Finite Group Databasebeta

← #37571772

Subgroups — #37571772

Order 24448.

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×Dic382C211yesmaximal, characteristic
2#27133068C211yesmaximal, characteristic, complemented
3#27132943C211yesmaximal, characteristic
4#27132874C211yesmaximal, characteristic, complemented
5#27132410C211yesmaximal, characteristic, complemented
6#27132395C211yesmaximal, characteristic, complemented
7D4⋊C8×C191C211yesmaximal, characteristic
8C22×Dic382—21no—
9C2×Dic764—21no—
10#20259322C2211yescharacteristic
11#20259322C2211yescharacteristic
12#20259321—21no—
13#20259301—21no—
14#20259299—22no—
15C4×Dic382C2211yescharacteristic
16D4×Dic191—21no—
17#20259256—22no—
18#20259255—21no—
19C4⋊(C191⋊C8)C2211yescharacteristic
20#20259247—21no—
21#20259243—21no—
22C4×D4×C191C2211yescharacteristic
23C4⋊C8×C191C2211yescharacteristic
24C4×C8×C191C2211yescharacteristic
25C2×Dic382—22no—
26C2×Dic382—21no—
27C2×Dic382—21no—
28C2×Dic382—42no—
29C2×Dic382—42no—
30C2×Dic382D411yescharacteristic
31C22×Dic191—22no—
32Dic764—42no—
33C2×C191⋊C8—21no—
34C4⋊Dic191—21no—
35C4⋊Dic191—21no—
36C4⋊Dic191—21no—
37C4⋊Dic191D411yescharacteristic
38C4⋊Dic191D412yes—
39Dic191⋊C4—22no—
40Dic191⋊C4—21no—
41C22⋊Dic191—22no—
42C4×Dic191—21no—
43C2×D4×C191D411yescharacteristic
44C22×C4×C191—21no—
45C2×C8×C191—21no—
46C2×C8×C191—21no—
47C4⋊C4×C191D411yescharacteristic
48C22⋊C4×C191—21no—
49C42×C191C2311yescharacteristic
50Dic382—21no—
51Dic382—44no—
52Dic382—41no—
53Dic382—41no—
54Dic382—41no—
55Dic382—41no—
56Dic382D812yes—
57C2×Dic191—21no—
58C2×Dic191—21no—
59C2×Dic191—22no—
60C2×Dic191—44no—
61C2×Dic191—41no—
62C191⋊C8—41no—
63C23×C191—21no—
64D4×C191—21no—
65D4×C191Q1612yescomplemented
66C2×C4×C191—21no—
67C2×C4×C191—41no—
68C2×C4×C191—41no—
69C2×C4×C191C2×D411yescharacteristic
70C2×C4×C191C2×D411yescharacteristic
71C2×C4×C191C2×D411yescharacteristic
72C8×C191—22no—
73C8×C191—41no—
74C22×C191—22no—
75C22×C191—41no—
76C22×C191—41no—
77C22×C191#12011yescharacteristic
78C4×C191—21no—
79C4×C191—41no—
80C4×C191C2×D811yescharacteristic
81C4×C191C2×Q1611yescharacteristic
82C4×C191AGL(1,ℤ/8)11yescharacteristic
83C4×C191#13711yescharacteristic
84Dic191—22no—
85Dic191—42no—
86Dic191—41no—
87Dic191—81no—
88C2×C191—42no—
89C2×C191#44711yescharacteristic
90C2×C191#45111yescharacteristic
91C2×C191#45611yescharacteristic
92C191#164911yescharacteristic, complemented, Sylow
93#1649—1911nomaximal, Sylow
94D4×Q8—1911no—
95#500—1911no—
96#474—1911no—
97#462—1911no—
98#332—1911no—
99Q8⋊C8—1911no—
100D4⋊C8D19111yescharacteristic
101C22×Q8—3821no—
102C2×Q16—3821no—
103C4⋊Q8—3821no—
104C4⋊Q8—1911no—
105C4⋊Q8—1911no—
106C22⋊Q8—3822no—
107C22⋊Q8—3821no—
108C4×Q8—1911no—
109C4×D4—3821no—
110C4×D4C2×D19111yescharacteristic
111#107—3822no—
112#107—3821no—
113C4⋊C8—1911no—
114C4⋊C8C2×D19111yescharacteristic
115#103—3821no—
116#102—3821no—
117C4×C8C2×D19111yescharacteristic
118C2×Q8—3822no—
119C2×Q8—3821no—
120C2×Q8—3821no—
121C2×Q8—1911no—
122C2×Q8—7642no—
123C2×Q8—7642no—
124C2×D4D76411yescharacteristic
125C22×C4—21no—
126C22×C4—3822no—
127Q16—7642no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—3821no—
131C4⋊C4—3822no—
132C4⋊C4—3821no—
133C4⋊C4—3821no—
134C4⋊C4—3821no—
135C4⋊C4—3821no—
136C4⋊C4—1911no—
137C4⋊C4—1912no—
138C4⋊C4D76411yescharacteristic
139C22⋊C4—21no—
140C22⋊C4—3822no—
141C42—3821no—
142C42C22×D19111yescharacteristic
143C23—21no—
144Q8—3821no—
145Q8—1912no—
146Q8—7644no—
147Q8—7641no—
148Q8—7641no—
149Q8—7641no—
150Q8—7641no—
151D4—21no—
152D4Dic76412yescomplemented
153C2×C4—21no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—3821no—
157C2×C4—3821no—
158C2×C4—3822no—
159C2×C4—7644no—
160C2×C4—7641no—
161C2×C4D4×D19111yescharacteristic
162C2×C4D4×D19111yescharacteristic
163C2×C4C2×D76411yescharacteristic
164C8—22no—
165C8—41no—
166C8—7641no—
167C22—22no—
168C22—41no—
169C22—41no—
170C22#2025929211yescharacteristic
171C4—21no—
172C4—41no—
173C4—3822no—
174C4—15281no—
175C4—7642no—
176C4—7641no—
177C4D8×D19111yescharacteristic
178C4C2×Dic76411yescharacteristic
179C4#2025934911yescharacteristic
180C4#2025935511yescharacteristic
181C2—42no—
182C2#2713279811yescharacteristic
183C2#2713282111yescharacteristic
184C2#2713284711yescharacteristic
subgroupG/H
centerC22#20259292
commutator, cosocleC42×C191C23
FrattiniC42C22×D191
FittingD4⋊C8×C191C2
socleC22×C191#120
prime radicalC2×D4×C191D4
hypercenterD4⋊C8D191
hypocenterC191#1649
LCSG ▷ C42×C191 ▷ C22×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #27133068C2semidirect
4. #27132874C2semidirect
5. #27132410C2semidirect
6. #27132395C2semidirect
65. D4×C191Q16semidirect
92. C191#1649semidirect
152. D4Dic764semidirect

Maximal subgroups

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

subgroupindex
D4×Dic3822
#271330682
#271329432
#271328742
#271324102
#271323952
D4⋊C8×C1912
#1649191