—
123C22×C97C2×#4111yescharacteristic
124C22×C97C2×#4112yes—
125C22×C97C2×#4111yescharacteristic
126C4×C97—24no—
127C4×C97—41no—
128C4×C97—42no—
129Dic97—22no—
130Dic97—22no—
131Dic97—24no—
132Dic97—41no—
133Dic97—41no—
134Dic97—42no—
135C2×C97—24no—
136C2×C97—81no—
137C2×C97C2×#12311yescharacteristic
138C2×C97#52511yescharacteristic
139C2×C97#52511yescharacteristic
140C2×C97#52912yes—
141C2×C97#52912yes—
142#2853—971nomaximal, Sylow
143C97#285311yescharacteristic, complemented, Sylow
144C2×#123—971no—
145C22×C42—971no—
146#388—972no—
147#388—972no—
148#387—972no—
149#387—972no—
150#386—971no—
151#386D9711yescharacteristic
152#385—971no—
153#385—971no—
154#381—971no—
155C22×D4C2×D9711yescharacteristic
156C23×C4—971no—
157C23×C4—971no—
158C23×C4C2×D9711yescharacteristic
159#123—1944no—
160C2×C4⋊C4—971no—
161C2×C4⋊C4—971no—
162C2×C4⋊C4—972no—
163C2×C4⋊C4—972no—
164C2×C22⋊C4—971no—
165C2×C22⋊C4—971no—
166C2×C22⋊C4—971no—
167C2×C22⋊C4—971no—
168C2×C22⋊C4—972no—
169C2×C22⋊C4C2×D9711yescharacteristic
170C2×C22⋊C4C2×D9712yes—
171C2×C42—972no—
172C2×C42—972no—
173C2×C42—1944no—
174#95—972no—
175#95—972no—
176#95—972no—
177#95—972no—
178#95—972no—
179#95C2×D9712yes—
180C24C22×D9711yescharacteristic
181C24C22×D9711yescharacteristic
182C2×D4—22no—
183C2×D4—22no—
184C2×D4—22no—
185C22×C4—22no—
186C22×C4—24no—
187C22×C4—974no—
188C22×C4—971no—
189C22×C4—971no—
190C22×C4—972no—
191C22×C4—972no—
192C22×C4—971no—
193C22×C4—971no—
194C22×C4—1944no—
195C22×C4—1942no—
196C22×C4—1942no—
197C22×C4—1942no—
198C22×C4C22×D9712yes—
199C22×C4C22×D9711yescharacteristic
200C22×C4C22×D9712yes—
201C4⋊C4—1942no—
202C4⋊C4—1942no—
203C4⋊C4—1944no—
204C4⋊C4—1944no—
205C22⋊C4—22no—
206C22⋊C4—24no—
207C22⋊C4—1944no—
208C22⋊C4—1942no—
209C22⋊C4—1942no—
210C22⋊C4—1942no—
211C22⋊C4—1942no—
212C42—1944no—
213C42—1944no—
214C23—22no—
215C23—22no—
216C23—21no—
217C23—22no—
218C23—22no—
219C23—21no—
220C23—21no—
221C23D4×D9712yes—
222C23#1309551012yes—
223C23#1309551112yes—
224C23C23×D9711yescharacteristic
225D4—44no—
226C2×C4—28no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—972no—
236C2×C4—972no—
237C2×C4—974no—
238C2×C4—1941no—
239C2×C4—1941no—
240C2×C4—1942no—
241C2×C4—1942no—
242C2×C4—1942no—
243C2×C4—1942no—
244C2×C4—1942no—
245C2×C4—1948no—
246C2×C4—1948no—
247C2×C4—1942no—
248C2×C4—1942no—
249C2×C4—1941no—
250C2×C4—1941no—
251C2×C4—1941no—
252C2×C4—1941no—
253C2×C4#1309551014yes—
254C22—22no—
255C22—22no—
256C22—24no—
257C22—24no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—42no—
262C22—42no—
263C22#1569079214yes—
264C22C2×D4×D9711yescharacteristic
265C22#41×D9712yes—
266C22C2×#1309551011yescharacteristic
267C22C2×#1309551012yes—
268C22C2×#1309551111yescharacteristic
269C4—24no—
270C4—41no—
271C4—42no—
272C4—3881no—
273C4—3881no—
274C4—3882no—
275C4—1942no—
276C4—1942no—
277C4—1944no—
278C2—24no—
279C2—81no—
280C2C2×#1569079211yescharacteristic
281C2#2026124111yescharacteristic
282C2#2026124311yescharacteristic
283C2#2026126512yes—
284C2#2026126612yes—
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, cosocleC23×C97C24
Fitting#386×C97C2
prime radicalC22×D4×C97C22
hypercenter#386D97
hypocenterC97#2853
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C22×C4×Dic97C2semidirect
3. #20260629C2semidirect
4. #20260628C2semidirect
5. #20260618C2semidirect
6. #20260617C2semidirect
8. #20260610C2semidirect
9. #20260608C2semidirect
10. #20260596C2semidirect
143. C97#2853semidirect

Maximal subgroups

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

subgroupindex
C2×#156907912
C22×C4×Dic972
#202606292
#202606282
#202606182
#202606172
#202606152
#202606102
#202606082
#202605962
#386×C972
#285397