Finite Group Databasebeta

← #27144973

Subgroups — #27144973

Order 12416.

696 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 151 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×#15690772C211yesmaximal, characteristic, complemented
2C2×#15690770C212yesmaximal, complemented
3C2×#15690766C211yesmaximal, characteristic, complemented
4#20260666C211yesmaximal, characteristic, complemented
5#20260659C212yesmaximal, complemented
6#20260653C211yesmaximal, characteristic, complemented
7#20260650C212yesmaximal, complemented
8#20260644C211yesmaximal, characteristic
9#20260586C211yesmaximal, characteristic, complemented
10#20260582C212yesmaximal, complemented
11#397×C97C211yesmaximal, characteristic, complemented
12C22×D388C2211yescharacteristic
13C22×C22⋊D97C2212yes—
14C23×Dic97C2211yescharacteristic
15C22×C4×D97C2212yes—
16#15690772—24no—
17#15690770—28no—
18#15690766—24no—
19C2×#13095490C2212yes—
20C2×#13095490C2211yescharacteristic
21C2×#13095490C2211yescharacteristic
22C2×#13095490C2212yes—
23C2×C22⋊Dic97C2211yescharacteristic
24C2×#13095488C2212yes—
25C2×#13095488C2212yes—
26C2×#13095488C2212yes—
27C2×#13095488C2212yes—
28C2×C4×Dic97C2211yescharacteristic
29C2×C4×Dic97C2212yes—
30C2×C4×Dic97C2211yescharacteristic
31#15690717C2211yescharacteristic
32#15690717C2212yes—
33C2×C4⋊C4×C97C2211yescharacteristic
34C2×C22⋊C4×C97C2211yescharacteristic
35C2×C22⋊C4×C97C2212yes—
36#95×C97C2211yescharacteristic
37#95×C97C2212yes—
38C23×D97C2312yes—
39C2×D388—24no—
40C2×D388—22no—
41C2×C22⋊D97—24no—
42C2×C22⋊D97—24no—
43C2×C22⋊D97—24no—
44C22×Dic97—22no—
45C22×Dic97—22no—
46C22×Dic97D414yescomplemented
47C22×Dic97C2311yescharacteristic
48C22×Dic97C2311yescharacteristic
49C22×Dic97C2311yescharacteristic
50C22×Dic97C2312yes—
51C22×Dic97C2311yescharacteristic
52C2×C4×D97—24no—
53C2×C4×D97—24no—
54C2×C4×D97—24no—
55#13095490—22no—
56#13095490—22no—
57#13095490—24no—
58#13095490—24no—
59C22⋊Dic97—22no—
60#13095488—24no—
61#13095488—24no—
62#13095488—24no—
63#13095488—24no—
64C4×Dic97—22no—
65C4×Dic97—24no—
66C4×Dic97D414yescomplemented
67C24×C97C2311yescharacteristic
68C22×C4×C97C2312yes—
69C22×C4×C97C2311yescharacteristic
70C22×C4×C97C2312yes—
71C22×C4×C97C2311yescharacteristic
72C4⋊C4×C97—22no—
73C22⋊C4×C97—24no—
74C22⋊C4×C97—22no—
75C23×C97—21no—
76C23×C97—21no—
77C23×C97—22no—
78C23×C97—22no—
79C23×C97C2×D412yes—
80C23×C97C2411yescharacteristic
81C22×D97—22no—
82C22×D97—22no—
83C22×D97—22no—
84C22×D97—22no—
85C22×D97—22no—
86C22×D97—22no—
87C22×D97#4114yes—
88C2×C4×C97—22no—
89C2×C4×C97—22no—
90C2×C4×C97—21no—
91C2×C4×C97—22no—
92C2×C4×C97—22no—
93C2×C4×C97—22no—
94C2×C4×C97—22no—
95C2×C4×C97—22no—
96C2×C4×C97—22no—
97C2×C4×C97C2×D412yes—
98C22⋊D97—216no—
99C2×Dic97—22no—
100C2×Dic97—21no—
101C2×Dic97—22no—
102C2×Dic97—21no—
103C2×Dic97—22no—
104C2×Dic97—28no—
105C2×Dic97—21no—
106C2×Dic97—22no—
107C2×Dic97C2×D412yes—
108C2×Dic97C2×D412yes—
109C2×Dic97C2×D412yes—
110C2×Dic97C2×D412yes—
111C2×Dic97#4114yes—
112C2×Dic97#4112yes—
113D388—28no—
114C4×D97—48no—
115C22×C97—22no—
116C22×C97—24no—
117C22×C97—41no—
118C22×C97—42no—
119C22×C97—41no—
120C22×C97C22×D411yescharacteristic
121C22×C97C22×D411yescharacteristic
122C22×C97C2×#4111yescharacteristic
123C22×C97C2×#4112yes—
124C22×C97C2×#4112yes—
125C2×D97—24no—
126C2×D97—24no—
127C2×D97—24no—
128C2×D97—42no—
129C2×D97—42no—
130C2×D97—42no—
131C2×D97—42no—
132C4×C97—22no—
133C4×C97—42no—
134C4×C97—42no—
135C4×C97—41no—
136Dic97—22no—
137Dic97—24no—
138Dic97—22no—
139Dic97—41no—
140Dic97#12114yes—
141C2×C97—42no—
142C2×C97C2×#12111yescharacteristic
143C2×C97#52512yes—
144C2×C97#52511yescharacteristic
145C2×C97#52912yes—
146C2×C97#53211yescharacteristic
147D97—44no—
148#2392—971nomaximal, Sylow
149C97#239211yescharacteristic, complemented, Sylow
150C2×C4×D4—971no—
151C2×C4×D4—972no—
152C2×#117—971no—
153#398—971no—
154#397D9711yescharacteristic, complemented
155#396—972no—
156#394—972no—
157#394—971no—
158#392—971no—
159C4×C4⋊C4—971no—
160C4×C22⋊C4—972no—
161C22×D4—972no—
162C22×D4—971no—
163C23×C4—971no—
164C23×C4—972no—
165C4×D4—1944no—
166C4×D4—1948no—
167#117—1944no—
168C2×C4⋊C4—971no—
169C2×C4⋊C4—971no—
170C2×C4⋊C4—972no—
171C2×C4⋊C4—972no—
172C2×C4⋊C4C2×D9711yescharacteristic
173C2×C22⋊C4—971no—
174C2×C22⋊C4—972no—
175C2×C22⋊C4—972no—
176C2×C22⋊C4—972no—
177C2×C22⋊C4—972no—
178C2×C22⋊C4C2×D9711yescharacteristic
179C2×C22⋊C4C2×D9712yes—
180C2×C42—971no—
181C2×C42—971no—
182C2×C42—972no—
183#95—972no—
184#95—971no—
185#95C2×D9712yes—
186#95C2×D9711yescharacteristic
187C24—972no—
188C24C22×D9711yescharacteristic
189C2×D4—1944no—
190C2×D4—1944no—
191C2×D4—1944no—
192C2×D4—1944no—
193C2×D4—1942no—
194C22×C4—971no—
195C22×C4—974no—
196C22×C4—972no—
197C22×C4—971no—
198C22×C4—971no—
199C22×C4—971no—
200C22×C4—1944no—
201C22×C4—1944no—
202C22×C4—1944no—
203C22×C4—1942no—
204C22×C4—1942no—
205C22×C4C22×D9712yes—
206C22×C4C22×D9712yes—
207C22×C4C22×D9711yescharacteristic
208C22×C4C22×D9711yescharacteristic
209C4⋊C4—22no—
210C4⋊C4—1944no—
211C4⋊C4—1942no—
212C4⋊C4—1944no—
213C4⋊C4—1942no—
214C22⋊C4—24no—
215C22⋊C4—22no—
216C22⋊C4—1944no—
217C22⋊C4—1944no—
218C22⋊C4—1944no—
219C22⋊C4—1944no—
220C22⋊C4—1942no—
221C42—974no—
222C42—1942no—
223C42—1944no—
224C23—21no—
225C23—21no—
226C23—22no—
227C23—22no—
228C23—974no—
229C23—1942no—
230C23—1942no—
231C23—1942no—
232C23—1942no—
233C23—1942no—
234C23—1942no—
235C23D4×D9712yes—
236C23C23×D9711yescharacteristic
237D4—1948no—
238D4—19416no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—972no—
249C2×C4—972no—
250C2×C4—972no—
251C2×C4—972no—
252C2×C4—972no—
253C2×C4—974no—
254C2×C4—3888no—
255C2×C4—1948no—
256C2×C4—1941no—
257C2×C4—1942no—
258C2×C4—1942no—
259C2×C4—1941no—
260C2×C4—1942no—
261C2×C4—1941no—
262C2×C4—1942no—
263C2×C4D4×D9712yes—
264C22—22no—
265C22—24no—
266C22—41no—
267C22—42no—
268C22—41no—
269C22—3882no—
270C22—3882no—
271C22—3882no—
272C22—3882no—
273C22—1944no—
274C22—1944no—
275C22—1944no—
276C22C2×D4×D9711yescharacteristic
277C22C2×D4×D9711yescharacteristic
278C22#41×D9711yescharacteristic
279C22#41×D9712yes—
280C22#41×D9712yes—
281C4—22no—
282C4—42no—
283C4—42no—
284C4—41no—
285C4—974no—
286C4—3881no—
287C4—1942no—
288C4—1944no—
289C4—1942no—
290C2—42no—
291C2—3884no—
292C2#121×D9711yescharacteristic
293C2#2026124311yescharacteristic
294C2#2026124312yes—
295C2#2026126812yes—
296C2#2026128411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, cosocleC23×C97C24
Fitting#397×C97C2
prime radical#20260644C2
hypercenter#397D97
hypocenterC97#2392
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#15690772C2semidirect
2. C2×#15690770C2semidirect
3. C2×#15690766C2semidirect
4. #20260666C2semidirect
5. #20260659C2semidirect
6. #20260653C2semidirect
7. #20260650C2semidirect
9. #20260586C2semidirect
10. #20260582C2semidirect
11. #397×C97C2semidirect
46. C22×Dic97D4semidirect
66. C4×Dic97D4semidirect
149. C97#2392semidirect
154. #397D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907722
C2×#156907702
C2×#156907662
#202606662
#202606592
#202606532
#202606502
#202606442
#202605862
#202605822
#397×C972
#239297