Finite Group Databasebeta

← #27140999

Subgroups — #27140999

Order 12416.

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

normal subgroups only

This group has 234 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#20261131C211yesmaximal, characteristic, complemented
2#20261020C211yesmaximal, characteristic, complemented
3#20260940C211yesmaximal, characteristic
4#20260481C211yesmaximal, characteristic, complemented
5Q8⋊#11715895C211yesmaximal, characteristic, complemented
6D4⋊#11715895C211yesmaximal, characteristic
7D4×Q8×C97C211yesmaximal, characteristic
8C2×#13095499—21no—
9#15690810C2211yescharacteristic
10#15690741—22no—
11#15690741—21no—
12C4⋊#11715895C2211yescharacteristic
13C4⋊#11715895C2211yescharacteristic
14#15690734—21no—
15#15690733—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—
26#13095499—42no—
27C2×#11715895—21no—
28C2×#11715895—21no—
29C2×#11715895—21no—
30C4⋊Dic97—21no—
31C4⋊Dic97—21no—
32C2×Q8×C97—22no—
33C2×Q8×C97—21no—
34C2×Q8×C97—42no—
35C2×Q8×C97—42no—
36C2×Q8×C97D411yescharacteristic
37C2×D4×C97D411yescharacteristic
38C22×C4×C97—22no—
39C22×C4×C97—21no—
40C4⋊C4×C97—22no—
41C4⋊C4×C97—21no—
42C4⋊C4×C97—21no—
43C4⋊C4×C97D411yescharacteristic
44C4⋊C4×C97D411yescharacteristic
45C4⋊C4×C97D412yes—
46C22⋊C4×C97—22no—
47C22⋊C4×C97—21no—
48C42×C97—21no—
49C42×C97C2311yescharacteristic
50C23×C97—21no—
51Q8×C97—21no—
52Q8×C97—44no—
53Q8×C97—41no—
54Q8×C97—41no—
55Q8×C97SD1612yescomplemented
56D4×C97—21no—
57D4×C97SD1612yes—
58C2×C4×C97—21no—
59C2×C4×C97—21no—
60C2×C4×C97—22no—
61C2×C4×C97—21no—
62C2×C4×C97—44no—
63C2×C4×C97—41no—
64C2×C4×C97—41no—
65C2×C4×C97C2×D411yescharacteristic
66C2×C4×C97C2×D411yescharacteristic
67C2×C4×C97C2×D411yescharacteristic
68C2×Dic97—41no—
69Dic194—41no—
70Dic194—41no—
71#11715895—22no—
72#11715895—41no—
73#11715895—41no—
74C22×C97—22no—
75C22×C97—41no—
76C22×C97—41no—
77C22×C97#12011yescharacteristic
78C4×C97—22no—
79C4×C97—21no—
80C4×C97—42no—
81C4×C97—41no—
82C4×C97—41no—
83C4×C97C2×SD1611yescharacteristic
84C4×C97C2×SD1611yescharacteristic
85C4×C97AGL(1,ℤ/8)11yescharacteristic
86C4×C97#13711yescharacteristic
87Dic97—81no—
88C2×C97—42no—
89C2×C97#44811yescharacteristic
90C2×C97#45011yescharacteristic
91C2×C97#45611yescharacteristic
92#1651—971nomaximal, Sylow
93C97#165111yescharacteristic, complemented, Sylow
94D4×Q8D9711yescharacteristic
95#498—971no—
96#475—971no—
97#461—971no—
98#332—971no—
99Q8⋊C8—971no—
100D4⋊C8—971no—
101C22×Q8—21no—
102C2×SD16—1941no—
103C4⋊Q8—21no—
104C4⋊Q8—971no—
105C4⋊Q8C2×D9711yescharacteristic
106C22⋊Q8—21no—
107C22⋊Q8—22no—
108C4×Q8C2×D9711yescharacteristic
109C4×D4—21no—
110C4×D4C2×D9711yescharacteristic
111#106—1941no—
112#106—1942no—
113C4⋊C8—971no—
114C4⋊C8—971no—
115#103—1941no—
116#103—1941no—
117C4×C8—971no—
118C2×Q8—22no—
119C2×Q8—21no—
120C2×Q8—42no—
121C2×Q8—42no—
122C2×Q8—1941no—
123C2×Q8C22⋊D9711yescharacteristic
124C2×D4C22⋊D9711yescharacteristic
125C22×C4—22no—
126C22×C4—21no—
127SD16—3882no—
128C2×C8—1941no—
129C2×C8—1941no—
130C2×C8—1941no—
131C4⋊C4—22no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—1941no—
135C4⋊C4—1941no—
136C4⋊C4C22⋊D9711yescharacteristic
137C4⋊C4C22⋊D9711yescharacteristic
138C4⋊C4C22⋊D9712yes—
139C22⋊C4—22no—
140C22⋊C4—21no—
141C42—21no—
142C42C22×D9711yescharacteristic
143C23—21no—
144Q8—21no—
145Q8—44no—
146Q8—41no—
147Q8—41no—
148Q8—3881no—
149Q8—3881no—
150Q8#1309549912yescomplemented
151D4—21no—
152D4#1309549812yes—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—22no—
156C2×C4—21no—
157C2×C4—44no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—3881no—
161C2×C4C2×C22⋊D9711yescharacteristic
162C2×C4C2×C22⋊D9711yescharacteristic
163C2×C4C2×C22⋊D9711yescharacteristic
164C8—3881no—
165C8—3881no—
166C8—1942no—
167C22—22no—
168C22—41no—
169C22—41no—
170C22#1569077911yescharacteristic
171C4—22no—
172C4—21no—
173C4—42no—
174C4—41no—
175C4—41no—
176C4—7761no—
177C4C2×#1309549811yescharacteristic
178C4C2×#1309549911yescharacteristic
179C4#1569083611yescharacteristic
180C4#1569084111yescharacteristic
181C2—42no—
182C2#2026087411yescharacteristic
183C2#2026088611yescharacteristic
184C2#2026091811yescharacteristic
subgroupG/H
centerC22#15690779
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
FittingD4×Q8×C97C2
socleC22×C97#120
prime radicalC2×D4×C97D4
hypercenterD4×Q8D97
hypocenterC97#1651
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. #20261131C2semidirect
2. #20261020C2semidirect
4. #20260481C2semidirect
5. Q8⋊#11715895C2semidirect
55. Q8×C97SD16semidirect
93. C97#1651semidirect
150. Q8#13095499semidirect

Maximal subgroups

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

subgroupindex
#202611312
#202610202
#202609402
#202604812
Q8⋊#117158952
D4⋊#117158952
D4×Q8×C972
#165197