Finite Group Databasebeta

← #27150091

Subgroups — #27150091

Order 12416.

752 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 181 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×#15690806C211yesmaximal, characteristic
3C2×#15690799C211yesmaximal, characteristic, complemented
4C2×#15690798C211yesmaximal, characteristic, complemented
5C2×#15690797C211yesmaximal, characteristic, complemented
6#20260635C218yesmaximal, complemented
7C42×Dic97C211yesmaximal, characteristic, complemented
8C2×#124×C97C211yesmaximal, characteristic, complemented
9C22×Dic194C2211yescharacteristic
10C22×D388C2211yescharacteristic
11C22×C22⋊D97C2214yes—
12#15690808—24no—
13#15690806—24no—
14#15690799—24no—
15#15690798—24no—
16#15690797—24no—
17C2×#13095490C2214yes—
18C2×C22⋊Dic97C2214yes—
19C2×#13095488C2214yes—
20C2×#13095488C2214yes—
21C2×C4×Dic97C2214yes—
22C2×C4×Dic97C2211yescharacteristic
23C2×C4×Dic97C2211yescharacteristic
24C22×Q8×C97C2211yescharacteristic
25C22×D4×C97C2211yescharacteristic
26#124×C97—24no—
27C2×C22⋊C4×C97C2214yes—
28C2×C42×C97C2211yescharacteristic
29C23×D97C2312yes—
30C2×Dic194—22no—
31C2×Dic194—22no—
32C2×Dic194—22no—
33C2×D388—22no—
34C2×D388—22no—
35C2×D388—22no—
36C2×C22⋊D97—28no—
37C2×C22⋊D97—216no—
38C22×Dic97C2312yes—
39C22×Dic97C2312yes—
40C22×Dic97C2312yes—
41#13095490—28no—
42C22⋊Dic97—28no—
43#13095488—28no—
44#13095488—28no—
45C4×Dic97—28no—
46C4×Dic97D414yes—
47C4×Dic97D414yescomplemented
48C24×C97C2312yes—
49C2×Q8×C97—22no—
50C2×Q8×C97—22no—
51C2×Q8×C97—22no—
52C2×D4×C97—22no—
53C2×D4×C97—22no—
54C2×D4×C97—22no—
55C22×C4×C97C2312yes—
56C22×C4×C97C2312yes—
57C22×C4×C97C2311yescharacteristic
58C22⋊C4×C97—28no—
59C42×C97D414yescomplemented
60C23×C97—22no—
61C23×C97—22no—
62C23×C97—22no—
63C23×C97—28no—
64C23×C97C2411yescharacteristic
65C22×D97—22no—
66C22×D97—22no—
67C22×D97—22no—
68C22×D97—28no—
69Q8×C97—44no—
70D4×C97—44no—
71C2×C4×C97—24no—
72C2×C4×C97—22no—
73C2×C4×C97—22no—
74C2×C4×C97—22no—
75C2×C4×C97C2×D412yes—
76C2×C4×C97C2×D412yes—
77C2×C4×C97C2×D414yes—
78C2×C4×C97C2×D412yes—
79C22⋊D97—416no—
80C2×Dic97—24no—
81C2×Dic97—22no—
82C2×Dic97—22no—
83C2×Dic97—22no—
84C2×Dic97C2×D414yes—
85C2×Dic97C2×D414yes—
86C2×Dic97#4118yes—
87D388—44no—
88Dic194—44no—
89C22×C97—44no—
90C22×C97—42no—
91C22×C97—44no—
92C22×C97—44no—
93C22×C97C22×D411yescharacteristic
94C22×C97C22×D411yescharacteristic
95C22×C97C22×D411yescharacteristic
96C22×C97C2×#4114yes—
97C2×D97—44no—
98C2×D97—42no—
99C2×D97—44no—
100C2×D97—44no—
101C4×C97—24no—
102C4×C97—42no—
103C4×C97#12714yes—
104Dic97—24no—
105Dic97—42no—
106Dic97#12418yescomplemented
107C2×C97—82no—
108C2×C97C2×#12412yes—
109C2×C97C2×#12711yescharacteristic
110C2×C97#53212yes—
111C2×C97#53212yes—
112D97—82no—
113#2858—971nomaximal, Sylow
114C97#285811yescharacteristic, complemented, Sylow
115C2×C4⋊Q8—971no—
116C2×#127—971no—
117C2×#124—971no—
118C2×#124—971no—
119C2×#124—971no—
120C2×#124D9711yescharacteristic, complemented
121#390—978no—
122C43—971no—
123C22×Q8—971no—
124C22×Q8C2×D9711yescharacteristic
125C22×D4—974no—
126C22×D4—971no—
127C22×D4C2×D9711yescharacteristic
128C4⋊Q8—1944no—
129#127—1944no—
130#124—24no—
131#124—1944no—
132#124—1944no—
133#124—1944no—
134C2×C4⋊C4—974no—
135C2×C22⋊C4—974no—
136C2×C22⋊C4—974no—
137C2×C22⋊C4—974no—
138C2×C22⋊C4C2×D9714yes—
139C2×C42—971no—
140C2×C42—971no—
141C2×C42—974no—
142C2×C42C2×D9711yescharacteristic
143C24—972no—
144C24C22×D9712yes—
145C2×Q8—22no—
146C2×Q8—22no—
147C2×Q8—22no—
148C2×Q8—1942no—
149C2×Q8—1942no—
150C2×Q8—1942no—
151C2×D4—22no—
152C2×D4—22no—
153C2×D4—22no—
154C2×D4—1948no—
155C2×D4—19416no—
156C2×D4—1942no—
157C2×D4—1942no—
158C2×D4—1942no—
159C22×C4—972no—
160C22×C4—972no—
161C22×C4—972no—
162C22×C4C22×D9712yes—
163C22×C4C22×D9712yes—
164C22×C4C22×D9711yescharacteristic
165C4⋊C4—1948no—
166C22⋊C4—28no—
167C22⋊C4—1948no—
168C22⋊C4—1948no—
169C22⋊C4—1948no—
170C42—974no—
171C42—974no—
172C42—1948no—
173C42C22⋊D9714yescomplemented
174C23—22no—
175C23—22no—
176C23—22no—
177C23—28no—
178C23—1942no—
179C23—1942no—
180C23—1942no—
181C23—1948no—
182C23C23×D9711yescharacteristic
183Q8—44no—
184Q8—3884no—
185D4—44no—
186D4—38816no—
187D4—3884no—
188C2×C4—24no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—974no—
193C2×C4—978no—
194C2×C4—974no—
195C2×C4—1942no—
196C2×C4—1942no—
197C2×C4—1942no—
198C2×C4—1944no—
199C2×C4D4×D9712yes—
200C2×C4D4×D9712yes—
201C2×C4C2×C22⋊D9714yes—
202C2×C4C2×C22⋊D9712yes—
203C22—44no—
204C22—42no—
205C22—44no—
206C22—44no—
207C22—3884no—
208C22—3882no—
209C22—3884no—
210C22—3884no—
211C22C2×D4×D9711yescharacteristic
212C22C2×D4×D9711yescharacteristic
213C22C22×C22⋊D9711yescharacteristic
214C22#41×D9714yes—
215C4—24no—
216C4—42no—
217C4—978no—
218C4—3882no—
219C4—1944no—
220C4#1569080614yes—
221C2—82no—
222C2—7762no—
223C2#124×D9712yes—
224C2C2×#1569080611yescharacteristic
225C2#2026128112yes—
226C2#2026128412yes—
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, cosocleC23×C97C24
FittingC2×#124×C97C2
prime radicalC2×#15690806C2
hypercenterC2×#124D97
hypocenterC97#2858
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×#15690808C2semidirect
3. C2×#15690799C2semidirect
4. C2×#15690798C2semidirect
5. C2×#15690797C2semidirect
6. #20260635C2semidirect
7. C42×Dic97C2semidirect
8. C2×#124×C97C2semidirect
47. C4×Dic97D4semidirect
59. C42×C97D4semidirect
106. Dic97#124semidirect
114. C97#2858semidirect
120. C2×#124D97semidirect
173. C42C22⋊D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908082
C2×#156908062
C2×#156907992
C2×#156907982
C2×#156907972
#202606352
C42×Dic972
C2×#124×C972
#285897