Finite Group Databasebeta

← #38741861

Subgroups — #38741861

Order 25216.

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

normal subgroups only

This group has 234 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#27161733C211yesmaximal, characteristic
2#27161429C211yesmaximal, characteristic, complemented
3#27161366C211yesmaximal, characteristic, complemented
4#27160826C211yesmaximal, characteristic, complemented
5D4⋊#13096802C211yesmaximal, characteristic, complemented
6#27160806C211yesmaximal, characteristic, complemented
7#476×C197C211yesmaximal, characteristic, complemented
8C2×#15691357—21no—
9#20263326C2211yescharacteristic
10#20263310—22no—
11#20263306—21no—
12#20263300—21no—
13C4×D788C2211yescharacteristic
14C2×C4⋊Dic197—21no—
15#20263257—21no—
16C4⋊#13096802C2211yescharacteristic
17#20263250—22no—
18#20263246—21no—
19C4×#13096802C2211yescharacteristic
20C4⋊Q8×C197C2211yescharacteristic
21C4×D4×C197C2211yescharacteristic
22#106×C197—21no—
23C4⋊C8×C197C2211yescharacteristic
24#102×C197—21no—
25#15691357—42no—
26#15691357—42no—
27C2×Dic394—21no—
28C2×D788D411yescharacteristic
29C2×C22⋊D197—22no—
30C22×Dic197—22no—
31C2×C4×D197—21no—
32C2×#13096802—21no—
33C2×#13096802—21no—
34C4⋊Dic197—22no—
35C4⋊Dic197—21no—
36C4⋊Dic197D412yes—
37C4⋊Dic197D411yescharacteristic
38#15691335—22no—
39#15691335—21no—
40C2×Q8×C197—21no—
41C2×D4×C197D411yescharacteristic
42C22×C4×C197—21no—
43C2×C8×C197—21no—
44C4⋊C4×C197—21no—
45C4⋊C4×C197—21no—
46C4⋊C4×C197D411yescharacteristic
47C22⋊C4×C197—21no—
48C42×C197C2311yescharacteristic
49C22×D197—21no—
50Dic394—41no—
51Dic394—41no—
52D788—21no—
53D788—21no—
54C22⋊D197—44no—
55C2×Dic197—21no—
56C2×Dic197—22no—
57C2×Dic197—44no—
58C4×D197—41no—
59C4×D197—41no—
60#13096802—41no—
61#13096802—41no—
62C23×C197—21no—
63Q8×C197—42no—
64D4×C197—21no—
65D4×C197SD1612yes—
66C2×C4×C197—21no—
67C2×C4×C197—41no—
68C2×C4×C197—41no—
69C2×C4×C197—41no—
70C2×C4×C197C2×D411yescharacteristic
71C2×C4×C197C2×D411yescharacteristic
72C2×C4×C197C2×D411yescharacteristic
73C8×C197—41no—
74C22×C197—22no—
75C22×C197—41no—
76C22×C197—41no—
77C22×C197#12011yescharacteristic
78C2×D197—41no—
79C2×D197—41no—
80C2×D197—41no—
81C4×C197—21no—
82C4×C197—41no—
83C4×C197—81no—
84C4×C197C2×SD1611yescharacteristic
85C4×C197#13511yescharacteristic
86C4×C197AGL(1,ℤ/8)11yescharacteristic
87C4×C197#13711yescharacteristic
88Dic197—41no—
89Dic197—42no—
90C2×C197—42no—
91C2×C197#45011yescharacteristic
92C2×C197#45211yescharacteristic
93C2×C197#45411yescharacteristic
94D197—81no—
95C197#163511yescharacteristic, complemented, Sylow
96#1635—1971nomaximal, Sylow
97#547—1971no—
98#487—1971no—
99#476—1971no—
100#476D19711yescharacteristic, complemented
101#332—1971no—
102D4⋊C8—1971no—
103D4⋊C8—1971no—
104C2×#41—3941no—
105C4⋊Q8—1971no—
106C4⋊Q8C2×D19711yescharacteristic
107#123—3942no—
108C22⋊Q8—3941no—
109#121—3941no—
110C4×D4—1971no—
111C4×D4C2×D19711yescharacteristic
112C2×C4⋊C4—3941no—
113#106—21no—
114#106—3941no—
115C4⋊C8—1971no—
116C4⋊C8C2×D19711yescharacteristic
117#103—3942no—
118#102—21no—
119#102—3941no—
120C4×C8—1971no—
121#41—7882no—
122#41—7882no—
123C2×Q8—21no—
124C2×Q8—3941no—
125C2×D4—3942no—
126C2×D4—1971no—
127C2×D4C22⋊D19711yescharacteristic
128C22×C4—21no—
129C22×C4—3942no—
130C22×C4—3941no—
131C2×C8—21no—
132C2×C8—3941no—
133C2×C8—3941no—
134C4⋊C4—21no—
135C4⋊C4—21no—
136C4⋊C4—3942no—
137C4⋊C4—3941no—
138C4⋊C4—1972no—
139C4⋊C4—1971no—
140C4⋊C4C22⋊D19711yescharacteristic
141C22⋊C4—21no—
142C22⋊C4—3942no—
143C22⋊C4—3941no—
144C42C22×D19711yescharacteristic
145C23—21no—
146C23—3941no—
147Q8—42no—
148Q8—7881no—
149Q8—7881no—
150D4—21no—
151D4—7884no—
152D4—3941no—
153D4—3941no—
154D4#1569134512yes—
155C2×C4—21no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—7884no—
160C2×C4—7881no—
161C2×C4—7881no—
162C2×C4—3941no—
163C2×C4—3942no—
164C2×C4D4×D19711yescharacteristic
165C2×C4D4×D19711yescharacteristic
166C2×C4C2×C22⋊D19711yescharacteristic
167C8—41no—
168C8—7881no—
169C8—7881no—
170C22—22no—
171C22—41no—
172C22—41no—
173C22—7881no—
174C22—7881no—
175C22—7881no—
176C22#2026329411yescharacteristic
177C4—21no—
178C4—41no—
179C4—81no—
180C4—7881no—
181C4—7882no—
182C4C2×#1569134511yescharacteristic
183C4#2026334711yescharacteristic
184C4#2026335111yescharacteristic
185C4#2026335511yescharacteristic
186C2—42no—
187C2—15761no—
188C2#2716122911yescharacteristic
189C2#2716123811yescharacteristic
190C2#2716125011yescharacteristic
subgroupG/H
centerC22#20263294
commutator, cosocleC42×C197C23
FrattiniC42C22×D197
Fitting#476×C197C2
socleC22×C197#120
prime radical#27161733C2
hypercenter#476D197
hypocenterC197#1635
LCSG ▷ C42×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
2. #27161429C2semidirect
3. #27161366C2semidirect
4. #27160826C2semidirect
5. D4⋊#13096802C2semidirect
6. #27160806C2semidirect
7. #476×C197C2semidirect
95. C197#1635semidirect
100. #476D197semidirect

Maximal subgroups

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

subgroupindex
#271617332
#271614292
#271613662
#271608262
D4⋊#130968022
#271608062
#476×C1972
#1635197