Finite Group Databasebeta

← #38767924

Subgroups — #38767924

Order 25472.

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

normal subgroups only

This group has 236 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, complemented
2#27164391C211yesmaximal, characteristic, complemented
3#27164219C211yesmaximal, characteristic, complemented
4#27164213C211yesmaximal, characteristic
5#27163766C211yesmaximal, characteristic, complemented
6#27163750C211yesmaximal, characteristic, complemented
7D4⋊C8×C199C211yesmaximal, characteristic, complemented
8C2×#15691875—21no—
9C2×#15691875—22no—
10C2×#15691871—21no—
11#20263955C2211yescharacteristic
12#20263952C2211yescharacteristic
13#20263946—21no—
14#20263934—21no—
15#20263932—22no—
16C4×Dic398C2211yescharacteristic
17D4×Dic199—21no—
18C2×C4⋊Dic199—21no—
19C4⋊(C199⋊C8)C2211yescharacteristic
20#20263877—21no—
21#20263875—21no—
22C4×D4×C199C2211yescharacteristic
23C4⋊C8×C199C2211yescharacteristic
24C4×C8×C199C2211yescharacteristic
25C2×Dic398D411yescharacteristic
26C2×Dic398D412yes—
27C2×D796—21no—
28C2×D796—21no—
29C2×D796—21no—
30C22×Dic199—22no—
31#15691875—44no—
32#15691875—42no—
33#15691871—42no—
34C2×C199⋊C8—21no—
35C4⋊Dic199—22no—
36C4⋊Dic199—21no—
37C4⋊Dic199—21no—
38C4⋊Dic199D411yescharacteristic
39Dic199⋊C4—22no—
40Dic199⋊C4—21no—
41C22⋊Dic199—22no—
42C4×Dic199—21no—
43C2×D4×C199D411yescharacteristic
44C22×C4×C199—21no—
45C2×C8×C199—21no—
46C2×C8×C199—21no—
47C4⋊C4×C199D411yescharacteristic
48C22⋊C4×C199—21no—
49C42×C199C2311yescharacteristic
50C22×D199—41no—
51Dic398—21no—
52Dic398—21no—
53Dic398—42no—
54D796—42no—
55D796—41no—
56D796—41no—
57D796—41no—
58D796—41no—
59C2×Dic199—21no—
60C2×Dic199—21no—
61C2×Dic199—22no—
62C2×Dic199—82no—
63C199⋊C8—41no—
64C23×C199—21no—
65D4×C199—21no—
66D4×C199SD1612yescomplemented
67C2×C4×C199—21no—
68C2×C4×C199—41no—
69C2×C4×C199—41no—
70C2×C4×C199C2×D411yescharacteristic
71C2×C4×C199C2×D411yescharacteristic
72C2×C4×C199C2×D411yescharacteristic
73C8×C199—22no—
74C8×C199—41no—
75C22×C199—22no—
76C22×C199—41no—
77C22×C199—41no—
78C22×C199#12011yescharacteristic
79C2×D199—81no—
80C2×D199—81no—
81C2×D199—81no—
82C4×C199—21no—
83C4×C199—41no—
84C4×C199C2×SD1611yescharacteristic
85C4×C199#13511yescharacteristic
86C4×C199AGL(1,ℤ/8)11yescharacteristic
87C4×C199#13711yescharacteristic
88Dic199—41no—
89Dic199—41no—
90Dic199—42no—
91C2×C199—42no—
92C2×C199#44811yescharacteristic
93C2×C199#44911yescharacteristic
94C2×C199#45311yescharacteristic
95D199—161no—
96C199#162211yescharacteristic, complemented, Sylow
97#1622—1991nomaximal, Sylow
98#554—1991no—
99#492—1991no—
100#460—1991no—
101#459—1991no—
102#333—1991no—
103Q8⋊C8—1991no—
104D4⋊C8D19911yescharacteristic, complemented
105C2×SD16—3981no—
106C2×SD16—3982no—
107C2×D8—3981no—
108C4⋊Q8—1991no—
109#127—1991no—
110#125—3981no—
111C22⋊Q8—3982no—
112C22⋊Q8—3981no—
113C4×Q8—1991no—
114C4×D4—3981no—
115C4×D4C2×D19911yescharacteristic
116C2×C4⋊C4—3981no—
117C4⋊C8—1991no—
118C4⋊C8C2×D19911yescharacteristic
119#102—3981no—
120#102—3981no—
121C4×C8C2×D19911yescharacteristic
122C2×Q8—1992no—
123C2×Q8—1991no—
124C2×D4—3981no—
125C2×D4—3981no—
126C2×D4—3981no—
127C2×D4D79611yescharacteristic
128C22×C4—21no—
129C22×C4—3982no—
130SD16—7964no—
131SD16—7962no—
132D8—7962no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—3981no—
136C4⋊C4—3982no—
137C4⋊C4—3982no—
138C4⋊C4—3981no—
139C4⋊C4—3981no—
140C4⋊C4—3981no—
141C4⋊C4—1991no—
142C4⋊C4D79611yescharacteristic
143C22⋊C4—21no—
144C22⋊C4—3982no—
145C42—3981no—
146C42C22×D19911yescharacteristic
147C23—21no—
148C23—7961no—
149Q8—7962no—
150Q8—3981no—
151Q8—3981no—
152D4—21no—
153D4—7962no—
154D4—7961no—
155D4—7961no—
156D4—7961no—
157D4—7961no—
158D4#1569187512yescomplemented
159C2×C4—21no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—15922no—
163C2×C4—3981no—
164C2×C4—3981no—
165C2×C4—3982no—
166C2×C4D4×D19911yescharacteristic
167C2×C4D4×D19911yescharacteristic
168C2×C4C2×D79611yescharacteristic
169C8—22no—
170C8—41no—
171C8—7961no—
172C22—22no—
173C22—41no—
174C22—41no—
175C22—15921no—
176C22—15921no—
177C22—15921no—
178C22#2026392511yescharacteristic
179C4—21no—
180C4—41no—
181C4—7961no—
182C4—7961no—
183C4—7962no—
184C4C2×#1569187511yescharacteristic
185C4#2026397611yescharacteristic
186C4#2026397911yescharacteristic
187C4#2026398811yescharacteristic
188C2—42no—
189C2—31841no—
190C2#2716415911yescharacteristic
191C2#2716416411yescharacteristic
192C2#2716418711yescharacteristic
subgroupG/H
centerC22#20263925
commutator, cosocleC42×C199C23
FrattiniC42C22×D199
FittingD4⋊C8×C199C2
socleC22×C199#120
prime radical#27164213C2
hypercenterD4⋊C8D199
hypocenterC199#1622
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
1. #27164725C2semidirect
2. #27164391C2semidirect
3. #27164219C2semidirect
5. #27163766C2semidirect
6. #27163750C2semidirect
7. D4⋊C8×C199C2semidirect
66. D4×C199SD16semidirect
96. C199#1622semidirect
104. D4⋊C8D199semidirect
158. D4#15691875semidirect

Maximal subgroups

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

subgroupindex
#271647252
#271643912
#271642192
#271642132
#271637662
#271637502
D4⋊C8×C1992
#1622199