Finite Group Databasebeta

← #38741845

Subgroups — #38741845

Order 25216.

214 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 55 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#27161789C211yesmaximal, characteristic
2#27161458C211yesmaximal, characteristic, complemented
3#27161366C211yesmaximal, characteristic
4#27160827C211yesmaximal, characteristic, complemented
5Q8⋊#13096802C211yesmaximal, characteristic, complemented
6#27160806C211yesmaximal, characteristic
7#477×C197C211yesmaximal, characteristic, complemented
8C2×#15691348—22no—
9#20263326C2211yescharacteristic
10#20263318—21no—
11#20263304—22no—
12#20263302—21no—
13C4×D788—21no—
14C4×D788C2211yescharacteristic
15C4⋊C4×D197—21no—
16#20263257—21no—
17C4⋊#13096802C2211yescharacteristic
18#20263246—21no—
19C4×#13096802C2211yescharacteristic
20C4⋊Q8×C197C2211yescharacteristic
21C4×Q8×C197C2211yescharacteristic
22#107×C197—21no—
23C4⋊C8×C197C2211yescharacteristic
24#103×C197—21no—
25C2×Dic394D412yes—
26C2×D788D411yescharacteristic
27C2×C4×D197—22no—
28C2×C4×D197—21no—
29#15691348—44no—
30C2×#13096802—21no—
31C2×#13096802—21no—
32C4⋊Dic197—21no—
33C4⋊Dic197—21no—
34C4⋊Dic197D411yescharacteristic
35#15691337—22no—
36#15691337—22no—
37#15691335—22no—
38#15691335—21no—
39C2×Q8×C197—21no—
40C2×Q8×C197D411yescharacteristic
41C2×C8×C197—21no—
42C4⋊C4×C197—21no—
43C4⋊C4×C197—21no—
44C4⋊C4×C197—21no—
45C4⋊C4×C197D411yescharacteristic
46C42×C197—21no—
47C42×C197C2311yescharacteristic
48C22×D197—21no—
49Dic394—42no—
50D788—21no—
51D788Q1612yescomplemented
52C2×Dic197—21no—
53C2×Dic197—22no—
54C4×D197—41no—
55C4×D197—41no—
56C4×D197—82no—
57#13096802—22no—
58#13096802—41no—
59Q8×C197—21no—
60Q8×C197—21no—
61Q8×C197—42no—
62C2×C4×C197—21no—
63C2×C4×C197—21no—
64C2×C4×C197—41no—
65C2×C4×C197C2×D411yescharacteristic
66C2×C4×C197C2×D411yescharacteristic
67C2×C4×C197C2×D411yescharacteristic
68C8×C197—41no—
69C22×C197#12011yescharacteristic
70C2×D197—22no—
71C2×D197—41no—
72C2×D197—41no—
73C4×C197—21no—
74C4×C197—41no—
75C4×C197—41no—
76C4×C197—81no—
77C4×C197C2×Q1611yescharacteristic
78C4×C197#13511yescharacteristic
79C4×C197#13711yescharacteristic
80C4×C197#13711yescharacteristic
81Dic197—41no—
82Dic197—42no—
83C2×C197#45111yescharacteristic
84C2×C197#45211yescharacteristic
85C2×C197#45311yescharacteristic
86D197—42no—
87C197#163211yescharacteristic, complemented, Sylow
88#1632—1971nomaximal, Sylow
89#554—1971no—
90#494—1971no—
91#477D19711yescharacteristic, complemented
92#476—1971no—
93#333—1971no—
94Q8⋊C8—1971no—
95D4⋊C8—1971no—
96C2×Q16—3942no—
97C4⋊Q8—1971no—
98C4⋊Q8C2×D19711yescharacteristic
99#125—3941no—
100C22⋊Q8—3942no—
101C22⋊Q8—3941no—
102C4×Q8C2×D19711yescharacteristic
103C4×D4—3941no—
104C4×D4—1971no—
105C2×C4⋊C4—3941no—
106#107—21no—
107#106—3941no—
108C4⋊C8—1971no—
109C4⋊C8C2×D19711yescharacteristic
110#103—21no—
111#102—3941no—
112C4×C8—1971no—
113C2×Q8—21no—
114C2×Q8—1972no—
115C2×Q8C22⋊D19711yescharacteristic
116C2×D4—1971no—
117C22×C4—3942no—
118C22×C4—3941no—
119Q16—7884no—
120C2×C8—21no—
121C2×C8—3941no—
122C2×C8—3941no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—3942no—
127C4⋊C4—3942no—
128C4⋊C4—3941no—
129C4⋊C4—3941no—
130C4⋊C4—1971no—
131C4⋊C4C22⋊D19711yescharacteristic
132C22⋊C4—3942no—
133C22⋊C4—3941no—
134C42—21no—
135C42C22×D19711yescharacteristic
136C23—3941no—
137Q8—21no—
138Q8—21no—
139Q8—42no—
140Q8—7882no—
141D4—3941no—
142D4—1972no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—41no—
146C2×C4—7881no—
147C2×C4—7881no—
148C2×C4—15762no—
149C2×C4—3941no—
150C2×C4—3942no—
151C2×C4D4×D19711yescharacteristic
152C2×C4D4×D19711yescharacteristic
153C2×C4C2×C22⋊D19711yescharacteristic
154C8—41no—
155C8—7881no—
156C8—3942no—
157C22—7881no—
158C22—7881no—
159C22—3942no—
160C22#2026329411yescharacteristic
161C4—21no—
162C4—41no—
163C4—41no—
164C4—81no—
165C4—7881no—
166C4—7882no—
167C4Q16×D19711yescharacteristic
168C4#2026334611yescharacteristic
169C4#2026335711yescharacteristic
170C4#2026335811yescharacteristic
171C2—7882no—
172C2#2716123311yescharacteristic
173C2#2716123911yescharacteristic
174C2#2716124511yescharacteristic
subgroupG/H
centerC22#20263294
commutator, cosocleC42×C197C23
FrattiniC42C22×D197
Fitting#477×C197C2
socleC22×C197#120
prime radicalC2×D788D4
hypercenter#477D197
hypocenterC197#1632
LCSG ▷ C42×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
2. #27161458C2semidirect
4. #27160827C2semidirect
5. Q8⋊#13096802C2semidirect
7. #477×C197C2semidirect
51. D788Q16semidirect
87. C197#1632semidirect
91. #477D197semidirect

Maximal subgroups

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

subgroupindex
#271617892
#271614582
#271613662
#271608272
Q8⋊#130968022
#271608062
#477×C1972
#1632197