Finite Group Databasebeta

← #38768115

Subgroups — #38768115

Order 25472.

234 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 59 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
1D4×Dic398C211yesmaximal, characteristic
2#27164416C211yesmaximal, characteristic, complemented
3#27164304C211yesmaximal, characteristic, complemented
4#27164223C211yesmaximal, characteristic
5#27163765C211yesmaximal, characteristic, complemented
6#27163750C211yesmaximal, characteristic, complemented
7D4⋊C8×C199C211yesmaximal, characteristic
8C22×Dic398—21no—
9C2×#15691874—21no—
10#20263955C2211yescharacteristic
11#20263955C2211yescharacteristic
12#20263954—21no—
13#20263934—21no—
14#20263932—22no—
15C4×Dic398C2211yescharacteristic
16D4×Dic199—21no—
17#20263887—22no—
18#20263887—21no—
19C4⋊(C199⋊C8)C2211yescharacteristic
20#20263880—21no—
21#20263878—21no—
22C4×D4×C199C2211yescharacteristic
23C4⋊C8×C199C2211yescharacteristic
24C4×C8×C199C2211yescharacteristic
25C2×Dic398—22no—
26C2×Dic398—21no—
27C2×Dic398—21no—
28C2×Dic398—42no—
29C2×Dic398—42no—
30C2×Dic398D411yescharacteristic
31C22×Dic199—22no—
32#15691874—42no—
33C2×C199⋊C8—21no—
34C4⋊Dic199—21no—
35C4⋊Dic199—21no—
36C4⋊Dic199—21no—
37C4⋊Dic199D412yes—
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
50Dic398—21no—
51Dic398—44no—
52Dic398—41no—
53Dic398—41no—
54Dic398—41no—
55Dic398—41no—
56Dic398SD1612yescomplemented
57C2×Dic199—21no—
58C2×Dic199—21no—
59C2×Dic199—22no—
60C2×Dic199—44no—
61C2×Dic199—41no—
62C199⋊C8—41no—
63C23×C199—21no—
64D4×C199—21no—
65D4×C199SD1612yes—
66C2×C4×C199—21no—
67C2×C4×C199—41no—
68C2×C4×C199—41no—
69C2×C4×C199C2×D411yescharacteristic
70C2×C4×C199C2×D411yescharacteristic
71C2×C4×C199C2×D411yescharacteristic
72C8×C199—22no—
73C8×C199—41no—
74C22×C199—22no—
75C22×C199—41no—
76C22×C199—41no—
77C22×C199#12011yescharacteristic
78C4×C199—21no—
79C4×C199—41no—
80C4×C199C2×SD1611yescharacteristic
81C4×C199C2×SD1611yescharacteristic
82C4×C199AGL(1,ℤ/8)11yescharacteristic
83C4×C199#13711yescharacteristic
84Dic199—22no—
85Dic199—42no—
86Dic199—41no—
87Dic199—81no—
88C2×C199—42no—
89C2×C199#44811yescharacteristic
90C2×C199#45011yescharacteristic
91C2×C199#45611yescharacteristic
92C199#165111yescharacteristic, complemented, Sylow
93#1651—1991nomaximal, Sylow
94D4×Q8—1991no—
95#498—1991no—
96#475—1991no—
97#461—1991no—
98#332—1991no—
99Q8⋊C8—1991no—
100D4⋊C8D19911yescharacteristic
101C22×Q8—3981no—
102C2×SD16—3981no—
103C4⋊Q8—3981no—
104C4⋊Q8—1991no—
105C4⋊Q8—1991no—
106C22⋊Q8—3982no—
107C22⋊Q8—3981no—
108C4×Q8—1991no—
109C4×D4—3981no—
110C4×D4C2×D19911yescharacteristic
111#106—3981no—
112#106—3982no—
113C4⋊C8—1991no—
114C4⋊C8C2×D19911yescharacteristic
115#103—3981no—
116#103—3981no—
117C4×C8C2×D19911yescharacteristic
118C2×Q8—7962no—
119C2×Q8—7962no—
120C2×Q8—3982no—
121C2×Q8—3981no—
122C2×Q8—3981no—
123C2×Q8—1991no—
124C2×D4D79611yescharacteristic
125C22×C4—21no—
126C22×C4—3982no—
127SD16—7962no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—3981no—
131C4⋊C4—3982no—
132C4⋊C4—3981no—
133C4⋊C4—3981no—
134C4⋊C4—3981no—
135C4⋊C4—3981no—
136C4⋊C4—1991no—
137C4⋊C4—1992no—
138C4⋊C4D79611yescharacteristic
139C22⋊C4—21no—
140C22⋊C4—3982no—
141C42—3981no—
142C42C22×D19911yescharacteristic
143C23—21no—
144Q8—7964no—
145Q8—7961no—
146Q8—7961no—
147Q8—7961no—
148Q8—7961no—
149Q8—3981no—
150Q8—1992no—
151D4—21no—
152D4#1569187512yes—
153C2×C4—21no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—7964no—
157C2×C4—7961no—
158C2×C4—3981no—
159C2×C4—3982no—
160C2×C4—3981no—
161C2×C4D4×D19911yescharacteristic
162C2×C4D4×D19911yescharacteristic
163C2×C4C2×D79611yescharacteristic
164C8—22no—
165C8—41no—
166C8—7961no—
167C22—22no—
168C22—41no—
169C22—41no—
170C22#2026392511yescharacteristic
171C4—21no—
172C4—41no—
173C4—7962no—
174C4—7961no—
175C4—15921no—
176C4—3982no—
177C4SD16×D19911yescharacteristic
178C4C2×#1569187511yescharacteristic
179C4#2026397911yescharacteristic
180C4#2026398811yescharacteristic
181C2—42no—
182C2#2716415911yescharacteristic
183C2#2716417011yescharacteristic
184C2#2716420211yescharacteristic
subgroupG/H
centerC22#20263925
commutator, cosocleC42×C199C23
FrattiniC42C22×D199
FittingD4⋊C8×C199C2
socleC22×C199#120
prime radicalC2×D4×C199D4
hypercenterD4⋊C8D199
hypocenterC199#1651
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
2. #27164416C2semidirect
3. #27164304C2semidirect
5. #27163765C2semidirect
6. #27163750C2semidirect
56. Dic398SD16semidirect
92. C199#1651semidirect

Maximal subgroups

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

subgroupindex
D4×Dic3982
#271644162
#271643042
#271642232
#271637652
#271637502
D4⋊C8×C1992
#1651199