Finite Group Databasebeta

← #38768018

Subgroups — #38768018

Order 25472.

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

normal subgroups only

This group has 228 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#27164369C211yesmaximal, characteristic, complemented
3#27164314C211yesmaximal, characteristic, complemented
4#27164308C211yesmaximal, characteristic
5#27163765C211yesmaximal, characteristic, complemented
6#27163750C211yesmaximal, characteristic, complemented
7D4⋊C8×C199C211yesmaximal, characteristic
8C22×Dic398—21no—
9#20263955C2211yescharacteristic
10#20263955C2211yescharacteristic
11#20263954—21no—
12#20263934—21no—
13#20263932—22no—
14C4×Dic398C2211yescharacteristic
15D4×Dic199—21no—
16#20263889—21no—
17#20263886—21no—
18C4⋊(C199⋊C8)C2211yescharacteristic
19#20263880—21no—
20#20263880—22no—
21#20263876—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—
32C2×C199⋊C8—21no—
33C4⋊Dic199—21no—
34C4⋊Dic199—21no—
35C4⋊Dic199—21no—
36C4⋊Dic199D412yes—
37C4⋊Dic199D411yescharacteristic
38Dic199⋊C4—22no—
39Dic199⋊C4—21no—
40C22⋊Dic199—22no—
41C4×Dic199—21no—
42C2×D4×C199D411yescharacteristic
43C22×C4×C199—21no—
44C2×C8×C199—21no—
45C2×C8×C199—21no—
46C4⋊C4×C199D411yescharacteristic
47C22⋊C4×C199—21no—
48C42×C199C2311yescharacteristic
49Dic398—21no—
50Dic398—44no—
51Dic398—41no—
52Dic398—41no—
53Dic398—42no—
54Dic398SD1612yes—
55C2×Dic199—21no—
56C2×Dic199—21no—
57C2×Dic199—22no—
58C2×Dic199—44no—
59C2×Dic199—41no—
60C199⋊C8—41no—
61C23×C199—21no—
62D4×C199—21no—
63D4×C199Q1612yes—
64C2×C4×C199—21no—
65C2×C4×C199—41no—
66C2×C4×C199—41no—
67C2×C4×C199C2×D411yescharacteristic
68C2×C4×C199C2×D411yescharacteristic
69C2×C4×C199C2×D411yescharacteristic
70C8×C199—41no—
71C8×C199—41no—
72C22×C199—22no—
73C22×C199—41no—
74C22×C199—41no—
75C22×C199#12011yescharacteristic
76C4×C199—21no—
77C4×C199—41no—
78C4×C199C2×SD1611yescharacteristic
79C4×C199C2×Q1611yescharacteristic
80C4×C199AGL(1,ℤ/8)11yescharacteristic
81C4×C199#13711yescharacteristic
82Dic199—22no—
83Dic199—42no—
84Dic199—41no—
85Dic199—81no—
86C2×C199—42no—
87C2×C199#45011yescharacteristic
88C2×C199#45111yescharacteristic
89C2×C199#45411yescharacteristic
90C199#163711yescharacteristic, complemented, Sylow
91#1637—1991nomaximal, Sylow
92D4×Q8—1991no—
93#487—1991no—
94#477—1991no—
95#476—1991no—
96#332—1991no—
97Q8⋊C8—1991no—
98D4⋊C8D19911yescharacteristic
99C22×Q8—3981no—
100C4⋊Q8—3981no—
101C4⋊Q8—1991no—
102C4⋊Q8—1991no—
103C22⋊Q8—3982no—
104C22⋊Q8—3981no—
105C4×Q8—1991no—
106C4×D4—3981no—
107C4×D4C2×D19911yescharacteristic
108#107—3981no—
109#106—3981no—
110C4⋊C8—1991no—
111C4⋊C8C2×D19911yescharacteristic
112#103—3981no—
113#103—3982no—
114#102—3981no—
115C4×C8C2×D19911yescharacteristic
116C2×Q8—7962no—
117C2×Q8—7962no—
118C2×Q8—3982no—
119C2×Q8—3981no—
120C2×Q8—3981no—
121C2×Q8—1991no—
122C2×D4D79611yescharacteristic
123C22×C4—21no—
124C22×C4—3982no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—3981no—
128C4⋊C4—3982no—
129C4⋊C4—3981no—
130C4⋊C4—3981no—
131C4⋊C4—3981no—
132C4⋊C4—3981no—
133C4⋊C4—1992no—
134C4⋊C4—1991no—
135C4⋊C4D79611yescharacteristic
136C22⋊C4—21no—
137C22⋊C4—3982no—
138C42—3981no—
139C42C22×D19911yescharacteristic
140C23—21no—
141Q8—7962no—
142Q8—7964no—
143Q8—7961no—
144Q8—7961no—
145Q8—3981no—
146Q8—1992no—
147D4—21no—
148D4Dic79612yes—
149C2×C4—21no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—7964no—
153C2×C4—7961no—
154C2×C4—3981no—
155C2×C4—3982no—
156C2×C4—3981no—
157C2×C4D4×D19911yescharacteristic
158C2×C4D4×D19911yescharacteristic
159C2×C4C2×D79611yescharacteristic
160C8—41no—
161C8—41no—
162C8—7961no—
163C22—22no—
164C22—41no—
165C22—41no—
166C22#2026392511yescharacteristic
167C4—21no—
168C4—41no—
169C4—7962no—
170C4—7961no—
171C4—15921no—
172C4—3982no—
173C4SD16×D19911yescharacteristic
174C4C2×Dic79611yescharacteristic
175C4#2026397911yescharacteristic
176C4#2026398811yescharacteristic
177C2—42no—
178C2#2716417011yescharacteristic
179C2#2716417611yescharacteristic
180C2#2716419211yescharacteristic
subgroupG/H
centerC22#20263925
commutator, cosocleC42×C199C23
FrattiniC42C22×D199
FittingD4⋊C8×C199C2
socleC22×C199#120
prime radicalC2×D4×C199D4
hypercenterD4⋊C8D199
hypocenterC199#1637
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. #27164369C2semidirect
3. #27164314C2semidirect
5. #27163765C2semidirect
6. #27163750C2semidirect
90. C199#1637semidirect

Maximal subgroups

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

subgroupindex
D4×Dic3982
#271643692
#271643142
#271643082
#271637652
#271637502
D4⋊C8×C1992
#1637199