Finite Group Databasebeta

← #27141001

Subgroups — #27141001

Order 12416.

234 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 59 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
1D4×Dic194C211yesmaximal, characteristic
2#20261133C211yesmaximal, characteristic, complemented
3#20261021C211yesmaximal, characteristic, complemented
4#20260940C211yesmaximal, characteristic
5#20260482C211yesmaximal, characteristic, complemented
6#20260467C211yesmaximal, characteristic, complemented
7D4⋊C8×C97C211yesmaximal, characteristic
8C22×Dic194—21no—
9C2×#13095499—21no—
10#15690810C2211yescharacteristic
11#15690810C2211yescharacteristic
12#15690809—21no—
13#15690789—21no—
14#15690787—22no—
15C4×Dic194C2211yescharacteristic
16D4×Dic97—21no—
17#15690742—22no—
18#15690742—21no—
19C4⋊#11715895C2211yescharacteristic
20#15690735—21no—
21#15690733—21no—
22C4×D4×C97C2211yescharacteristic
23C4⋊C8×C97C2211yescharacteristic
24C4×C8×C97C2211yescharacteristic
25C2×Dic194—22no—
26C2×Dic194—21no—
27C2×Dic194—21no—
28C2×Dic194—42no—
29C2×Dic194—42no—
30C2×Dic194D411yescharacteristic
31C22×Dic97—22no—
32#13095499—42no—
33C2×#11715895—21no—
34C4⋊Dic97—21no—
35C4⋊Dic97—21no—
36C4⋊Dic97—21no—
37C4⋊Dic97D411yescharacteristic
38C4⋊Dic97D412yes—
39#13095490—22no—
40#13095490—21no—
41C22⋊Dic97—22no—
42C4×Dic97—21no—
43C2×D4×C97D411yescharacteristic
44C22×C4×C97—21no—
45C2×C8×C97—21no—
46C2×C8×C97—21no—
47C4⋊C4×C97D411yescharacteristic
48C22⋊C4×C97—21no—
49C42×C97C2311yescharacteristic
50C23×C97—21no—
51D4×C97—21no—
52D4×C97SD1612yes—
53C2×C4×C97—21no—
54C2×C4×C97—41no—
55C2×C4×C97—41no—
56C2×C4×C97C2×D411yescharacteristic
57C2×C4×C97C2×D411yescharacteristic
58C2×C4×C97C2×D411yescharacteristic
59C2×Dic97—21no—
60C2×Dic97—21no—
61C2×Dic97—22no—
62C2×Dic97—44no—
63C2×Dic97—41no—
64Dic194—21no—
65Dic194—44no—
66Dic194—41no—
67Dic194—41no—
68Dic194—41no—
69Dic194—41no—
70Dic194SD1612yescomplemented
71C8×C97—22no—
72C8×C97—41no—
73#11715895—41no—
74C22×C97—22no—
75C22×C97—41no—
76C22×C97—41no—
77C22×C97#12011yescharacteristic
78C4×C97—21no—
79C4×C97—41no—
80C4×C97C2×SD1611yescharacteristic
81C4×C97C2×SD1611yescharacteristic
82C4×C97AGL(1,ℤ/8)11yescharacteristic
83C4×C97#13711yescharacteristic
84Dic97—22no—
85Dic97—42no—
86Dic97—41no—
87Dic97—81no—
88C2×C97—42no—
89C2×C97#44811yescharacteristic
90C2×C97#45011yescharacteristic
91C2×C97#45611yescharacteristic
92#1651—971nomaximal, Sylow
93C97#165111yescharacteristic, complemented, Sylow
94D4×Q8—971no—
95#498—971no—
96#475—971no—
97#461—971no—
98#332—971no—
99Q8⋊C8—971no—
100D4⋊C8D9711yescharacteristic
101C22×Q8—1941no—
102C2×SD16—1941no—
103C4⋊Q8—971no—
104C4⋊Q8—971no—
105C4⋊Q8—1941no—
106C22⋊Q8—1942no—
107C22⋊Q8—1941no—
108C4×Q8—971no—
109C4×D4—1941no—
110C4×D4C2×D9711yescharacteristic
111#106—1941no—
112#106—1942no—
113C4⋊C8—971no—
114C4⋊C8C2×D9711yescharacteristic
115#103—1941no—
116#103—1941no—
117C4×C8C2×D9711yescharacteristic
118C2×Q8—971no—
119C2×Q8—3882no—
120C2×Q8—3882no—
121C2×Q8—1942no—
122C2×Q8—1941no—
123C2×Q8—1941no—
124C2×D4D38811yescharacteristic
125C22×C4—21no—
126C22×C4—1942no—
127SD16—3882no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—1941no—
131C4⋊C4—971no—
132C4⋊C4—972no—
133C4⋊C4—1942no—
134C4⋊C4—1941no—
135C4⋊C4—1941no—
136C4⋊C4—1941no—
137C4⋊C4—1941no—
138C4⋊C4D38811yescharacteristic
139C22⋊C4—21no—
140C22⋊C4—1942no—
141C42—1941no—
142C42C22×D9711yescharacteristic
143C23—21no—
144Q8—972no—
145Q8—3884no—
146Q8—3881no—
147Q8—3881no—
148Q8—3881no—
149Q8—3881no—
150Q8—1941no—
151D4—21no—
152D4#1309550012yes—
153C2×C4—21no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—3881no—
157C2×C4—3884no—
158C2×C4—1941no—
159C2×C4—1942no—
160C2×C4—1941no—
161C2×C4D4×D9711yescharacteristic
162C2×C4D4×D9711yescharacteristic
163C2×C4C2×D38811yescharacteristic
164C8—22no—
165C8—41no—
166C8—3881no—
167C22—22no—
168C22—41no—
169C22—41no—
170C22#1569078011yescharacteristic
171C4—21no—
172C4—41no—
173C4—7761no—
174C4—3882no—
175C4—3881no—
176C4—1942no—
177C4SD16×D9711yescharacteristic
178C4C2×#1309550011yescharacteristic
179C4#1569083411yescharacteristic
180C4#1569084311yescharacteristic
181C2—42no—
182C2#2026087611yescharacteristic
183C2#2026088711yescharacteristic
184C2#2026091911yescharacteristic
subgroupG/H
centerC22#15690780
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
FittingD4⋊C8×C97C2
socleC22×C97#120
prime radicalC2×D4×C97D4
hypercenterD4⋊C8D97
hypocenterC97#1651
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #20261133C2semidirect
3. #20261021C2semidirect
5. #20260482C2semidirect
6. #20260467C2semidirect
70. Dic194SD16semidirect
93. C97#1651semidirect

Maximal subgroups

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

subgroupindex
D4×Dic1942
#202611332
#202610212
#202609402
#202604822
#202604672
D4⋊C8×C972
#165197