Finite Group Databasebeta

← #27145026

Subgroups — #27145026

Order 12416.

504 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 167 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×Q8×Dic97C211yesmaximal, characteristic
2C2×#15690775C211yesmaximal, characteristic
3#20260668C212yesmaximal
4#20260667C211yesmaximal, characteristic
5#20260665C211yesmaximal, characteristic
6#20260656C212yesmaximal
7#20260654C211yesmaximal, characteristic
8C4⋊C4×Dic97C212yesmaximal
9#20260586C212yesmaximal
10#20260580C211yesmaximal, characteristic
11#395×C97C211yesmaximal, characteristic
12C22×Dic194C2211yescharacteristic
13Q8×Dic97—24no—
14#15690775—24no—
15C2×C4⋊Dic97C2212yes—
16C2×C4⋊Dic97C2212yes—
17C2×C4⋊Dic97C2211yescharacteristic
18C2×#13095490C2211yescharacteristic
19C2×#13095490C2212yes—
20C2×#13095490C2212yes—
21C2×#13095490C2211yescharacteristic
22C2×C4×Dic97C2211yescharacteristic
23C2×C4×Dic97C2211yescharacteristic
24C2×C4×Dic97C2212yes—
25C2×C4×Dic97C2211yescharacteristic
26C2×C4×Dic97C2212yes—
27#15690717C2212yes—
28#15690717C2211yescharacteristic
29#15690717C2212yes—
30#15690717C2212yes—
31#15690717C2212yes—
32C22×Q8×C97C2211yescharacteristic
33C2×C4⋊C4×C97C2211yescharacteristic
34C2×C4⋊C4×C97C2212yes—
35#95×C97C2212yes—
36#95×C97C2211yescharacteristic
37C2×Dic194—22no—
38C2×Dic194—22no—
39C2×Dic194—22no—
40C22×Dic97C2311yescharacteristic
41C22×Dic97C2312yes—
42C22×Dic97C2311yescharacteristic
43C22×Dic97C2312yes—
44C22×Dic97C2311yescharacteristic
45C22×Dic97C2311yescharacteristic
46C4⋊Dic97—22no—
47C4⋊Dic97—24no—
48C4⋊Dic97—24no—
49#13095490—22no—
50#13095490—24no—
51#13095490—24no—
52#13095490—22no—
53C4×Dic97—22no—
54C4×Dic97—22no—
55C4×Dic97—24no—
56C4×Dic97D414yes—
57C4×Dic97Q818yescomplemented
58C2×Q8×C97—22no—
59C2×Q8×C97—24no—
60C22×C4×C97C2311yescharacteristic
61C22×C4×C97C2311yescharacteristic
62C22×C4×C97C2312yes—
63C22×C4×C97C2311yescharacteristic
64C22×C4×C97C2312yes—
65C4⋊C4×C97—22no—
66C4⋊C4×C97—24no—
67C23×C97C2411yescharacteristic
68Q8×C97—28no—
69C2×C4×C97—21no—
70C2×C4×C97—22no—
71C2×C4×C97—22no—
72C2×C4×C97—22no—
73C2×C4×C97—22no—
74C2×C4×C97—22no—
75C2×C4×C97—21no—
76C2×C4×C97—21no—
77C2×C4×C97—22no—
78C2×C4×C97C2×D412yes—
79C2×C4×C97C2×Q814yes—
80C2×C4×C97#4114yes—
81C2×C4×C97#4112yes—
82C2×Dic97—21no—
83C2×Dic97—22no—
84C2×Dic97—22no—
85C2×Dic97—21no—
86C2×Dic97—21no—
87C2×Dic97—21no—
88C2×Dic97—21no—
89C2×Dic97—22no—
90C2×Dic97—22no—
91C2×Dic97—22no—
92C2×Dic97—22no—
93C2×Dic97C2×D412yes—
94C2×Dic97C2×D412yes—
95C2×Dic97C2×Q814yes—
96C2×Dic97C2×Q814yes—
97C2×Dic97#4112yes—
98Dic194—28no—
99C22×C97C22×D411yescharacteristic
100C22×C97C22×Q812yes—
101C22×C97C2×#4111yescharacteristic
102C22×C97C2×#4112yes—
103C22×C97C2×#4111yescharacteristic
104C4×C97—22no—
105C4×C97—22no—
106C4×C97—24no—
107C4×C97—24no—
108C4×C97—41no—
109Dic97—22no—
110Dic97—22no—
111Dic97—24no—
112Dic97—41no—
113Dic97—42no—
114Dic97C4⋊Q814yes—
115C2×C97#52511yescharacteristic
116C2×C97#52511yescharacteristic
117C2×C97C2×C4⋊Q811yescharacteristic
118C2×C97#53312yes—
119C2×C97#53312yes—
120#2399—971nomaximal, Sylow
121C97#239911yescharacteristic, complemented, Sylow
122C2×C4×Q8—971no—
123C2×C4×Q8—971no—
124#398—971no—
125#398—972no—
126#398—971no—
127#395—971no—
128#395—972no—
129#395D9711yescharacteristic
130C4×C4⋊C4—972no—
131C4×C4⋊C4—972no—
132#376—971no—
133C22×Q8—971no—
134C22×Q8C2×D9711yescharacteristic
135C4×Q8—1944no—
136C4×Q8—1944no—
137C2×C4⋊C4—971no—
138C2×C4⋊C4—971no—
139C2×C4⋊C4—972no—
140C2×C4⋊C4—971no—
141C2×C4⋊C4—972no—
142C2×C4⋊C4—972no—
143C2×C4⋊C4—972no—
144C2×C4⋊C4C2×D9711yescharacteristic
145C2×C4⋊C4C2×D9712yes—
146C2×C42—971no—
147C2×C42—972no—
148C2×C42—971no—
149C2×C42—972no—
150C2×C42—971no—
151#95—971no—
152#95—972no—
153#95—972no—
154#95—972no—
155#95—972no—
156#95C2×D9711yescharacteristic
157#95C2×D9712yes—
158C2×Q8—22no—
159C2×Q8—24no—
160C2×Q8—1942no—
161C2×Q8—1942no—
162C2×Q8—1942no—
163C22×C4—971no—
164C22×C4—972no—
165C22×C4—971no—
166C22×C4—971no—
167C22×C4—972no—
168C22×C4—971no—
169C22×C4C22×D9711yescharacteristic
170C22×C4C22×D9711yescharacteristic
171C22×C4C22×D9712yes—
172C22×C4C22×D9712yes—
173C22×C4C22×D9711yescharacteristic
174C4⋊C4—22no—
175C4⋊C4—24no—
176C4⋊C4—1942no—
177C4⋊C4—1944no—
178C4⋊C4—1944no—
179C4⋊C4—1942no—
180C4⋊C4—1942no—
181C4⋊C4—1944no—
182C4⋊C4—1944no—
183C42—974no—
184C42—978no—
185C42—1942no—
186C42—1944no—
187C42—1942no—
188C23C23×D9711yescharacteristic
189Q8—28no—
190Q8—1948no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—972no—
201C2×C4—974no—
202C2×C4—972no—
203C2×C4—974no—
204C2×C4—972no—
205C2×C4—1941no—
206C2×C4—1942no—
207C2×C4—1942no—
208C2×C4—1942no—
209C2×C4—1942no—
210C2×C4—1941no—
211C2×C4—1941no—
212C2×C4—1942no—
213C2×C4—1942no—
214C2×C4—1941no—
215C2×C4—1941no—
216C2×C4D4×D9712yes—
217C2×C4Q8×D9714yes—
218C2×C4#1309551014yes—
219C2×C4#1309551212yes—
220C22C2×D4×D9711yescharacteristic
221C22C2×Q8×D9712yes—
222C22#41×D9711yescharacteristic
223C22C2×#1309551012yes—
224C22C2×#1309551211yescharacteristic
225C4—22no—
226C4—24no—
227C4—22no—
228C4—24no—
229C4—41no—
230C4—974no—
231C4—3881no—
232C4—3882no—
233C4—1942no—
234C4—1942no—
235C4—1944no—
236C2#2026124111yescharacteristic
237C2#2026124811yescharacteristic
238C2C4⋊Q8×D9711yescharacteristic
239C2#2026128712yes—
240C2#2026129012yes—
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, prime radical, cosocleC23×C97C24
Fitting#395×C97C2
hypercenter#395D97
hypocenterC97#2399
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
57. C4×Dic97Q8semidirect
121. C97#2399semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×Dic972
C2×#156907752
#202606682
#202606672
#202606652
#202606562
#202606542
C4⋊C4×Dic972
#202605862
#202605802
#395×C972
#239997