Finite Group Databasebeta

← #27157594

Subgroups — #27157594

Order 12416.

312 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 92 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×#15690877C211yesmaximal, characteristic, complemented
2C2×C4⋊#11715897C211yesmaximal, characteristic, complemented
3C2×C22⋊#11715897C212yesmaximal
4#20261601C212yesmaximal, complemented
5C2×#15690766C211yesmaximal, characteristic
6C22×#11715897C2212yes—
7C22×#11715897C2212yes—
8#15690877—22no—
9C4⋊#11715897—22no—
10C22⋊#11715897—24no—
11C23×Dic97C2211yescharacteristic
12#15690766—22no—
13#15690766—22no—
14C2×#13095490C411yescharacteristic
15C2×#13095490C411yescharacteristic
16C2×C22⋊Dic97C411yescharacteristic
17C2×C4×Dic97C2211yescharacteristic
18C2×C4×Dic97C2211yescharacteristic
19C2×C22⋊C4×C97C411yescharacteristic
20C2×#11715897—22no—
21C2×#11715897—22no—
22C2×#11715897—22no—
23C2×#11715897—22no—
24C2×#11715897—22no—
25C2×#11715897—22no—
26C22×Dic97—21no—
27C22×Dic97—21no—
28C22×Dic97—21no—
29C22×Dic97—21no—
30C22×Dic97C2×C411yescharacteristic
31C22×Dic97C2×C411yescharacteristic
32C22×Dic97C2×C411yescharacteristic
33C22×Dic97D412yes—
34C22×Dic97Q812yes—
35C22×Dic97C2311yescharacteristic
36#13095490—22no—
37#13095490—22no—
38C22⋊Dic97—22no—
39C4×Dic97—21no—
40C4×Dic97—21no—
41C4×Dic97D412yes—
42C4×Dic97D412yes—
43C24×C97C2×C411yescharacteristic
44C22×C4×C97C2×C411yescharacteristic
45C22×C4×C97C2×C411yescharacteristic
46C22⋊C4×C97—22no—
47C23×C97—21no—
48C23×C97—21no—
49C23×C97—22no—
50C23×C97—22no—
51C23×C97C4⋊C412yes—
52C23×C97C22×C411yescharacteristic
53C2×C4×C97—22no—
54C2×C4×C97—21no—
55C2×C4×C97—22no—
56C2×C4×C97C22⋊C412yes—
57C2×Dic97—21no—
58C2×Dic97—21no—
59C2×Dic97—22no—
60C2×Dic97—22no—
61C2×Dic97—22no—
62C2×Dic97—22no—
63C2×Dic97—22no—
64C2×Dic97—21no—
65C2×Dic97—22no—
66C2×Dic97C22⋊C412yes—
67C2×Dic97C4⋊C412yes—
68C2×Dic97C2×D411yescharacteristic
69C2×Dic97C2×D411yescharacteristic
70C2×Dic97C2×D411yescharacteristic
71C2×Dic97C2×Q811yescharacteristic
72C2×Dic97#4111yescharacteristic
73C2×Dic97#4111yescharacteristic
74#11715897—42no—
75#11715897—42no—
76C22×C97—22no—
77C22×C97—24no—
78C22×C97—41no—
79C22×C97—41no—
80C22×C97—42no—
81C22×C97C2×C22⋊C411yescharacteristic
82C22×C97C2×C4⋊C411yescharacteristic
83C22×C97#11711yescharacteristic
84C22×C97#13112yes—
85C22×C97#13112yes—
86C4×C97—22no—
87C4×C97—41no—
88Dic97—22no—
89Dic97—22no—
90Dic97—41no—
91Dic97#12111yescharacteristic
92Dic97#12111yescharacteristic
93Dic97C22⋊Q811yescharacteristic
94Dic97C22⋊Q811yescharacteristic
95C2×C97—42no—
96C2×C97#38011yescharacteristic
97C2×C97#40811yescharacteristic
98C2×C97#40811yescharacteristic
99C2×C97#42412yes—
100C2×C97#43312yes—
101#1798—971nomaximal, Sylow
102C97#179811yescharacteristic, complemented, Sylow
103C2×#117—971no—
104C2×C4⋊C8—971no—
105C2×C4⋊C8—971no—
106C2×C22⋊C8—972no—
107#336—972no—
108C23×C4—971no—
109C22×C8—972no—
110C22×C8—972no—
111#117—1942no—
112#117—1942no—
113C2×C4⋊C4—971no—
114C2×C4⋊C4—971no—
115C2×C22⋊C4—971no—
116C2×C22⋊C4#8760411yescharacteristic
117C2×C42—971no—
118C2×C42—971no—
119C4⋊C8—1942no—
120C4⋊C8—1942no—
121C22⋊C8—1944no—
122C24C2×#8760411yescharacteristic
123C22×C4—971no—
124C22×C4—972no—
125C22×C4—972no—
126C22×C4—971no—
127C22×C4—971no—
128C22×C4—971no—
129C22×C4—1941no—
130C22×C4—1941no—
131C22×C4—1941no—
132C22×C4—1941no—
133C22×C4C2×#8760411yescharacteristic
134C22×C4C2×#8760411yescharacteristic
135C2×C8—1942no—
136C2×C8—1942no—
137C2×C8—1942no—
138C2×C8—1942no—
139C2×C8—1942no—
140C2×C8—1942no—
141C4⋊C4—1942no—
142C4⋊C4—1942no—
143C22⋊C4—22no—
144C22⋊C4—1942no—
145C42—972no—
146C42—972no—
147C42—1941no—
148C42—1941no—
149C23—21no—
150C23—21no—
151C23—22no—
152C23—22no—
153C23C4⋊#8760412yes—
154C23C22×#8760411yescharacteristic
155C2×C4—22no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—971no—
159C2×C4—971no—
160C2×C4—971no—
161C2×C4—971no—
162C2×C4—971no—
163C2×C4—971no—
164C2×C4—972no—
165C2×C4—972no—
166C2×C4—1942no—
167C2×C4—1942no—
168C2×C4—1942no—
169C2×C4—1942no—
170C2×C4—1941no—
171C2×C4—1942no—
172C2×C4—1941no—
173C2×C4—1941no—
174C2×C4—1942no—
175C2×C4C22⋊#8760412yes—
176C8—3882no—
177C8—3882no—
178C22—22no—
179C22—24no—
180C22—41no—
181C22—41no—
182C22—42no—
183C22C2×C22⋊#8760411yescharacteristic
184C22C2×C4⋊#8760411yescharacteristic
185C22#1569088911yescharacteristic
186C22#1569089712yes—
187C22#1569089712yes—
188C4—22no—
189C4—41no—
190C4—971no—
191C4—971no—
192C4—971no—
193C4—971no—
194C4—3881no—
195C4—1942no—
196C4—1942no—
197C2—42no—
198C2#2026167711yescharacteristic
199C2#2026170811yescharacteristic
200C2#2026170911yescharacteristic
201C2#2026174112yes—
202C2#2026175912yes—
subgroupG/H
center, FrattiniC23C22×#87604
commutator, socleC23×C97C22×C4
FittingC2×C22⋊C4×C97C4
prime radicalC24×C97C2×C4
cosocleC22×Dic97C23
hypercenterC2×C22⋊C4#87604
hypocenterC97#1798
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#15690877C2semidirect
2. C2×C4⋊#11715897C2semidirect
4. #20261601C2semidirect
102. C97#1798semidirect

Maximal subgroups

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

subgroupindex
C2×#156908772
C2×C4⋊#117158972
C2×C22⋊#117158972
#202616012
C2×#156907662
#179897