Finite Group Databasebeta

← #38741784

Subgroups — #38741784

Order 25216.

236 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 61 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#27161453C211yesmaximal, characteristic, complemented
2#27161279C211yesmaximal, characteristic, complemented
3#27161274C211yesmaximal, characteristic
4#27160828C211yesmaximal, characteristic, complemented
5Q8⋊#13096802C211yesmaximal, characteristic, complemented
6D4⋊#13096802C211yesmaximal, characteristic, complemented
7#554×C197C211yesmaximal, characteristic, complemented
8C2×#15691345—22no—
9C2×#15691345—21no—
10C2×#15691343—21no—
11#20263323C2211yescharacteristic
12C4⋊#13096802C2211yescharacteristic
13C4⋊#13096802C2211yescharacteristic
14#20263246—21no—
15#20263246—21no—
16C4×#13096802C2211yescharacteristic
17C4⋊Q8×C197C2211yescharacteristic
18#125×C197—21no—
19C22⋊Q8×C197—21no—
20C22⋊Q8×C197—22no—
21C4×Q8×C197C2211yescharacteristic
22C4×D4×C197—21no—
23C4×D4×C197C2211yescharacteristic
24C2×C4⋊C4×C197—21no—
25C2×D788—21no—
26C2×D788—21no—
27C2×D788—21no—
28#15691345—42no—
29#15691345—44no—
30#15691343—42no—
31C2×#13096802—21no—
32C2×#13096802—21no—
33C2×#13096802—21no—
34C2×Q8×C197D411yescharacteristic
35C2×Q8×C197D412yes—
36C2×D4×C197D411yescharacteristic
37C22×C4×C197—22no—
38C22×C4×C197—21no—
39C4⋊C4×C197—22no—
40C4⋊C4×C197—22no—
41C4⋊C4×C197—21no—
42C4⋊C4×C197—21no—
43C4⋊C4×C197—21no—
44C4⋊C4×C197D411yescharacteristic
45C4⋊C4×C197D411yescharacteristic
46C22⋊C4×C197—22no—
47C22⋊C4×C197—21no—
48C42×C197—21no—
49C42×C197C2311yescharacteristic
50C22×D197—41no—
51D788—42no—
52D788—41no—
53D788—41no—
54D788—41no—
55D788—41no—
56#13096802—22no—
57#13096802—41no—
58#13096802—41no—
59C23×C197—21no—
60Q8×C197—21no—
61Q8×C197—21no—
62Q8×C197—42no—
63D4×C197—21no—
64D4×C197SD1612yescomplemented
65C2×C4×C197—21no—
66C2×C4×C197—21no—
67C2×C4×C197—21no—
68C2×C4×C197—22no—
69C2×C4×C197—41no—
70C2×C4×C197—41no—
71C2×C4×C197—82no—
72C2×C4×C197C2×D411yescharacteristic
73C2×C4×C197C2×D411yescharacteristic
74C2×C4×C197C2×D411yescharacteristic
75C22×C197—22no—
76C22×C197—41no—
77C22×C197—41no—
78C22×C197#12011yescharacteristic
79C2×D197—81no—
80C2×D197—81no—
81C2×D197—81no—
82C4×C197—21no—
83C4×C197—41no—
84C4×C197—41no—
85C4×C197—41no—
86C4×C197—42no—
87C4×C197C2×SD1611yescharacteristic
88C4×C197#13511yescharacteristic
89C4×C197AGL(1,ℤ/8)11yescharacteristic
90C4×C197#13711yescharacteristic
91C2×C197—42no—
92C2×C197#44811yescharacteristic
93C2×C197#44911yescharacteristic
94C2×C197#45311yescharacteristic
95D197—161no—
96C197#162211yescharacteristic, complemented, Sylow
97#1622—1971nomaximal, Sylow
98#554D19711yescharacteristic, complemented
99#492—1971no—
100#460—1971no—
101#459—1971no—
102#333—1971no—
103Q8⋊C8—1971no—
104D4⋊C8—1971no—
105C2×SD16—3941no—
106C2×SD16—3942no—
107C2×D8—3941no—
108C4⋊Q8C2×D19711yescharacteristic
109#127—1971no—
110#125—21no—
111C22⋊Q8—21no—
112C22⋊Q8—22no—
113C4×Q8C2×D19711yescharacteristic
114C4×D4—21no—
115C4×D4C2×D19711yescharacteristic
116C2×C4⋊C4—21no—
117C4⋊C8—1971no—
118C4⋊C8—1971no—
119#102—3941no—
120#102—3941no—
121C4×C8—1971no—
122C2×Q8C22⋊D19711yescharacteristic
123C2×Q8C22⋊D19712yes—
124C2×D4—3941no—
125C2×D4—3941no—
126C2×D4—3941no—
127C2×D4C22⋊D19711yescharacteristic
128C22×C4—22no—
129C22×C4—21no—
130SD16—7882no—
131SD16—7884no—
132D8—7882no—
133C2×C8—3941no—
134C2×C8—3941no—
135C2×C8—3941no—
136C4⋊C4—22no—
137C4⋊C4—22no—
138C4⋊C4—21no—
139C4⋊C4—21no—
140C4⋊C4—21no—
141C4⋊C4C22⋊D19711yescharacteristic
142C4⋊C4C22⋊D19711yescharacteristic
143C22⋊C4—22no—
144C22⋊C4—21no—
145C42—21no—
146C42C22×D19711yescharacteristic
147C23—21no—
148C23—7881no—
149Q8—21no—
150Q8—21no—
151Q8—42no—
152D4—21no—
153D4—7882no—
154D4—7881no—
155D4—7881no—
156D4—7881no—
157D4—7881no—
158D4#1569134512yescomplemented
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—22no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—82no—
166C2×C4C2×C22⋊D19711yescharacteristic
167C2×C4C2×C22⋊D19711yescharacteristic
168C2×C4C2×C22⋊D19711yescharacteristic
169C8—7881no—
170C8—7881no—
171C8—3942no—
172C22—22no—
173C22—41no—
174C22—41no—
175C22—15761no—
176C22—15761no—
177C22—15761no—
178C22#2026329511yescharacteristic
179C4—21no—
180C4—41no—
181C4—41no—
182C4—41no—
183C4—42no—
184C4C2×#1569134511yescharacteristic
185C4#2026334611yescharacteristic
186C4#2026335111yescharacteristic
187C4#2026335711yescharacteristic
188C2—42no—
189C2—31521no—
190C2#2716121811yescharacteristic
191C2#2716122411yescharacteristic
192C2#2716124711yescharacteristic
subgroupG/H
centerC22#20263295
commutator, cosocleC42×C197C23
FrattiniC42C22×D197
Fitting#554×C197C2
socleC22×C197#120
prime radical#27161274C2
hypercenter#554D197
hypocenterC197#1622
LCSG ▷ C42×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #554

Complements

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

NH ≅ G/Ntype
1. #27161453C2semidirect
2. #27161279C2semidirect
4. #27160828C2semidirect
5. Q8⋊#13096802C2semidirect
6. D4⋊#13096802C2semidirect
7. #554×C197C2semidirect
64. D4×C197SD16semidirect
96. C197#1622semidirect
98. #554D197semidirect
158. D4#15691345semidirect

Maximal subgroups

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

subgroupindex
#271614532
#271612792
#271612742
#271608282
Q8⋊#130968022
D4⋊#130968022
#554×C1972
#1622197