Finite Group Databasebeta

← #38741972

Subgroups — #38741972

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#27161744C211yesmaximal, characteristic
2#27161361C211yesmaximal, characteristic, complemented
3#27161281C211yesmaximal, characteristic
4#27160825C211yesmaximal, characteristic, complemented
5#27160811C211yesmaximal, characteristic, complemented
6#27160808C211yesmaximal, characteristic
7#498×C197C211yesmaximal, characteristic, complemented
8C2×Q8×D197—21no—
9C2×#15691345—21no—
10#20263325—21no—
11#20263304—22no—
12#20263302—21no—
13C4×Dic394C2211yescharacteristic
14C4×D788C2211yescharacteristic
15#20263288—21no—
16#20263257—21no—
17C4⋊#13096802C2211yescharacteristic
18C4⋊#13096802C2211yescharacteristic
19#20263250—21no—
20#20263249—21no—
21C4⋊Q8×C197C2211yescharacteristic
22C4⋊Q8×C197C2211yescharacteristic
23#106×C197—22no—
24C4×C8×C197C2211yescharacteristic
25C2×Dic394—22no—
26C2×Dic394D411yescharacteristic
27Q8×D197—42no—
28Q8×D197—42no—
29C2×D788D411yescharacteristic
30C2×C4×D197—22no—
31C2×C4×D197—21no—
32#15691345—42no—
33C2×#13096802—21no—
34C2×#13096802—21no—
35C4⋊Dic197D411yescharacteristic
36C4⋊Dic197D411yescharacteristic
37#15691337—22no—
38#15691337—21no—
39#15691335—22no—
40#15691335—21no—
41C4×Dic197—21no—
42C2×Q8×C197—21no—
43C2×Q8×C197—21no—
44C2×C8×C197—21no—
45C4⋊C4×C197—21no—
46C4⋊C4×C197—21no—
47C4⋊C4×C197—21no—
48C4⋊C4×C197D412yes—
49C42×C197C2311yescharacteristic
50C22×D197—21no—
51Dic394—21no—
52Dic394—44no—
53Dic394SD1612yescomplemented
54D788—21no—
55D788SD1612yes—
56C2×Dic197—21no—
57C2×Dic197—21no—
58C2×Dic197—21no—
59C4×D197—44no—
60C4×D197—41no—
61C4×D197—41no—
62#13096802—41no—
63#13096802—41no—
64Q8×C197—41no—
65Q8×C197—41no—
66Q8×C197—41no—
67Q8×C197—41no—
68C2×C4×C197—22no—
69C2×C4×C197—41no—
70C2×C4×C197C2×D411yescharacteristic
71C2×C4×C197C2×D411yescharacteristic
72C2×C4×C197C2×D411yescharacteristic
73C8×C197—22no—
74C22×C197#12011yescharacteristic
75C2×D197—22no—
76C2×D197—41no—
77C2×D197—41no—
78C4×C197—21no—
79C4×C197—42no—
80C4×C197—81no—
81C4×C197C2×SD1611yescharacteristic
82C4×C197C2×SD1611yescharacteristic
83C4×C197AGL(1,ℤ/8)11yescharacteristic
84C4×C197#13711yescharacteristic
85Dic197—22no—
86Dic197—41no—
87Dic197—41no—
88C2×C197#44811yescharacteristic
89C2×C197#45011yescharacteristic
90C2×C197#45611yescharacteristic
91D197—42no—
92C197#165111yescharacteristic, complemented, Sylow
93#1651—1971nomaximal, Sylow
94D4×Q8—1971no—
95#498D19711yescharacteristic, complemented
96#475—1971no—
97#461—1971no—
98#332—1971no—
99Q8⋊C8—1971no—
100D4⋊C8—1971no—
101C22×Q8—3941no—
102C2×SD16—3941no—
103C4⋊Q8—3941no—
104C4⋊Q8C2×D19711yescharacteristic
105C4⋊Q8C2×D19711yescharacteristic
106C22⋊Q8—3941no—
107C22⋊Q8—3942no—
108C4×Q8—1971no—
109C4×D4—3941no—
110C4×D4—1971no—
111#106—22no—
112#106—3941no—
113C4⋊C8—1971no—
114C4⋊C8—1971no—
115#103—3941no—
116#103—3941no—
117C4×C8C2×D19711yescharacteristic
118C2×Q8—21no—
119C2×Q8—21no—
120C2×Q8—7882no—
121C2×Q8—7882no—
122C2×Q8—3942no—
123C2×Q8—1971no—
124C2×D4—1971no—
125C22×C4—3942no—
126C22×C4—3941no—
127SD16—7882no—
128C2×C8—21no—
129C2×C8—3941no—
130C2×C8—3941no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—3942no—
135C4⋊C4—3941no—
136C4⋊C4—1971no—
137C4⋊C4—1971no—
138C4⋊C4C22⋊D19712yes—
139C22⋊C4—3942no—
140C22⋊C4—3941no—
141C42—3941no—
142C42C22×D19711yescharacteristic
143C23—3941no—
144Q8—41no—
145Q8—41no—
146Q8—41no—
147Q8—41no—
148Q8—7884no—
149Q8—3941no—
150Q8—1972no—
151D4—3941no—
152D4—1972no—
153C2×C4—22no—
154C2×C4—41no—
155C2×C4—7884no—
156C2×C4—7881no—
157C2×C4—7881no—
158C2×C4—3941no—
159C2×C4—3941no—
160C2×C4—3941no—
161C2×C4D4×D19711yescharacteristic
162C2×C4D4×D19711yescharacteristic
163C2×C4C2×C22⋊D19711yescharacteristic
164C8—22no—
165C8—7881no—
166C8—7881no—
167C22—7881no—
168C22—7881no—
169C22—3942no—
170C22#2026329411yescharacteristic
171C4—21no—
172C4—42no—
173C4—81no—
174C4—7881no—
175C4—7881no—
176C4—3942no—
177C4SD16×D19711yescharacteristic
178C4SD16×D19711yescharacteristic
179C4#2026335311yescharacteristic
180C4#2026335511yescharacteristic
181C2—7882no—
182C2#2716121611yescharacteristic
183C2#2716122811yescharacteristic
184C2#2716126111yescharacteristic
subgroupG/H
centerC22#20263294
commutator, cosocleC42×C197C23
FrattiniC42C22×D197
Fitting#498×C197C2
socleC22×C197#120
prime radicalC2×D788D4
hypercenter#498D197
hypocenterC197#1651
LCSG ▷ C42×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #498

Complements

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

NH ≅ G/Ntype
2. #27161361C2semidirect
4. #27160825C2semidirect
5. #27160811C2semidirect
7. #498×C197C2semidirect
53. Dic394SD16semidirect
92. C197#1651semidirect
95. #498D197semidirect

Maximal subgroups

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

subgroupindex
#271617442
#271613612
#271612812
#271608252
#271608112
#271608082
#498×C1972
#1651197