Finite Group Databasebeta

← #38768065

Subgroups — #38768065

Order 25472.

208 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 55 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#27164729C211yesmaximal, characteristic
2#27164368C211yesmaximal, characteristic, complemented
3#27164299C211yesmaximal, characteristic
4#27164229C211yesmaximal, characteristic, complemented
5#27163748C211yesmaximal, characteristic, complemented
6#27163745C211yesmaximal, characteristic
7#333×C199C211yesmaximal, characteristic, complemented
8C2×Dic796—21no—
9#20263955C2211yescharacteristic
10#20263946—21no—
11#20263935—21no—
12#20263931—22no—
13C4×Dic398C2211yescharacteristic
14C4×D796C2211yescharacteristic
15#20263917—21no—
16C4⋊C4×D199—21no—
17#20263889—21no—
18#20263880—21no—
19#20263879—22no—
20#20263877—21no—
21C4×C199⋊C8C2211yescharacteristic
22C4⋊Q8×C199C2211yescharacteristic
23C4⋊C8×C199C2211yescharacteristic
24C4⋊C8×C199C2211yescharacteristic
25C2×Dic398—21no—
26C2×Dic398D411yescharacteristic
27C2×D796D411yescharacteristic
28C2×C4×D199—22no—
29C2×C4×D199—21no—
30Dic796—42no—
31C2×C199⋊C8—21no—
32C4⋊Dic199—22no—
33C4⋊Dic199—21no—
34C4⋊Dic199—21no—
35C4⋊Dic199D411yescharacteristic
36C4⋊Dic199D411yescharacteristic
37Dic199⋊C4—22no—
38Dic199⋊C4—21no—
39(C2×D199)⋊C4—22no—
40(C2×D199)⋊C4—21no—
41C4×Dic199—21no—
42C2×Q8×C199D412yes—
43C2×C8×C199—21no—
44C2×C8×C199—21no—
45C4⋊C4×C199—21no—
46C4⋊C4×C199—21no—
47C42×C199C2311yescharacteristic
48C22×D199—21no—
49Dic398—21no—
50Dic398—21no—
51Dic398—41no—
52Dic398—41no—
53D796—21no—
54D796Q1612yescomplemented
55C2×Dic199—21no—
56C2×Dic199—21no—
57C2×Dic199—21no—
58C2×Dic199—41no—
59C4×D199—41no—
60C4×D199—41no—
61C4×D199—82no—
62C199⋊C8—41no—
63Q8×C199—42no—
64C2×C4×C199—22no—
65C2×C4×C199C2×D411yescharacteristic
66C2×C4×C199C2×D411yescharacteristic
67C2×C4×C199C2×D411yescharacteristic
68C8×C199—41no—
69C8×C199—41no—
70C22×C199#12011yescharacteristic
71C2×D199—22no—
72C2×D199—41no—
73C2×D199—41no—
74C4×C199—21no—
75C4×C199—42no—
76C4×C199C2×Q1611yescharacteristic
77C4×C199#13511yescharacteristic
78C4×C199AGL(1,ℤ/8)11yescharacteristic
79C4×C199#13711yescharacteristic
80Dic199—41no—
81Dic199—41no—
82Dic199—41no—
83Dic199—81no—
84C2×C199#44911yescharacteristic
85C2×C199#45111yescharacteristic
86C2×C199#45511yescharacteristic
87D199—42no—
88C199#164411yescharacteristic, complemented, Sylow
89#1644—1991nomaximal, Sylow
90#554—1991no—
91#487—1991no—
92#474—1991no—
93#462—1991no—
94#333D19911yescharacteristic, complemented
95Q8⋊C8—1991no—
96D4⋊C8—1991no—
97C2×Q16—3981no—
98C4⋊Q8—1991no—
99C4⋊Q8C2×D19911yescharacteristic
100#125—3981no—
101C22⋊Q8—3981no—
102C22⋊Q8—3982no—
103C4×Q8—1991no—
104C4×D4—3981no—
105C4×D4—1991no—
106C2×C4⋊C4—3981no—
107#107—3981no—
108C4⋊C8C2×D19911yescharacteristic
109C4⋊C8C2×D19911yescharacteristic
110#103—3982no—
111#103—3981no—
112#102—3981no—
113C4×C8—1991no—
114C2×Q8—3981no—
115C2×Q8—1991no—
116C2×Q8D79612yes—
117C2×D4—1991no—
118C22×C4—3982no—
119C22×C4—3981no—
120Q16—7962no—
121C2×C8—21no—
122C2×C8—21no—
123C2×C8—3981no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—3982no—
127C4⋊C4—3982no—
128C4⋊C4—3981no—
129C4⋊C4—3981no—
130C4⋊C4—3981no—
131C4⋊C4—1991no—
132C4⋊C4—1991no—
133C22⋊C4—3982no—
134C22⋊C4—3981no—
135C42—3981no—
136C42C22×D19911yescharacteristic
137C23—3981no—
138Q8—42no—
139Q8—7961no—
140Q8—7961no—
141Q8—3981no—
142Q8—3981no—
143D4—3981no—
144D4—1992no—
145C2×C4—22no—
146C2×C4—7961no—
147C2×C4—7961no—
148C2×C4—7961no—
149C2×C4—15922no—
150C2×C4—3981no—
151C2×C4—3981no—
152C2×C4—3981no—
153C2×C4D4×D19911yescharacteristic
154C2×C4D4×D19911yescharacteristic
155C2×C4C2×D79611yescharacteristic
156C8—41no—
157C8—41no—
158C8—7961no—
159C22—7961no—
160C22—7961no—
161C22—3982no—
162C22#2026392511yescharacteristic
163C4—21no—
164C4—42no—
165C4—7961no—
166C4—7961no—
167C4—7961no—
168C4—15921no—
169C4Q16×D19911yescharacteristic
170C4#2026397811yescharacteristic
171C4#2026398211yescharacteristic
172C4#2026398411yescharacteristic
173C2—7962no—
174C2#2716416611yescharacteristic
175C2#2716417511yescharacteristic
176C2#2716419811yescharacteristic
subgroupG/H
centerC22#20263925
commutator, cosocleC42×C199C23
FrattiniC42C22×D199
Fitting#333×C199C2
socleC22×C199#120
prime radicalC2×D796D4
hypercenter#333D199
hypocenterC199#1644
LCSG ▷ C42×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C42 ◁ #333

Complements

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

NH ≅ G/Ntype
2. #27164368C2semidirect
4. #27164229C2semidirect
5. #27163748C2semidirect
7. #333×C199C2semidirect
54. D796Q16semidirect
88. C199#1644semidirect
94. #333D199semidirect

Maximal subgroups

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

subgroupindex
#271647292
#271643682
#271642992
#271642292
#271637482
#271637452
#333×C1992
#1644199