Finite Group Databasebeta

← #38767905

Subgroups — #38767905

Order 25472.

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

normal subgroups only

This group has 336 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#27164653C211yesmaximal, characteristic, complemented
2#27164394C211yesmaximal, characteristic, complemented
3#27164214C212yesmaximal, complemented
4#27163745C212yesmaximal, complemented
5#331×C199C211yesmaximal, characteristic, complemented
6C2×D4×D199—22no—
7C2×D1592—22no—
8C2×#15691871—22no—
9#20263952C2211yescharacteristic, complemented
10#20263926—22no—
11#20263923—22no—
12C4×D796C2212yes—
13#20263877—22no—
14C4×C199⋊C8C2211yescharacteristic, complemented
15#127×C199C2211yescharacteristic
16C4⋊C8×C199C2212yescomplemented
17C23×D199—22no—
18C2×D796—21no—
19C2×D796—22no—
20C2×D796D412yes—
21C2×C22⋊D199—24no—
22D4×D199—44no—
23D4×D199—44no—
24C2×C4×D199—22no—
25D1592—44no—
26#15691871—44no—
27C2×C199⋊C8—21no—
28C4⋊Dic199D412yes—
29C22⋊Dic199—22no—
30(C2×D199)⋊C4—22no—
31C2×D4×C199—22no—
32C2×D4×C199D412yes—
33C2×C8×C199—22no—
34C42×C199C2311yescharacteristic
35C22×D199—22no—
36C22×D199—48no—
37C22×D199—41no—
38C22×D199—82no—
39D796—22no—
40D796—42no—
41D796—42no—
42D796—42no—
43D796D814yescomplemented
44C22⋊D199—48no—
45C2×Dic199—22no—
46C4×D199—42no—
47C4×D199—42no—
48C199⋊C8—22no—
49C23×C199—22no—
50D4×C199—42no—
51D4×C199—42no—
52D4×C199—42no—
53C2×C4×C199C2×D412yes—
54C2×C4×C199C2×D411yescharacteristic
55C8×C199—42no—
56C22×C199—44no—
57C22×C199—42no—
58C22×C199#12011yescharacteristic
59C2×D199—24no—
60C2×D199—42no—
61C2×D199—42no—
62C2×D199—88no—
63C2×D199—82no—
64C2×D199—81no—
65C4×C199—21no—
66C4×C199C2×D812yes—
67C4×C199AGL(1,ℤ/8)12yes—
68Dic199—42no—
69C2×C199—82no—
70C2×C199#44712yes—
71C2×C199#45311yescharacteristic
72D199—44no—
73D199—161no—
74C199#161911yescharacteristic, complemented, Sylow
75#1619—1991nomaximal, Sylow
76D42—1991no—
77#493—1991no—
78#459—1992no—
79#331D19911yescharacteristic, complemented
80D4⋊C8—1992no—
81C22×D4—3982no—
82C2×D8—3982no—
83C2×D8—3982no—
84#127—1991no—
85#127C2×D19911yescharacteristic
86#121—3982no—
87#120—3982no—
88C4×D4—1992no—
89C4⋊C8C2×D19912yescomplemented
90#102—3982no—
91C4×C8—1991no—
92C24—3982no—
93C2×D4—22no—
94C2×D4—7964no—
95C2×D4—7964no—
96C2×D4—3984no—
97C2×D4—3982no—
98C2×D4—3981no—
99C2×D4—1992no—
100C2×D4D79612yes—
101C22×C4—3982no—
102D8—7964no—
103D8—7964no—
104C2×C8—22no—
105C2×C8—3981no—
106C4⋊C4—1992no—
107C22⋊C4—3982no—
108C22⋊C4—3982no—
109C42C22×D19911yescharacteristic
110C23—22no—
111C23—7968no—
112C23—7961no—
113C23—15922no—
114C23—3982no—
115D4—42no—
116D4—42no—
117D4—42no—
118D4—7962no—
119D4—7968no—
120D4—7962no—
121D4—7962no—
122D4—3982no—
123D4—1994no—
124C2×C4—7962no—
125C2×C4—7962no—
126C2×C4—3982no—
127C2×C4D4×D19912yes—
128C2×C4C2×D79611yescharacteristic
129C8—42no—
130C8—3982no—
131C22—42no—
132C22—44no—
133C22—7962no—
134C22—7962no—
135C22—15928no—
136C22—15921no—
137C22—15922no—
138C22—3984no—
139C22#2026392511yescharacteristic
140C4—21no—
141C4—7962no—
142C4D8×D19912yes—
143C4#2026398212yes—
144C2—82no—
145C2—7964no—
146C2—31841no—
147C2#2716415312yes—
148C2#2716418811yescharacteristic
subgroupG/H
centerC22#20263925
commutator, cosocleC42×C199C23
FrattiniC42C22×D199
Fitting#331×C199C2
socleC22×C199#120
hypercenter#331D199
hypocenterC199#1619
LCSG ▷ C42×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
1. #27164653C2semidirect
2. #27164394C2semidirect
3. #27164214C2semidirect
4. #27163745C2semidirect
5. #331×C199C2semidirect
9. #20263952C22semidirect
14. C4×C199⋊C8C22semidirect
16. C4⋊C8×C199C22semidirect
43. D796D8semidirect
74. C199#1619semidirect
79. #331D199semidirect
89. C4⋊C8C2×D199semidirect

Maximal subgroups

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

subgroupindex
#271646532
#271643942
#271642142
#271637452
#331×C1992
#1619199