Finite Group Databasebeta

← #27141937

Subgroups — #27141937

Order 12416.

316 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 113 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×#15690769C211yesmaximal, characteristic
2C2×#15690744C211yesmaximal, characteristic, complemented
3C2×#15690742C211yesmaximal, characteristic, complemented
4#20260595C211yesmaximal, characteristic
5#20260498C211yesmaximal, characteristic, complemented
6#20260498C211yesmaximal, characteristic, complemented
7C2×C22⋊C8×C97C211yesmaximal, characteristic
8#15690769—22no—
9#15690769—22no—
10C2×C4⋊Dic97C2211yescharacteristic
11C2×C4⋊Dic97C2211yescharacteristic
12C2×C4⋊Dic97C2211yescharacteristic
13C2×C4⋊Dic97C2211yescharacteristic
14C2×C22⋊Dic97—21no—
15C2×C22⋊Dic97—21no—
16C2×C4×Dic97—21no—
17#15690744—22no—
18#15690742—22no—
19#15690717—21no—
20C23×C4×C97C2211yescharacteristic
21C22×C8×C97C2211yescharacteristic
22C22×C8×C97C2211yescharacteristic
23C22⋊C8×C97C414yescomplemented
24C22×Dic97—21no—
25C22×Dic97—21no—
26C22×Dic97—21no—
27C22×Dic97—21no—
28C4⋊Dic97—21no—
29C4⋊Dic97—21no—
30C4⋊Dic97—21no—
31C4⋊Dic97—21no—
32C4⋊Dic97—21no—
33C4⋊Dic97—21no—
34C4⋊Dic97D412yes—
35C4⋊Dic97D412yes—
36C22⋊Dic97—24no—
37C22⋊Dic97—42no—
38C4×Dic97—24no—
39C24×C97D411yescharacteristic
40C22×C4×C97—21no—
41C22×C4×C97—21no—
42C22×C4×C97—21no—
43C22×C4×C97—21no—
44C22×C4×C97C2×C412yes—
45C22×C4×C97D411yescharacteristic
46C22×C4×C97Q812yes—
47C22×C4×C97C2311yescharacteristic
48C2×C8×C97—21no—
49C2×C8×C97—21no—
50C2×C8×C97—21no—
51C2×C8×C97—21no—
52C2×C8×C97C2×C412yes—
53C2×C8×C97C2×C412yes—
54C23×C97—21no—
55C23×C97—21no—
56C23×C97—21no—
57C23×C97—21no—
58C23×C97—21no—
59C23×C97—21no—
60C23×C97C4⋊C412yes—
61C23×C97C2×D411yescharacteristic
62C2×C4×C97—21no—
63C2×C4×C97—21no—
64C2×C4×C97—22no—
65C2×C4×C97—22no—
66C2×C4×C97—22no—
67C2×C4×C97—22no—
68C2×C4×C97C4⋊C412yes—
69C2×C4×C97C22×C411yescharacteristic
70C2×C4×C97C2×D411yescharacteristic
71C2×C4×C97C2×D411yescharacteristic
72C2×C4×C97C2×Q811yescharacteristic
73C2×C4×C97#4111yescharacteristic
74C2×C4×C97#4111yescharacteristic
75C2×Dic97—22no—
76C2×Dic97—22no—
77C2×Dic97—41no—
78C2×Dic97—41no—
79C2×Dic97—41no—
80C2×Dic97—41no—
81C2×Dic97—41no—
82C2×Dic97—41no—
83C2×Dic97—41no—
84C2×Dic97—41no—
85C2×Dic97—41no—
86C2×Dic97—41no—
87C8×C97—22no—
88C8×C97—22no—
89C22×C97—22no—
90C22×C97—22no—
91C22×C97—22no—
92C22×C97—41no—
93C22×C97—41no—
94C22×C97—41no—
95C22×C97—41no—
96C22×C97C2×C4⋊C411yescharacteristic
97C22×C97#12011yescharacteristic
98C22×C97#12311yescharacteristic
99C22×C97#13511yescharacteristic
100C22×C97#13511yescharacteristic
101C22×C97AGL(1,ℤ/8)11yescharacteristic
102C22×C97#13711yescharacteristic
103C4×C97—22no—
104C4×C97C4×D411yescharacteristic
105C4×C97C4×D411yescharacteristic
106C4×C97C22⋊Q811yescharacteristic
107C4×C97C22⋊Q811yescharacteristic
108Dic97—42no—
109Dic97—42no—
110Dic97—81no—
111Dic97—81no—
112C2×C97—42no—
113C2×C97#38511yescharacteristic
114C2×C97#42711yescharacteristic
115C2×C97#42811yescharacteristic
116C2×C97#44911yescharacteristic
117C2×C97#45211yescharacteristic
118C2×C97#48211yescharacteristic
119C2×C97#48511yescharacteristic
120#1854—971nomaximal, Sylow
121C97#185411yescharacteristic, complemented, Sylow
122C2×#117—971no—
123C2×#107—971no—
124C2×#106—971no—
125C2×C22⋊C8D9711yescharacteristic
126#380—971no—
127#340—971no—
128#340—971no—
129C23×C4C2×D9711yescharacteristic
130C22×C8C2×D9711yescharacteristic
131C22×C8C2×D9711yescharacteristic
132#117—1942no—
133#117—1942no—
134C2×C4⋊C4—971no—
135C2×C4⋊C4—971no—
136C2×C4⋊C4—971no—
137C2×C4⋊C4—971no—
138C2×C22⋊C4—1941no—
139C2×C22⋊C4—1941no—
140C2×C42—1941no—
141#107—1942no—
142#106—1942no—
143C22⋊C8Dic9714yescomplemented
144#95—1941no—
145C24D38811yescharacteristic
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—1941no—
151C22×C4—1941no—
152C22×C4—1941no—
153C22×C4—1941no—
154C22×C4Dic19412yes—
155C22×C4D38811yescharacteristic
156C22×C4C2×Dic9712yes—
157C22×C4C22×D9711yescharacteristic
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8C2×Dic9712yes—
163C2×C8C2×Dic9712yes—
164C4⋊C4—972no—
165C4⋊C4—972no—
166C4⋊C4—1941no—
167C4⋊C4—1941no—
168C4⋊C4—1941no—
169C4⋊C4—1941no—
170C4⋊C4—1941no—
171C4⋊C4—1941no—
172C22⋊C4—3882no—
173C22⋊C4—1944no—
174C42—1944no—
175C23—21no—
176C23—21no—
177C23—21no—
178C23—21no—
179C23—21no—
180C23—21no—
181C23C4⋊Dic9712yes—
182C23C2×D38811yescharacteristic
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—3881no—
190C2×C4—3881no—
191C2×C4—3881no—
192C2×C4—3881no—
193C2×C4—3881no—
194C2×C4—3881no—
195C2×C4—3881no—
196C2×C4—3881no—
197C2×C4—3881no—
198C2×C4—3881no—
199C2×C4—1942no—
200C2×C4—1942no—
201C2×C4C4⋊Dic9712yes—
202C2×C4C22×Dic9711yescharacteristic
203C2×C4D4×D9711yescharacteristic
204C2×C4D4×D9711yescharacteristic
205C2×C4C2×Dic19411yescharacteristic
206C2×C4#1309551011yescharacteristic
207C2×C4#1309551011yescharacteristic
208C8—22no—
209C8—22no—
210C22—22no—
211C22—22no—
212C22—22no—
213C22—41no—
214C22—41no—
215C22—41no—
216C22—41no—
217C22C2×C4⋊Dic9711yescharacteristic
218C22#1569078011yescharacteristic
219C22#1569079411yescharacteristic
220C22#1569083211yescharacteristic
221C22#1569083211yescharacteristic
222C22#1569083811yescharacteristic
223C22#1569084311yescharacteristic
224C4—22no—
225C4—7761no—
226C4—7761no—
227C4—3882no—
228C4—3882no—
229C4D4×Dic9711yescharacteristic
230C4D4×Dic9711yescharacteristic
231C4#1569078711yescharacteristic
232C4#1569078711yescharacteristic
233C2—42no—
234C2#2026061011yescharacteristic
235C2#2026077311yescharacteristic
236C2#2026077711yescharacteristic
237C2#2026088211yescharacteristic
238C2#2026089911yescharacteristic
239C2#2026106011yescharacteristic
240C2#2026107711yescharacteristic
subgroupG/H
centerC23C2×D388
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
FittingC2×C22⋊C8×C97C2
socleC23×C97C2×D4
prime radicalC24×C97D4
cosocleC22×C4×C97C23
hypercenterC2×C22⋊C8D97
hypocenterC97#1854
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×#15690744C2semidirect
3. C2×#15690742C2semidirect
5. #20260498C2semidirect
6. #20260498C2semidirect
23. C22⋊C8×C97C4semidirect
121. C97#1854semidirect
143. C22⋊C8Dic97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907692
C2×#156907442
C2×#156907422
#202605952
#202604982
C2×C22⋊C8×C972
#185497