Finite Group Databasebeta

← #27140901

Subgroups — #27140901

Order 12416.

228 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 65 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#20261087C211yesmaximal, characteristic, complemented
2#20261030C211yesmaximal, characteristic, complemented
3#20261025C211yesmaximal, characteristic
4#20260481C211yesmaximal, characteristic, complemented
5Q8⋊#11715895C211yesmaximal, characteristic, complemented
6D4⋊#11715895C211yesmaximal, characteristic
7D4×Q8×C97C211yesmaximal, characteristic
8#15690810C2211yescharacteristic
9#15690743—21no—
10#15690741—21no—
11C4⋊#11715895C2211yescharacteristic
12C4⋊#11715895C2211yescharacteristic
13#15690734—21no—
14#15690733—22no—
15#15690731—21no—
16C4×#11715895C2211yescharacteristic
17C22×Q8×C97—21no—
18C4⋊Q8×C97—21no—
19C4⋊Q8×C97C2211yescharacteristic
20C22⋊Q8×C97—21no—
21C22⋊Q8×C97—22no—
22C4×Q8×C97C2211yescharacteristic
23C4×D4×C97—21no—
24C4×D4×C97C2211yescharacteristic
25C2×Dic194—21no—
26C2×#11715895—21no—
27C2×#11715895—21no—
28C2×#11715895—21no—
29C4⋊Dic97—21no—
30C4⋊Dic97—21no—
31C2×Q8×C97—22no—
32C2×Q8×C97—21no—
33C2×Q8×C97—42no—
34C2×Q8×C97—42no—
35C2×Q8×C97D411yescharacteristic
36C2×D4×C97D411yescharacteristic
37C22×C4×C97—22no—
38C22×C4×C97—21no—
39C4⋊C4×C97—22no—
40C4⋊C4×C97—21no—
41C4⋊C4×C97—21no—
42C4⋊C4×C97D411yescharacteristic
43C4⋊C4×C97D411yescharacteristic
44C4⋊C4×C97D412yes—
45C22⋊C4×C97—22no—
46C22⋊C4×C97—21no—
47C42×C97—21no—
48C42×C97C2311yescharacteristic
49C23×C97—21no—
50Q8×C97—21no—
51Q8×C97—44no—
52Q8×C97—41no—
53Q8×C97—41no—
54Q8×C97SD1612yes—
55D4×C97—21no—
56D4×C97Q1612yes—
57C2×C4×C97—21no—
58C2×C4×C97—21no—
59C2×C4×C97—22no—
60C2×C4×C97—21no—
61C2×C4×C97—44no—
62C2×C4×C97—41no—
63C2×C4×C97—41no—
64C2×C4×C97C2×D411yescharacteristic
65C2×C4×C97C2×D411yescharacteristic
66C2×C4×C97C2×D411yescharacteristic
67C2×Dic97—41no—
68Dic194—42no—
69#11715895—41no—
70#11715895—41no—
71#11715895—41no—
72C22×C97—22no—
73C22×C97—41no—
74C22×C97—41no—
75C22×C97#12011yescharacteristic
76C4×C97—22no—
77C4×C97—21no—
78C4×C97—42no—
79C4×C97—41no—
80C4×C97—41no—
81C4×C97C2×SD1611yescharacteristic
82C4×C97C2×Q1611yescharacteristic
83C4×C97AGL(1,ℤ/8)11yescharacteristic
84C4×C97#13711yescharacteristic
85Dic97—81no—
86C2×C97—42no—
87C2×C97#45011yescharacteristic
88C2×C97#45111yescharacteristic
89C2×C97#45411yescharacteristic
90#1637—971nomaximal, Sylow
91C97#163711yescharacteristic, complemented, Sylow
92D4×Q8D9711yescharacteristic
93#487—971no—
94#477—971no—
95#476—971no—
96#332—971no—
97Q8⋊C8—971no—
98D4⋊C8—971no—
99C22×Q8—21no—
100C4⋊Q8—21no—
101C4⋊Q8—971no—
102C4⋊Q8C2×D9711yescharacteristic
103C22⋊Q8—21no—
104C22⋊Q8—22no—
105C4×Q8C2×D9711yescharacteristic
106C4×D4—21no—
107C4×D4C2×D9711yescharacteristic
108#107—1941no—
109#106—1941no—
110C4⋊C8—971no—
111C4⋊C8—971no—
112#103—1941no—
113#103—1942no—
114#102—1941no—
115C4×C8—971no—
116C2×Q8—22no—
117C2×Q8—21no—
118C2×Q8—42no—
119C2×Q8—42no—
120C2×Q8—1941no—
121C2×Q8C22⋊D9711yescharacteristic
122C2×D4C22⋊D9711yescharacteristic
123C22×C4—22no—
124C22×C4—21no—
125C2×C8—1941no—
126C2×C8—1941no—
127C2×C8—1941no—
128C4⋊C4—22no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4—1941no—
132C4⋊C4—1941no—
133C4⋊C4C22⋊D9712yes—
134C4⋊C4C22⋊D9711yescharacteristic
135C4⋊C4C22⋊D9711yescharacteristic
136C22⋊C4—22no—
137C22⋊C4—21no—
138C42—21no—
139C42C22×D9711yescharacteristic
140C23—21no—
141Q8—21no—
142Q8—44no—
143Q8—41no—
144Q8—41no—
145Q8—3882no—
146Q8#1309549912yes—
147D4—21no—
148D4#1309550112yes—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—22no—
152C2×C4—21no—
153C2×C4—44no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—3881no—
157C2×C4C2×C22⋊D9711yescharacteristic
158C2×C4C2×C22⋊D9711yescharacteristic
159C2×C4C2×C22⋊D9711yescharacteristic
160C8—3881no—
161C8—3881no—
162C8—3881no—
163C22—22no—
164C22—41no—
165C22—41no—
166C22#1569077911yescharacteristic
167C4—22no—
168C4—21no—
169C4—42no—
170C4—41no—
171C4—41no—
172C4—7761no—
173C4C2×#1309549911yescharacteristic
174C4C2×#1309550111yescharacteristic
175C4#1569083611yescharacteristic
176C4#1569084111yescharacteristic
177C2—42no—
178C2#2026088611yescharacteristic
179C2#2026089111yescharacteristic
180C2#2026090811yescharacteristic
subgroupG/H
centerC22#15690779
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
FittingD4×Q8×C97C2
socleC22×C97#120
prime radicalC2×D4×C97D4
hypercenterD4×Q8D97
hypocenterC97#1637
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. #20261087C2semidirect
2. #20261030C2semidirect
4. #20260481C2semidirect
5. Q8⋊#11715895C2semidirect
91. C97#1637semidirect

Maximal subgroups

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

subgroupindex
#202610872
#202610302
#202610252
#202604812
Q8⋊#117158952
D4⋊#117158952
D4×Q8×C972
#163797