Finite Group Databasebeta

← #38767986

Subgroups — #38767986

Order 25472.

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#27164725C211yesmaximal, characteristic
2#27164398C211yesmaximal, characteristic, complemented
3#27164314C211yesmaximal, characteristic, complemented
4#27164308C211yesmaximal, characteristic
5#27163766C211yesmaximal, characteristic, complemented
6#27163750C211yesmaximal, characteristic, complemented
7D4⋊C8×C199C211yesmaximal, characteristic
8C2×Dic796—22no—
9#20263955C2211yescharacteristic
10#20263955C2211yescharacteristic
11#20263946—21no—
12#20263934—21no—
13#20263932—22no—
14C4×Dic398C2211yescharacteristic
15D4×Dic199—21no—
16C2×C4⋊Dic199—21no—
17#20263889—21no—
18#20263886—21no—
19C4⋊(C199⋊C8)C2211yescharacteristic
20#20263880—21no—
21#20263876—21no—
22C4×D4×C199C2211yescharacteristic
23C4⋊C8×C199C2211yescharacteristic
24C4×C8×C199C2211yescharacteristic
25C2×Dic398—21no—
26C2×Dic398D411yescharacteristic
27C2×Dic398D412yes—
28C22×Dic199—22no—
29Dic796—44no—
30C2×C199⋊C8—21no—
31C4⋊Dic199—22no—
32C4⋊Dic199—21no—
33C4⋊Dic199—21no—
34C4⋊Dic199—21no—
35C4⋊Dic199—21no—
36C4⋊Dic199D411yescharacteristic
37Dic199⋊C4—22no—
38Dic199⋊C4—21no—
39C22⋊Dic199—22no—
40C4×Dic199—21no—
41C2×D4×C199D411yescharacteristic
42C22×C4×C199—21no—
43C2×C8×C199—21no—
44C2×C8×C199—21no—
45C4⋊C4×C199D411yescharacteristic
46C22⋊C4×C199—21no—
47C42×C199C2311yescharacteristic
48Dic398—21no—
49Dic398—21no—
50Dic398—42no—
51Dic398—42no—
52C2×Dic199—21no—
53C2×Dic199—21no—
54C2×Dic199—22no—
55C2×Dic199—41no—
56C2×Dic199—82no—
57C199⋊C8—41no—
58C23×C199—21no—
59D4×C199—21no—
60D4×C199Q1612yescomplemented
61C2×C4×C199—21no—
62C2×C4×C199—41no—
63C2×C4×C199—41no—
64C2×C4×C199C2×D411yescharacteristic
65C2×C4×C199C2×D411yescharacteristic
66C2×C4×C199C2×D411yescharacteristic
67C8×C199—22no—
68C8×C199—41no—
69C22×C199—22no—
70C22×C199—41no—
71C22×C199—41no—
72C22×C199#12011yescharacteristic
73C4×C199—21no—
74C4×C199—41no—
75C4×C199C2×Q1611yescharacteristic
76C4×C199#13511yescharacteristic
77C4×C199#13711yescharacteristic
78C4×C199#13711yescharacteristic
79Dic199—41no—
80Dic199—41no—
81Dic199—42no—
82Dic199—81no—
83C2×C199—42no—
84C2×C199#45111yescharacteristic
85C2×C199#45211yescharacteristic
86C2×C199#45311yescharacteristic
87C199#163211yescharacteristic, complemented, Sylow
88#1632—1991nomaximal, Sylow
89#554—1991no—
90#494—1991no—
91#477—1991no—
92#476—1991no—
93#333—1991no—
94Q8⋊C8—1991no—
95D4⋊C8D19911yescharacteristic
96C2×Q16—3982no—
97C4⋊Q8—1991no—
98C4⋊Q8—1991no—
99#125—3981no—
100C22⋊Q8—3982no—
101C22⋊Q8—3981no—
102C4×Q8—1991no—
103C4×D4—3981no—
104C4×D4C2×D19911yescharacteristic
105C2×C4⋊C4—3981no—
106#107—3981no—
107#106—3981no—
108C4⋊C8—1991no—
109C4⋊C8C2×D19911yescharacteristic
110#103—3981no—
111#102—3981no—
112C4×C8C2×D19911yescharacteristic
113C2×Q8—3981no—
114C2×Q8—1992no—
115C2×Q8—1991no—
116C2×D4D79611yescharacteristic
117C22×C4—21no—
118C22×C4—3982no—
119Q16—7964no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—3981no—
123C4⋊C4—3982no—
124C4⋊C4—3982no—
125C4⋊C4—3981no—
126C4⋊C4—3981no—
127C4⋊C4—3981no—
128C4⋊C4—3981no—
129C4⋊C4—3981no—
130C4⋊C4—1991no—
131C4⋊C4D79611yescharacteristic
132C22⋊C4—21no—
133C22⋊C4—3982no—
134C42—3981no—
135C42C22×D19911yescharacteristic
136C23—21no—
137Q8—7962no—
138Q8—7962no—
139Q8—3981no—
140Q8—3981no—
141D4—21no—
142D4Dic79612yescomplemented
143C2×C4—21no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—7961no—
147C2×C4—15922no—
148C2×C4—3981no—
149C2×C4—3981no—
150C2×C4—3982no—
151C2×C4D4×D19911yescharacteristic
152C2×C4D4×D19911yescharacteristic
153C2×C4C2×D79611yescharacteristic
154C8—22no—
155C8—41no—
156C8—7961no—
157C22—22no—
158C22—41no—
159C22—41no—
160C22#2026392511yescharacteristic
161C4—21no—
162C4—41no—
163C4—7961no—
164C4—7961no—
165C4—7962no—
166C4—15921no—
167C4C2×Dic79611yescharacteristic
168C4#2026397411yescharacteristic
169C4#2026398411yescharacteristic
170C4#2026398811yescharacteristic
171C2—42no—
172C2#2716417611yescharacteristic
173C2#2716418111yescharacteristic
174C2#2716418711yescharacteristic
subgroupG/H
centerC22#20263925
commutator, cosocleC42×C199C23
FrattiniC42C22×D199
FittingD4⋊C8×C199C2
socleC22×C199#120
prime radicalC2×D4×C199D4
hypercenterD4⋊C8D199
hypocenterC199#1632
LCSG ▷ C42×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #27164398C2semidirect
3. #27164314C2semidirect
5. #27163766C2semidirect
6. #27163750C2semidirect
60. D4×C199Q16semidirect
87. C199#1632semidirect
142. D4Dic796semidirect

Maximal subgroups

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

subgroupindex
#271647252
#271643982
#271643142
#271643082
#271637662
#271637502
D4⋊C8×C1992
#1632199