Finite Group Databasebeta

← #27140712

Subgroups — #27140712

Order 12416.

238 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 53 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#20261366C211yesmaximal, characteristic
2#20260893C212yesmaximal
3#20260655C211yesmaximal, characteristic
4#20260473C212yesmaximal
5#324×C97C211yesmaximal, characteristic
6C22×Dic194C2211yescharacteristic
7C2×Dic388—22no—
8#15690809—22no—
9#15690801—22no—
10#15690789—21no—
11#15690789C2212yes—
12#15690769—21no—
13C2×#13095490—21no—
14#15690735—22no—
15#15690717—21no—
16#15690717C2211yescharacteristic
17C2×C4⋊C4×C97C2211yescharacteristic
18C22⋊C8×C97C2212yes—
19C2×Dic194—22no—
20C2×Dic194—21no—
21C2×Dic194—42no—
22C2×Dic194D412yes—
23C22×Dic97—21no—
24C22×Dic97—21no—
25Dic388—44no—
26C4⋊Dic97—21no—
27C4⋊Dic97—21no—
28C4⋊Dic97—21no—
29C4⋊Dic97D412yes—
30#13095490—22no—
31#13095490—22no—
32#13095490—42no—
33C22⋊Dic97—22no—
34C4×Dic97—22no—
35C22×C4×C97D411yescharacteristic
36C22×C4×C97D411yescharacteristic
37C22×C4×C97C2311yescharacteristic
38C2×C8×C97—22no—
39C4⋊C4×C97—22no—
40C23×C97C2×D411yescharacteristic
41C2×C4×C97—21no—
42C2×C4×C97—21no—
43C2×C4×C97—21no—
44C2×C4×C97—41no—
45C2×C4×C97—41no—
46C2×C4×C97Q1612yescomplemented
47C2×C4×C97C2×D412yes—
48C2×Dic97—22no—
49C2×Dic97—22no—
50C2×Dic97—22no—
51C2×Dic97—22no—
52C2×Dic97—41no—
53C2×Dic97—41no—
54C2×Dic97—41no—
55C2×Dic97—41no—
56Dic194—24no—
57Dic194—42no—
58Dic194—42no—
59Dic194—81no—
60C8×C97—42no—
61C22×C97—21no—
62C22×C97—21no—
63C22×C97#12011yescharacteristic
64C22×C97C2×Q1611yescharacteristic
65C22×C97#13711yescharacteristic
66C4×C97—22no—
67C4×C97—22no—
68C4×C97—41no—
69Dic97—42no—
70Dic97—42no—
71Dic97—42no—
72Dic97—42no—
73C2×C97—21no—
74C2×C97—21no—
75C2×C97#45111yescharacteristic
76C2×C97#45611yescharacteristic
77C2×C97B(4,𝔽2)11yescharacteristic
78#1605—971nomaximal, Sylow
79C97#160511yescharacteristic, complemented, Sylow
80#544—971no—
81#451—972no—
82#395—971no—
83#328—972no—
84#324D9711yescharacteristic
85C22×Q8—971no—
86C2×Q16—1942no—
87C4⋊Q8—1942no—
88#125—1942no—
89C22⋊Q8—972no—
90C22⋊Q8—1941no—
91#117—1941no—
92C2×C4⋊C4—1941no—
93C2×C4⋊C4C2×D9711yescharacteristic
94#103—1942no—
95C22⋊C8C2×D9712yes—
96#95—971no—
97#95—1941no—
98C2×Q8—972no—
99C2×Q8—3882no—
100C2×Q8—1941no—
101C2×Q8—1942no—
102C22×C4—1941no—
103C22×C4—1941no—
104C22×C4D38811yescharacteristic
105C22×C4D38811yescharacteristic
106C22×C4C22×D9711yescharacteristic
107Q16—3884no—
108C2×C8—22no—
109C4⋊C4—22no—
110C4⋊C4—972no—
111C4⋊C4—3882no—
112C4⋊C4—1942no—
113C4⋊C4—1941no—
114C4⋊C4—1941no—
115C4⋊C4—1941no—
116C4⋊C4—1942no—
117C22⋊C4—1942no—
118C42—1942no—
119C23C2×D38811yescharacteristic
120Q8—7761no—
121Q8—3882no—
122Q8—3882no—
123Q8—1944no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—21no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4—3881no—
130C2×C4—3881no—
131C2×C4—3881no—
132C2×C4—3881no—
133C2×C4—1942no—
134C2×C4—1942no—
135C2×C4—1942no—
136C2×C4—1942no—
137C2×C4Dic38812yescomplemented
138C2×C4D4×D9712yes—
139C8—42no—
140C22—21no—
141C22—21no—
142C22#1569078011yescharacteristic
143C22C2×Dic38811yescharacteristic
144C22#1569084311yescharacteristic
145C4—22no—
146C4—22no—
147C4—41no—
148C4—3882no—
149C4—3882no—
150C4—3882no—
151C4—3882no—
152C2—21no—
153C2—21no—
154C2#2026089311yescharacteristic
155C2#2026092011yescharacteristic
156C2#2026092311yescharacteristic
subgroupG/H
centerC22#15690780
commutator, cosocleC22×C4×C97C23
FrattiniC22×C4C22×D97
Fitting#324×C97C2
socleC22×C97#120
prime radicalC23×C97C2×D4
hypercenter#324D97
hypocenterC97#1605
LCSG ▷ C22×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
46. C2×C4×C97Q16semidirect
79. C97#1605semidirect
137. C2×C4Dic388semidirect

Maximal subgroups

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

subgroupindex
#202613662
#202608932
#202606552
#202604732
#324×C972
#160597