Finite Group Databasebeta

← #27150085

Subgroups — #27150085

Order 12416.

644 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 189 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×#15690808C211yesmaximal, characteristic, complemented
2C2×#15690790C211yesmaximal, characteristic
3C2×#15690786C211yesmaximal, characteristic
4C2×C42×D97C211yesmaximal, characteristic
5#20260641C211yesmaximal, characteristic, complemented
6#20260632C214yesmaximal, complemented
7#20260604C211yesmaximal, characteristic, complemented
8#20260594C214yesmaximal
9#391×C97C211yesmaximal, characteristic, complemented
10C22×Dic194C2211yescharacteristic
11C22×C4×D97C2211yescharacteristic
12C22×C4×D97C2212yes—
13#15690808—24no—
14#15690790—24no—
15#15690786—24no—
16C2×C4⋊Dic97C2212yes—
17C2×C4⋊Dic97C2211yescharacteristic
18C2×#13095490C2212yes—
19C2×#13095490C2214yes—
20C2×#13095490C2214yes—
21C2×#13095488C2212yes—
22C2×#13095488C2212yes—
23C2×C4×Dic97C2212yes—
24C2×C4×Dic97C2211yescharacteristic
25C42×D97—24no—
26#15690717C2214yes—
27C22×Q8×C97C2211yescharacteristic
28C2×C4⋊C4×C97C2211yescharacteristic
29C2×C42×C97C2211yescharacteristic
30#95×C97C2214yes—
31C23×D97C2311yescharacteristic
32C2×Dic194—22no—
33C2×Dic194—22no—
34C2×Dic194—22no—
35C22×Dic97C2312yes—
36C22×Dic97C2312yes—
37C22×Dic97C2311yescharacteristic
38C22×Dic97C2312yes—
39C2×C4×D97—22no—
40C2×C4×D97—22no—
41C2×C4×D97—24no—
42C2×C4×D97—24no—
43C2×C4×D97D414yes—
44C2×C4×D97Q818yescomplemented
45C4⋊Dic97—22no—
46C4⋊Dic97—24no—
47#13095490—24no—
48#13095490—28no—
49#13095490—28no—
50#13095488—24no—
51#13095488—24no—
52C4×Dic97—24no—
53C4×Dic97D414yes—
54C2×Q8×C97—22no—
55C2×Q8×C97—22no—
56C2×Q8×C97—22no—
57C22×C4×C97C2312yes—
58C22×C4×C97C2312yes—
59C22×C4×C97C2312yes—
60C22×C4×C97C2311yescharacteristic
61C4⋊C4×C97—22no—
62C42×C97—22no—
63C23×C97C2411yescharacteristic
64C22×D97—21no—
65C22×D97—22no—
66C22×D97—21no—
67C22×D97C2×D412yes—
68C22×D97C2×Q814yes—
69Q8×C97—44no—
70C2×C4×C97—22no—
71C2×C4×C97—22no—
72C2×C4×C97—22no—
73C2×C4×C97—22no—
74C2×C4×C97—22no—
75C2×C4×C97—22no—
76C2×C4×C97—24no—
77C2×C4×C97C2×D412yes—
78C2×C4×C97C2×D412yes—
79C2×C4×C97C2×Q814yes—
80C2×C4×C97#4112yes—
81C2×Dic97—21no—
82C2×Dic97—21no—
83C2×Dic97—22no—
84C2×Dic97—22no—
85C2×Dic97—22no—
86C2×Dic97—24no—
87C2×Dic97—22no—
88C2×Dic97C2×D412yes—
89C2×Dic97C2×D414yes—
90C2×Dic97C2×Q814yes—
91C2×Dic97#4114yes—
92C4×D97—28no—
93C4×D97—216no—
94Dic194—44no—
95C22×C97C22×D411yescharacteristic
96C22×C97C22×D411yescharacteristic
97C22×C97C22×Q812yes—
98C22×C97C2×#4112yes—
99C22×C97C2×#4111yescharacteristic
100C2×D97—22no—
101C2×D97—24no—
102C2×D97—22no—
103C2×D97—24no—
104C2×D97C4⋊Q814yescomplemented
105C4×C97—24no—
106C4×C97—42no—
107C4×C97—42no—
108C4×C97#12114yes—
109Dic97—22no—
110Dic97—42no—
111Dic97—42no—
112Dic97C22⋊Q818yescomplemented
113C2×C97C2×#12111yescharacteristic
114C2×C97C2×C22⋊Q812yes—
115C2×C97C2×C4⋊Q811yescharacteristic
116C2×C97#53211yescharacteristic
117C2×C97#53312yes—
118D97—24no—
119#2857—971nomaximal, Sylow
120C97#285711yescharacteristic, complemented, Sylow
121C2×C4⋊Q8—971no—
122C2×C22⋊Q8—971no—
123C2×C22⋊Q8—971no—
124C22×C42—971no—
125#391—971no—
126#391D9711yescharacteristic, complemented
127#389—974no—
128#384—971no—
129#380—974no—
130C22×Q8—971no—
131C22×Q8C2×D9711yescharacteristic
132C23×C4—971no—
133C23×C4—972no—
134C4⋊Q8—1944no—
135C22⋊Q8—1944no—
136C22⋊Q8—1944no—
137C2×C4⋊C4—972no—
138C2×C4⋊C4—971no—
139C2×C4⋊C4—972no—
140C2×C4⋊C4—974no—
141C2×C4⋊C4—974no—
142C2×C4⋊C4C2×D9711yescharacteristic
143C2×C22⋊C4—972no—
144C2×C22⋊C4—972no—
145C2×C42—971no—
146C2×C42—972no—
147C2×C42—1944no—
148C2×C42C2×D9711yescharacteristic
149#95—974no—
150#95C2×D9714yes—
151C24—971no—
152C2×Q8—22no—
153C2×Q8—22no—
154C2×Q8—22no—
155C2×Q8—1942no—
156C2×Q8—1942no—
157C2×Q8—1942no—
158C22×C4—972no—
159C22×C4—971no—
160C22×C4—974no—
161C22×C4—978no—
162C22×C4—972no—
163C22×C4—972no—
164C22×C4—1944no—
165C22×C4—1944no—
166C22×C4—1942no—
167C22×C4—1942no—
168C22×C4C22×D9712yes—
169C22×C4C22×D9712yes—
170C22×C4C22×D9712yes—
171C22×C4C22×D9711yescharacteristic
172C4⋊C4—22no—
173C4⋊C4—1948no—
174C4⋊C4—1944no—
175C4⋊C4—1948no—
176C4⋊C4—1944no—
177C4⋊C4—1942no—
178C22⋊C4—1944no—
179C22⋊C4—1944no—
180C42—22no—
181C42—974no—
182C42—1944no—
183C23—972no—
184C23—974no—
185C23—1941no—
186C23—1942no—
187C23—1941no—
188C23C23×D9711yescharacteristic
189Q8—44no—
190Q8—3884no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—24no—
197C2×C4—22no—
198C2×C4—974no—
199C2×C4—974no—
200C2×C4—974no—
201C2×C4—972no—
202C2×C4—19416no—
203C2×C4—1942no—
204C2×C4—1942no—
205C2×C4—1942no—
206C2×C4—1944no—
207C2×C4—1942no—
208C2×C4—1941no—
209C2×C4—1941no—
210C2×C4—1948no—
211C2×C4D4×D9712yes—
212C2×C4D4×D9712yes—
213C2×C4Q8×D9714yes—
214C2×C4#1309551112yes—
215C22—974no—
216C22—1942no—
217C22—1944no—
218C22—1942no—
219C22—1944no—
220C22C2×D4×D9711yescharacteristic
221C22C2×D4×D9711yescharacteristic
222C22C2×Q8×D9712yes—
223C22#41×D9712yes—
224C22C2×#1309551111yescharacteristic
225C4—24no—
226C4—42no—
227C4—42no—
228C4—978no—
229C4—3882no—
230C4—3882no—
231C4—1942no—
232C4#1569078314yes—
233C2—1944no—
234C2C2×#1569078311yescharacteristic
235C2C22⋊Q8×D9712yes—
236C2C4⋊Q8×D9711yescharacteristic
237C2#2026128411yescharacteristic
238C2#2026128912yes—
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, cosocleC23×C97C24
Fitting#391×C97C2
prime radicalC23×D97C23
hypercenter#391D97
hypocenterC97#2857
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C2×#15690808C2semidirect
5. #20260641C2semidirect
6. #20260632C2semidirect
7. #20260604C2semidirect
9. #391×C97C2semidirect
44. C2×C4×D97Q8semidirect
104. C2×D97C4⋊Q8semidirect
112. Dic97C22⋊Q8semidirect
120. C97#2857semidirect
126. #391D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908082
C2×#156907902
C2×#156907862
C2×C42×D972
#202606412
#202606322
#202606042
#202605942
#391×C972
#285797