Finite Group Databasebeta

← #38768016

Subgroups — #38768016

Order 25472.

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

normal subgroups only

This group has 228 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#27164683C211yesmaximal, characteristic
2#27164370C211yesmaximal, characteristic, complemented
3#27164314C211yesmaximal, characteristic, complemented
4#27164309C211yesmaximal, characteristic
5#27163748C211yesmaximal, characteristic, complemented
6#27163745C211yesmaximal, characteristic
7#332×C199C211yesmaximal, characteristic, complemented
8C2×Q8×D199—21no—
9#20263955C2211yescharacteristic
10#20263954—21no—
11#20263933—22no—
12#20263931—21no—
13C4×Dic398C2211yescharacteristic
14C4×D796C2211yescharacteristic
15#20263917—21no—
16#20263889—21no—
17#20263887—21no—
18#20263880—21no—
19#20263878—22no—
20#20263877—21no—
21C4×C199⋊C8C2211yescharacteristic
22C4⋊Q8×C199C2211yescharacteristic
23C4⋊C8×C199C2211yescharacteristic
24C4⋊C8×C199C2211yescharacteristic
25C2×Dic398—22no—
26C2×Dic398—21no—
27C2×Dic398D411yescharacteristic
28Q8×D199—42no—
29Q8×D199—42no—
30C2×D796D411yescharacteristic
31C2×C4×D199—22no—
32C2×C4×D199—21no—
33C2×C199⋊C8—21no—
34C4⋊Dic199—21no—
35C4⋊Dic199—21no—
36C4⋊Dic199D411yescharacteristic
37C4⋊Dic199D411yescharacteristic
38Dic199⋊C4—22no—
39Dic199⋊C4—21no—
40(C2×D199)⋊C4—22no—
41(C2×D199)⋊C4—21no—
42C4×Dic199—21no—
43C2×Q8×C199—21no—
44C2×C8×C199—21no—
45C2×C8×C199—21no—
46C4⋊C4×C199—21no—
47C4⋊C4×C199D412yes—
48C42×C199C2311yescharacteristic
49C22×D199—21no—
50Dic398—21no—
51Dic398—44no—
52Dic398—42no—
53Dic398SD1612yes—
54D796—21no—
55D796Q1612yes—
56C2×Dic199—21no—
57C2×Dic199—21no—
58C2×Dic199—21no—
59C2×Dic199—41no—
60C4×D199—44no—
61C4×D199—41no—
62C4×D199—41no—
63C199⋊C8—41no—
64Q8×C199—41no—
65Q8×C199—41no—
66C2×C4×C199—22no—
67C2×C4×C199C2×D411yescharacteristic
68C2×C4×C199C2×D411yescharacteristic
69C2×C4×C199C2×D411yescharacteristic
70C8×C199—41no—
71C8×C199—41no—
72C22×C199#12011yescharacteristic
73C2×D199—22no—
74C2×D199—41no—
75C2×D199—41no—
76C4×C199—21no—
77C4×C199—42no—
78C4×C199C2×SD1611yescharacteristic
79C4×C199C2×Q1611yescharacteristic
80C4×C199AGL(1,ℤ/8)11yescharacteristic
81C4×C199#13711yescharacteristic
82Dic199—22no—
83Dic199—41no—
84Dic199—41no—
85Dic199—81no—
86C2×C199#45011yescharacteristic
87C2×C199#45111yescharacteristic
88C2×C199#45411yescharacteristic
89D199—42no—
90C199#163711yescharacteristic, complemented, Sylow
91#1637—1991nomaximal, Sylow
92D4×Q8—1991no—
93#487—1991no—
94#477—1991no—
95#476—1991no—
96#332D19911yescharacteristic, complemented
97Q8⋊C8—1991no—
98D4⋊C8—1991no—
99C22×Q8—3981no—
100C4⋊Q8—3981no—
101C4⋊Q8—1991no—
102C4⋊Q8C2×D19911yescharacteristic
103C22⋊Q8—3981no—
104C22⋊Q8—3982no—
105C4×Q8—1991no—
106C4×D4—3981no—
107C4×D4—1991no—
108#107—3981no—
109#106—3981no—
110C4⋊C8C2×D19911yescharacteristic
111C4⋊C8C2×D19911yescharacteristic
112#103—3981no—
113#103—3982no—
114#102—3981no—
115C4×C8—1991no—
116C2×Q8—21no—
117C2×Q8—7962no—
118C2×Q8—7962no—
119C2×Q8—3982no—
120C2×Q8—3981no—
121C2×Q8—1991no—
122C2×D4—1991no—
123C22×C4—3982no—
124C22×C4—3981no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—3981no—
128C4⋊C4—21no—
129C4⋊C4—3982no—
130C4⋊C4—3981no—
131C4⋊C4—3981no—
132C4⋊C4—3981no—
133C4⋊C4—1991no—
134C4⋊C4—1991no—
135C4⋊C4D79612yes—
136C22⋊C4—3982no—
137C22⋊C4—3981no—
138C42—3981no—
139C42C22×D19911yescharacteristic
140C23—3981no—
141Q8—41no—
142Q8—41no—
143Q8—7962no—
144Q8—7964no—
145Q8—3981no—
146Q8—1992no—
147D4—3981no—
148D4—1992no—
149C2×C4—22no—
150C2×C4—7964no—
151C2×C4—7961no—
152C2×C4—7961no—
153C2×C4—7961no—
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—7961no—
164C22—7961no—
165C22—3982no—
166C22#2026392511yescharacteristic
167C4—21no—
168C4—42no—
169C4—7961no—
170C4—7961no—
171C4—15921no—
172C4—3982no—
173C4SD16×D19911yescharacteristic
174C4Q16×D19911yescharacteristic
175C4#2026397911yescharacteristic
176C4#2026398411yescharacteristic
177C2—7962no—
178C2#2716417011yescharacteristic
179C2#2716417511yescharacteristic
180C2#2716419311yescharacteristic
subgroupG/H
centerC22#20263925
commutator, cosocleC42×C199C23
FrattiniC42C22×D199
Fitting#332×C199C2
socleC22×C199#120
prime radicalC2×D796D4
hypercenter#332D199
hypocenterC199#1637
LCSG ▷ C42×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
2. #27164370C2semidirect
3. #27164314C2semidirect
5. #27163748C2semidirect
7. #332×C199C2semidirect
90. C199#1637semidirect
96. #332D199semidirect

Maximal subgroups

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

subgroupindex
#271646832
#271643702
#271643142
#271643092
#271637482
#271637452
#332×C1992
#1637199