Finite Group Databasebeta

← #27140870

Subgroups — #27140870

Order 12416.

214 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 61 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
1#20261114C211yesmaximal, characteristic, complemented
2#20261030C211yesmaximal, characteristic, complemented
3#20261025C211yesmaximal, characteristic
4#20260484C211yesmaximal, characteristic, complemented
5Q8⋊#11715895C211yesmaximal, characteristic, complemented
6D4⋊#11715895C211yesmaximal, characteristic
7#554×C97C211yesmaximal, characteristic
8C2×#13095501—22no—
9#15690810C2211yescharacteristic
10#15690743—21no—
11#15690741—21no—
12C4⋊#11715895C2211yescharacteristic
13C4⋊#11715895C2211yescharacteristic
14#15690734—21no—
15#15690731—21no—
16C4×#11715895C2211yescharacteristic
17C4⋊Q8×C97C2211yescharacteristic
18#125×C97—21no—
19C22⋊Q8×C97—21no—
20C22⋊Q8×C97—22no—
21C4×Q8×C97C2211yescharacteristic
22C4×D4×C97—21no—
23C4×D4×C97C2211yescharacteristic
24C2×C4⋊C4×C97—21no—
25C2×Dic194—21no—
26#13095501—44no—
27C2×#11715895—21no—
28C2×#11715895—21no—
29C2×#11715895—21no—
30C4⋊Dic97—21no—
31C4⋊Dic97—21no—
32C2×Q8×C97D411yescharacteristic
33C2×Q8×C97D412yes—
34C2×D4×C97D411yescharacteristic
35C22×C4×C97—22no—
36C22×C4×C97—21no—
37C4⋊C4×C97—22no—
38C4⋊C4×C97—22no—
39C4⋊C4×C97—21no—
40C4⋊C4×C97—21no—
41C4⋊C4×C97—21no—
42C4⋊C4×C97D411yescharacteristic
43C4⋊C4×C97D411yescharacteristic
44C22⋊C4×C97—22no—
45C22⋊C4×C97—21no—
46C42×C97—21no—
47C42×C97C2311yescharacteristic
48C23×C97—21no—
49Q8×C97—21no—
50Q8×C97—21no—
51Q8×C97—42no—
52D4×C97—21no—
53D4×C97Q1612yescomplemented
54C2×C4×C97—21no—
55C2×C4×C97—21no—
56C2×C4×C97—21no—
57C2×C4×C97—22no—
58C2×C4×C97—41no—
59C2×C4×C97—41no—
60C2×C4×C97—82no—
61C2×C4×C97C2×D411yescharacteristic
62C2×C4×C97C2×D411yescharacteristic
63C2×C4×C97C2×D411yescharacteristic
64C2×Dic97—41no—
65Dic194—42no—
66#11715895—22no—
67#11715895—41no—
68#11715895—41no—
69C22×C97—22no—
70C22×C97—41no—
71C22×C97—41no—
72C22×C97#12011yescharacteristic
73C4×C97—21no—
74C4×C97—41no—
75C4×C97—41no—
76C4×C97—41no—
77C4×C97—42no—
78C4×C97C2×Q1611yescharacteristic
79C4×C97#13511yescharacteristic
80C4×C97#13711yescharacteristic
81C4×C97#13711yescharacteristic
82Dic97—81no—
83C2×C97—42no—
84C2×C97#45111yescharacteristic
85C2×C97#45211yescharacteristic
86C2×C97#45311yescharacteristic
87#1632—971nomaximal, Sylow
88C97#163211yescharacteristic, complemented, Sylow
89#554D9711yescharacteristic
90#494—971no—
91#477—971no—
92#476—971no—
93#333—971no—
94Q8⋊C8—971no—
95D4⋊C8—971no—
96C2×Q16—1942no—
97C4⋊Q8—971no—
98C4⋊Q8C2×D9711yescharacteristic
99#125—21no—
100C22⋊Q8—21no—
101C22⋊Q8—22no—
102C4×Q8C2×D9711yescharacteristic
103C4×D4—21no—
104C4×D4C2×D9711yescharacteristic
105C2×C4⋊C4—21no—
106#107—1941no—
107#106—1941no—
108C4⋊C8—971no—
109C4⋊C8—971no—
110#103—1941no—
111#102—1941no—
112C4×C8—971no—
113C2×Q8—1941no—
114C2×Q8C22⋊D9711yescharacteristic
115C2×Q8C22⋊D9712yes—
116C2×D4C22⋊D9711yescharacteristic
117C22×C4—22no—
118C22×C4—21no—
119Q16—3884no—
120C2×C8—1941no—
121C2×C8—1941no—
122C2×C8—1941no—
123C4⋊C4—22no—
124C4⋊C4—22no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—1941no—
129C4⋊C4—1941no—
130C4⋊C4C22⋊D9711yescharacteristic
131C4⋊C4C22⋊D9711yescharacteristic
132C22⋊C4—22no—
133C22⋊C4—21no—
134C42—21no—
135C42C22×D9711yescharacteristic
136C23—21no—
137Q8—21no—
138Q8—21no—
139Q8—42no—
140Q8—3882no—
141D4—21no—
142D4#1309550112yescomplemented
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—22no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—82no—
150C2×C4—3881no—
151C2×C4C2×C22⋊D9711yescharacteristic
152C2×C4C2×C22⋊D9711yescharacteristic
153C2×C4C2×C22⋊D9711yescharacteristic
154C8—3881no—
155C8—3881no—
156C8—1942no—
157C22—22no—
158C22—41no—
159C22—41no—
160C22#1569077911yescharacteristic
161C4—21no—
162C4—41no—
163C4—41no—
164C4—41no—
165C4—42no—
166C4—7761no—
167C4C2×#1309550111yescharacteristic
168C4#1569083011yescharacteristic
169C4#1569084011yescharacteristic
170C4#1569084111yescharacteristic
171C2—42no—
172C2#2026089111yescharacteristic
173C2#2026089711yescharacteristic
174C2#2026090311yescharacteristic
subgroupG/H
centerC22#15690779
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting#554×C97C2
socleC22×C97#120
prime radicalC2×D4×C97D4
hypercenter#554D97
hypocenterC97#1632
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #554

Complements

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

NH ≅ G/Ntype
1. #20261114C2semidirect
2. #20261030C2semidirect
4. #20260484C2semidirect
5. Q8⋊#11715895C2semidirect
53. D4×C97Q16semidirect
88. C97#1632semidirect
142. D4#13095501semidirect

Maximal subgroups

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

subgroupindex
#202611142
#202610302
#202610252
#202604842
Q8⋊#117158952
D4⋊#117158952
#554×C972
#163297