Finite Group Databasebeta

← #27157223

Subgroups — #27157223

Order 12416.

200 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 54 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#20261756C211yesmaximal, characteristic, complemented
2#20261755C211yesmaximal, characteristic, complemented
3#20261592C212yesmaximal, complemented
4#20261591C212yesmaximal, complemented
5#20261271C211yesmaximal, characteristic
6#15690877—21no—
7C4⋊#11715897—21no—
8C22⋊#11715897—21no—
9C22⋊#11715897—21no—
10#15690865C2212yes—
11#15690865C2212yes—
12#15690805—21no—
13#15690801C2211yescharacteristic
14#15690799C2211yescharacteristic
15#15690794—21no—
16#15690784C411yescharacteristic
17Q8×Dic97C411yescharacteristic
18#15690772C411yescharacteristic
19#15690770—21no—
20#15690768—21no—
21C2×C4×Dic97C2211yescharacteristic
22C22⋊Q8×C97C411yescharacteristic
23C2×#11715897—22no—
24C2×#11715897—22no—
25C2×D388C2×C411yescharacteristic
26C2×C22⋊D97—21no—
27C22×Dic97D411yescharacteristic
28C22×Dic97D411yescharacteristic
29C2×C4×D97—21no—
30C4⋊Dic97—21no—
31C4⋊Dic97—21no—
32C4⋊Dic97C2×C411yescharacteristic
33#13095490—21no—
34#13095488—21no—
35#13095488—21no—
36#13095488—21no—
37#13095488—21no—
38C4×Dic97—21no—
39C4×Dic97C2×C411yescharacteristic
40C4×Dic97D411yescharacteristic
41C4×Dic97D411yescharacteristic
42C4×Dic97C2311yescharacteristic
43C2×Q8×C97C2×C411yescharacteristic
44C22×C4×C97C2×C411yescharacteristic
45C4⋊C4×C97—21no—
46C4⋊C4×C97C2×C411yescharacteristic
47C22⋊C4×C97—21no—
48C23×C97C22⋊C411yescharacteristic
49C22×D97—21no—
50Q8×C97—41no—
51C2×C4×C97—21no—
52C2×C4×C97—21no—
53C2×C4×C97—41no—
54C2×C4×C97C22⋊C411yescharacteristic
55C2×C4×C97C22×C411yescharacteristic
56C22⋊D97—42no—
57C2×Dic97—21no—
58C2×Dic97—41no—
59C2×Dic97—41no—
60C2×Dic97C22⋊C411yescharacteristic
61C2×Dic97C22⋊C411yescharacteristic
62C2×Dic97C2×D411yescharacteristic
63C2×Dic97C2×D411yescharacteristic
64D388—41no—
65C4×D97—42no—
66#11715897—42no—
67#11715897—42no—
68C22×C97—21no—
69C22×C97—21no—
70C22×C97—21no—
71C22×C97C2×C22⋊C411yescharacteristic
72C2×D97—41no—
73C2×D97—41no—
74C2×D97—41no—
75C4×C97—21no—
76C4×C97—21no—
77C4×C97—41no—
78C4×C97—41no—
79Dic97—21no—
80Dic97—21no—
81Dic97—21no—
82Dic97—21no—
83Dic97—41no—
84C2×C97—41no—
85C2×C97#41311yescharacteristic
86C2×C97#42111yescharacteristic
87C2×C97#42111yescharacteristic
88D97—81no—
89#1535—971nomaximal, Sylow
90C97#153511yescharacteristic, complemented, Sylow
91#529—971no—
92#432—971no—
93#432—971no—
94#330—972no—
95#329—972no—
96#126—1941no—
97#125—971no—
98#124—971no—
99#123—1941no—
100C22⋊Q8#8760411yescharacteristic
101#121—971no—
102C4×Q8—971no—
103C4×D4—971no—
104C4×D4—1941no—
105#117—1941no—
106C2×C42—971no—
107C4⋊C8—1941no—
108C4⋊C8—1941no—
109C22⋊C8—1941no—
110C22⋊C8—1941no—
111#97—972no—
112#97—972no—
113C2×Q8C2×#8760411yescharacteristic
114C2×D4—971no—
115C2×D4—1941no—
116C22×C4—971no—
117C22×C4—971no—
118C22×C4—1941no—
119C22×C4C2×#8760411yescharacteristic
120C2×C8—1942no—
121C2×C8—1942no—
122C4⋊C4—21no—
123C4⋊C4—971no—
124C4⋊C4—1941no—
125C4⋊C4—1941no—
126C4⋊C4—1941no—
127C4⋊C4C2×#8760411yescharacteristic
128C22⋊C4—21no—
129C22⋊C4—1941no—
130C22⋊C4—1941no—
131C22⋊C4—1941no—
132C22⋊C4—1941no—
133C42—971no—
134C42—971no—
135C42—971no—
136C42—971no—
137C42—1941no—
138C23—1941no—
139C23C22⋊#8760411yescharacteristic
140Q8—41no—
141D4—3881no—
142D4—3882no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—41no—
146C2×C4—971no—
147C2×C4—971no—
148C2×C4—971no—
149C2×C4—971no—
150C2×C4—3882no—
151C2×C4—3881no—
152C2×C4—3881no—
153C2×C4—1941no—
154C2×C4C22⋊#8760411yescharacteristic
155C2×C4C22×#8760411yescharacteristic
156C8—3882no—
157C8—3882no—
158C22—21no—
159C22—21no—
160C22—21no—
161C22—3881no—
162C22—3881no—
163C22—3881no—
164C22C2×C22⋊#8760411yescharacteristic
165C4—21no—
166C4—21no—
167C4—41no—
168C4—41no—
169C4—3881no—
170C4—1941no—
171C4—1941no—
172C4—1941no—
173C4—1941no—
174C2—41no—
175C2—7761no—
176C2#2026172211yescharacteristic
177C2#2026173211yescharacteristic
178C2#2026173411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC22⋊Q8×C97C4
socleC22×C97C2×C22⋊C4
prime radical#15690784C4
cosocleC4×Dic97C23
hypercenterC22⋊Q8#87604
hypocenterC97#1535
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #20261756C2semidirect
2. #20261755C2semidirect
3. #20261592C2semidirect
4. #20261591C2semidirect
90. C97#1535semidirect

Maximal subgroups

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

subgroupindex
#202617562
#202617552
#202615922
#202615912
#202612712
#153597