Finite Group Databasebeta

← #38768037

Subgroups — #38768037

Order 25472.

212 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 53 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#27164754C211yesmaximal, characteristic
2#27164364C211yesmaximal, characteristic
3#27164304C211yesmaximal, characteristic, complemented
4#27164229C211yesmaximal, characteristic, complemented
5#27163748C211yesmaximal, characteristic, complemented
6#27163748C211yesmaximal, characteristic, complemented
7#331×C199C211yesmaximal, characteristic
8C2×Dic796—21no—
9#20263955C2211yescharacteristic
10#20263942—21no—
11#20263942—21no—
12#20263936—22no—
13C4×Dic398C2211yescharacteristic
14C4×Dic398C2211yescharacteristic
15D4×Dic199—21no—
16D4×Dic199—21no—
17#20263887—21no—
18#20263880—21no—
19#20263880—21no—
20#20263876—22no—
21C4×C199⋊C8C2211yescharacteristic
22#127×C199C2211yescharacteristic
23C4⋊C8×C199C2211yescharacteristic
24C4⋊C8×C199C2211yescharacteristic
25C2×Dic398—21no—
26C2×Dic398D411yescharacteristic
27C2×Dic398D411yescharacteristic
28C22×Dic199—22no—
29Dic796—42no—
30C2×C199⋊C8—21no—
31C4⋊Dic199—21no—
32C4⋊Dic199—21no—
33C4⋊Dic199D411yescharacteristic
34C4⋊Dic199D411yescharacteristic
35Dic199⋊C4—21no—
36Dic199⋊C4—21no—
37C22⋊Dic199—22no—
38C22⋊Dic199—22no—
39C22⋊Dic199—22no—
40C4×Dic199—21no—
41C4×Dic199—21no—
42C2×D4×C199—21no—
43C2×D4×C199—21no—
44C2×D4×C199D412yes—
45C2×C8×C199—21no—
46C2×C8×C199—21no—
47C42×C199C2311yescharacteristic
48Dic398—21no—
49Dic398—21no—
50Dic398—21no—
51Dic398—21no—
52Dic398—41no—
53Dic398—41no—
54C2×Dic199—21no—
55C2×Dic199—21no—
56C2×Dic199—21no—
57C2×Dic199—21no—
58C2×Dic199—41no—
59C2×Dic199—82no—
60C199⋊C8—41no—
61C23×C199—22no—
62D4×C199—42no—
63D4×C199—41no—
64D4×C199—41no—
65D4×C199—41no—
66D4×C199—41no—
67C2×C4×C199C2×D411yescharacteristic
68C2×C4×C199C2×D411yescharacteristic
69C2×C4×C199C2×D411yescharacteristic
70C8×C199—41no—
71C8×C199—41no—
72C22×C199—42no—
73C22×C199—42no—
74C22×C199—42no—
75C22×C199#12011yescharacteristic
76C4×C199—21no—
77C4×C199#13511yescharacteristic
78C4×C199#13511yescharacteristic
79C4×C199AGL(1,ℤ/8)11yescharacteristic
80C4×C199#13711yescharacteristic
81Dic199—41no—
82Dic199—41no—
83Dic199—41no—
84Dic199—41no—
85Dic199—81no—
86C2×C199—82no—
87C2×C199#44911yescharacteristic
88C2×C199#45211yescharacteristic
89C2×C199#45511yescharacteristic
90C199#164011yescharacteristic, complemented, Sylow
91#1640—1991nomaximal, Sylow
92#559—1991no—
93#486—1991no—
94#475—1991no—
95#462—1991no—
96#331D19911yescharacteristic
97Q8⋊C8—1991no—
98Q8⋊C8—1991no—
99C2×Q16—3981no—
100C4⋊Q8—1991no—
101#127C2×D19911yescharacteristic
102#124—3981no—
103#124—3981no—
104#123—3982no—
105C4×Q8—1991no—
106C4×Q8—1991no—
107C4×D4—3981no—
108C4×D4—3981no—
109#106—3981no—
110C4⋊C8C2×D19911yescharacteristic
111C4⋊C8C2×D19911yescharacteristic
112#103—3981no—
113#103—3981no—
114#102—3982no—
115C4×C8—1991no—
116C2×Q8—3981no—
117C2×Q8—1991no—
118C2×Q8—1991no—
119C2×D4—21no—
120C2×D4—21no—
121C2×D4D79612yes—
122C22×C4—3982no—
123Q16—7962no—
124C2×C8—21no—
125C2×C8—21no—
126C2×C8—3981no—
127C4⋊C4—3981no—
128C4⋊C4—3981no—
129C4⋊C4—3981no—
130C4⋊C4—3981no—
131C4⋊C4—1991no—
132C4⋊C4—1991no—
133C22⋊C4—3982no—
134C22⋊C4—3982no—
135C22⋊C4—3982no—
136C42—3981no—
137C42—3981no—
138C42C22×D19911yescharacteristic
139C23—22no—
140Q8—7961no—
141Q8—7961no—
142Q8—3981no—
143Q8—3981no—
144Q8—3981no—
145Q8—3981no—
146D4—42no—
147D4—41no—
148D4—41no—
149D4—41no—
150D4—41no—
151C2×C4—7961no—
152C2×C4—15922no—
153C2×C4—3981no—
154C2×C4—3981no—
155C2×C4—3981no—
156C2×C4—3981no—
157C2×C4D4×D19911yescharacteristic
158C2×C4D4×D19911yescharacteristic
159C2×C4C2×D79611yescharacteristic
160C8—41no—
161C8—41no—
162C8—7961no—
163C22—42no—
164C22—42no—
165C22—42no—
166C22#2026392511yescharacteristic
167C4—21no—
168C4—7961no—
169C4—7961no—
170C4—7961no—
171C4—7961no—
172C4—15921no—
173C4#2026397411yescharacteristic
174C4#2026397611yescharacteristic
175C4#2026397911yescharacteristic
176C4#2026398411yescharacteristic
177C2—82no—
178C2#2716416411yescharacteristic
179C2#2716418111yescharacteristic
180C2#2716419811yescharacteristic
subgroupG/H
centerC22#20263925
commutator, cosocleC42×C199C23
FrattiniC42C22×D199
Fitting#331×C199C2
socleC22×C199#120
prime radical#127×C199C22
hypercenter#331D199
hypocenterC199#1640
LCSG ▷ C42×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
3. #27164304C2semidirect
4. #27164229C2semidirect
5. #27163748C2semidirect
6. #27163748C2semidirect
90. C199#1640semidirect

Maximal subgroups

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

subgroupindex
#271647542
#271643642
#271643042
#271642292
#271637482
#331×C1992
#1640199