Finite Group Databasebeta

← #37571656

Subgroups — #37571656

Order 24448.

214 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 57 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#27133370C211yesmaximal, characteristic
2#27133043C211yesmaximal, characteristic, complemented
3#27132959C211yesmaximal, characteristic, complemented
4#27132953C211yesmaximal, characteristic
5#27132411C211yesmaximal, characteristic, complemented
6#27132395C211yesmaximal, characteristic, complemented
7D4⋊C8×C191C211yesmaximal, characteristic
8C2×Dic764—22no—
9#20259322C2211yescharacteristic
10#20259322C2211yescharacteristic
11#20259313—21no—
12#20259301—21no—
13#20259299—22no—
14C4×Dic382C2211yescharacteristic
15D4×Dic191—21no—
16C2×C4⋊Dic191—21no—
17#20259256—21no—
18#20259253—21no—
19C4⋊(C191⋊C8)C2211yescharacteristic
20#20259247—21no—
21#20259243—21no—
22C4×D4×C191C2211yescharacteristic
23C4⋊C8×C191C2211yescharacteristic
24C4×C8×C191C2211yescharacteristic
25C2×Dic382—21no—
26C2×Dic382D411yescharacteristic
27C2×Dic382D412yes—
28C22×Dic191—22no—
29Dic764—44no—
30C2×C191⋊C8—21no—
31C4⋊Dic191—22no—
32C4⋊Dic191—21no—
33C4⋊Dic191—21no—
34C4⋊Dic191—21no—
35C4⋊Dic191—21no—
36C4⋊Dic191D411yescharacteristic
37Dic191⋊C4—22no—
38Dic191⋊C4—21no—
39C22⋊Dic191—22no—
40C4×Dic191—21no—
41C2×D4×C191D411yescharacteristic
42C22×C4×C191—21no—
43C2×C8×C191—21no—
44C2×C8×C191—21no—
45C4⋊C4×C191D411yescharacteristic
46C22⋊C4×C191—21no—
47C42×C191C2311yescharacteristic
48Dic382—21no—
49Dic382—21no—
50Dic382—42no—
51Dic382—42no—
52C2×Dic191—21no—
53C2×Dic191—21no—
54C2×Dic191—22no—
55C2×Dic191—41no—
56C2×Dic191—82no—
57C191⋊C8—41no—
58C23×C191—21no—
59D4×C191—21no—
60D4×C191Q1612yescomplemented
61C2×C4×C191—21no—
62C2×C4×C191—41no—
63C2×C4×C191—41no—
64C2×C4×C191C2×D411yescharacteristic
65C2×C4×C191C2×D411yescharacteristic
66C2×C4×C191C2×D411yescharacteristic
67C8×C191—22no—
68C8×C191—41no—
69C22×C191—22no—
70C22×C191—41no—
71C22×C191—41no—
72C22×C191#12011yescharacteristic
73C4×C191—21no—
74C4×C191—41no—
75C4×C191C2×Q1611yescharacteristic
76C4×C191#13511yescharacteristic
77C4×C191#13711yescharacteristic
78C4×C191#13711yescharacteristic
79Dic191—41no—
80Dic191—41no—
81Dic191—42no—
82Dic191—81no—
83C2×C191—42no—
84C2×C191#45111yescharacteristic
85C2×C191#45211yescharacteristic
86C2×C191#45311yescharacteristic
87C191#163211yescharacteristic, complemented, Sylow
88#1632—1911nomaximal, Sylow
89#554—1911no—
90#494—1911no—
91#477—1911no—
92#476—1911no—
93#333—1911no—
94Q8⋊C8—1911no—
95D4⋊C8D19111yescharacteristic
96C2×Q16—3822no—
97C4⋊Q8—1911no—
98C4⋊Q8—1911no—
99#125—3821no—
100C22⋊Q8—3822no—
101C22⋊Q8—3821no—
102C4×Q8—1911no—
103C4×D4—3821no—
104C4×D4C2×D19111yescharacteristic
105C2×C4⋊C4—3821no—
106#107—3821no—
107#106—3821no—
108C4⋊C8—1911no—
109C4⋊C8C2×D19111yescharacteristic
110#103—3821no—
111#102—3821no—
112C4×C8C2×D19111yescharacteristic
113C2×Q8—3821no—
114C2×Q8—1911no—
115C2×Q8—1912no—
116C2×D4D76411yescharacteristic
117C22×C4—21no—
118C22×C4—3822no—
119Q16—7644no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—3821no—
123C4⋊C4—3822no—
124C4⋊C4—3822no—
125C4⋊C4—3821no—
126C4⋊C4—3821no—
127C4⋊C4—3821no—
128C4⋊C4—3821no—
129C4⋊C4—3821no—
130C4⋊C4—1911no—
131C4⋊C4D76411yescharacteristic
132C22⋊C4—21no—
133C22⋊C4—3822no—
134C42—3821no—
135C42C22×D19111yescharacteristic
136C23—21no—
137Q8—3821no—
138Q8—3821no—
139Q8—7642no—
140Q8—7642no—
141D4—21no—
142D4Dic76412yescomplemented
143C2×C4—21no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—3821no—
147C2×C4—3821no—
148C2×C4—3822no—
149C2×C4—15282no—
150C2×C4—7641no—
151C2×C4D4×D19111yescharacteristic
152C2×C4D4×D19111yescharacteristic
153C2×C4C2×D76411yescharacteristic
154C8—22no—
155C8—41no—
156C8—7641no—
157C22—22no—
158C22—41no—
159C22—41no—
160C22#2025929211yescharacteristic
161C4—21no—
162C4—41no—
163C4—15281no—
164C4—7641no—
165C4—7641no—
166C4—7642no—
167C4C2×Dic76411yescharacteristic
168C4#2025934111yescharacteristic
169C4#2025935111yescharacteristic
170C4#2025935511yescharacteristic
171C2—42no—
172C2#2713282111yescharacteristic
173C2#2713282611yescharacteristic
174C2#2713283211yescharacteristic
subgroupG/H
centerC22#20259292
commutator, cosocleC42×C191C23
FrattiniC42C22×D191
FittingD4⋊C8×C191C2
socleC22×C191#120
prime radicalC2×D4×C191D4
hypercenterD4⋊C8D191
hypocenterC191#1632
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. #27133043C2semidirect
3. #27132959C2semidirect
5. #27132411C2semidirect
6. #27132395C2semidirect
60. D4×C191Q16semidirect
87. C191#1632semidirect
142. D4Dic764semidirect

Maximal subgroups

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

subgroupindex
#271333702
#271330432
#271329592
#271329532
#271324112
#271323952
D4⋊C8×C1912
#1632191