Finite Group Databasebeta

← #27157092

Subgroups — #27157092

Order 12416.

216 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 60 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#20261753C211yesmaximal, characteristic, complemented
2#20261739C211yesmaximal, characteristic, complemented
3Q8⋊#11715897C212yesmaximal, complemented
4D4⋊#11715897C212yesmaximal, complemented
5#20261264C211yesmaximal, characteristic
6C2×#13095521—21no—
7#15690877—21no—
8C4⋊#11715897—21no—
9C4⋊#11715897C2212yes—
10#15690867—21no—
11C4×#11715897C2212yes—
12#15690803—21no—
13#15690802C411yescharacteristic
14#15690792—21no—
15#15690786C411yescharacteristic
16#15690781C411yescharacteristic
17Q8×Dic97C2211yescharacteristic
18D4×Dic97C2211yescharacteristic
19#15690770—21no—
20#15690766—21no—
21C42×D97C2211yescharacteristic
22#124×C97C411yescharacteristic
23#13095521—42no—
24C2×#11715897—22no—
25C2×#11715897—22no—
26C2×C22⋊D97—21no—
27C22×Dic97—21no—
28C2×C4×D97C2×C411yescharacteristic
29C2×C4×D97D411yescharacteristic
30C2×C4×D97D411yescharacteristic
31C4⋊Dic97—21no—
32C4⋊Dic97C2×C411yescharacteristic
33C4⋊Dic97C2×C411yescharacteristic
34#13095490—21no—
35#13095490—21no—
36#13095490—21no—
37C22⋊Dic97—21no—
38C22⋊Dic97—21no—
39#13095488—21no—
40C4×Dic97—21no—
41C4×Dic97D411yescharacteristic
42C4×Dic97D411yescharacteristic
43C4×Dic97C2311yescharacteristic
44C2×Q8×C97C2×C411yescharacteristic
45C2×D4×C97C2×C411yescharacteristic
46C22⋊C4×C97—21no—
47C22⋊C4×C97—21no—
48C42×C97C2×C411yescharacteristic
49C23×C97—21no—
50C22×D97C22⋊C411yescharacteristic
51Q8×C97—22no—
52D4×C97—22no—
53C2×C4×C97—21no—
54C2×C4×C97C22⋊C411yescharacteristic
55C2×C4×C97C22⋊C411yescharacteristic
56C2×C4×C97C22×C411yescharacteristic
57C22⋊D97—42no—
58C2×Dic97—21no—
59C2×Dic97—21no—
60C2×Dic97—41no—
61C2×Dic97—41no—
62C2×Dic97C22⋊C411yescharacteristic
63C2×Dic97C2×D411yescharacteristic
64C2×Dic97C2×D411yescharacteristic
65C4×D97—22no—
66C4×D97—21no—
67C4×D97—21no—
68C4×D97—41no—
69#11715897—42no—
70#11715897—42no—
71C22×C97—41no—
72C22×C97—41no—
73C22×C97—41no—
74C22×C97C2×C22⋊C411yescharacteristic
75C2×D97—21no—
76C2×D97—21no—
77C2×D97—21no—
78C4×C97—21no—
79C4×C97—21no—
80C4×C97—41no—
81C4×C97#13112yes—
82Dic97—21no—
83Dic97—21no—
84Dic97—41no—
85Dic97—41no—
86Dic97AGL(1,ℤ/8)11yescharacteristic
87Dic97#13711yescharacteristic
88C2×C97—81no—
89C2×C97#40811yescharacteristic
90C2×C97#41711yescharacteristic
91C2×C97#42111yescharacteristic
92D97—41no—
93#1495—971nomaximal, Sylow
94C97#149511yescharacteristic, complemented, Sylow
95#529—971no—
96#431—971no—
97#423—971no—
98Q8⋊C8—972no—
99D4⋊C8—972no—
100C2×M16—1941no—
101#126—1941no—
102#125—971no—
103#124#8760411yescharacteristic
104#123—1941no—
105C22⋊Q8—971no—
106#121—971no—
107C4×Q8—971no—
108C4×D4—971no—
109C4×D4—1941no—
110#117—1941no—
111C2×C42—971no—
112C4⋊C8—972no—
113C4⋊C8—1941no—
114C4⋊C8—1941no—
115C22⋊C8—1941no—
116C4×C8—972no—
117C2×Q8C2×#8760411yescharacteristic
118C2×D4—1941no—
119C2×D4C2×#8760411yescharacteristic
120C22×C4—971no—
121C22×C4—971no—
122C22×C4—971no—
123C22×C4—1941no—
124M16—3882no—
125C2×C8—1942no—
126C2×C8—1942no—
127C4⋊C4—971no—
128C4⋊C4—971no—
129C4⋊C4—1941no—
130C4⋊C4—1941no—
131C4⋊C4—1941no—
132C4⋊C4—1941no—
133C22⋊C4—21no—
134C22⋊C4—21no—
135C22⋊C4—1941no—
136C22⋊C4—1941no—
137C22⋊C4—1941no—
138C42—971no—
139C42—971no—
140C42—971no—
141C42—1941no—
142C42C2×#8760411yescharacteristic
143C23—21no—
144C23—971no—
145Q8—22no—
146D4—22no—
147D4—3882no—
148C2×C4—21no—
149C2×C4—971no—
150C2×C4—971no—
151C2×C4—971no—
152C2×C4—3881no—
153C2×C4—3881no—
154C2×C4—3881no—
155C2×C4—1941no—
156C2×C4—1941no—
157C2×C4—1941no—
158C2×C4—1941no—
159C2×C4—1942no—
160C2×C4C22⋊#8760411yescharacteristic
161C2×C4C22⋊#8760411yescharacteristic
162C2×C4C22×#8760411yescharacteristic
163C8—3882no—
164C8—3882no—
165C22—41no—
166C22—41no—
167C22—41no—
168C22—1941no—
169C22—1941no—
170C22—1941no—
171C22C2×C22⋊#8760411yescharacteristic
172C4—21no—
173C4—21no—
174C4—41no—
175C4—971no—
176C4—971no—
177C4—3881no—
178C4—3881no—
179C4—1941no—
180C4—1941no—
181C4#1569089712yes—
182C2—81no—
183C2—3881no—
184C2#2026170911yescharacteristic
185C2#2026172611yescharacteristic
186C2#2026173211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
Fitting#124×C97C4
socleC22×C97C2×C22⋊C4
prime radical#15690781C4
cosocleC4×Dic97C23
hypercenter#124#87604
hypocenterC97#1495
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #124

Complements

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

NH ≅ G/Ntype
1. #20261753C2semidirect
2. #20261739C2semidirect
3. Q8⋊#11715897C2semidirect
4. D4⋊#11715897C2semidirect
94. C97#1495semidirect

Maximal subgroups

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

subgroupindex
#202617532
#202617392
Q8⋊#117158972
D4⋊#117158972
#202612642
#149597