Finite Group Databasebeta

← #27170265

Subgroups — #27170265

Order 12928.

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#20265887C211yesmaximal, characteristic
2#20265556C211yesmaximal, characteristic, complemented
3#20265464C211yesmaximal, characteristic
4#20264925C211yesmaximal, characteristic, complemented
5Q8⋊#11717579C211yesmaximal, characteristic, complemented
6#20264904C211yesmaximal, characteristic
7#477×C101C211yesmaximal, characteristic, complemented
8C2×#13108064—22no—
9#15692255C2211yescharacteristic
10#15692247—21no—
11#15692233—22no—
12#15692231—21no—
13C4×D404—21no—
14C4×D404C2211yescharacteristic
15C4⋊C4×D101—21no—
16#15692186—21no—
17C4⋊#11717579C2211yescharacteristic
18#15692175—21no—
19C4×#11717579C2211yescharacteristic
20C4⋊Q8×C101C2211yescharacteristic
21C4×Q8×C101C2211yescharacteristic
22#107×C101—21no—
23C4⋊C8×C101C2211yescharacteristic
24#103×C101—21no—
25C2×Dic202D412yes—
26C2×D404D411yescharacteristic
27C2×C4×D101—21no—
28C2×C4×D101—22no—
29#13108064—44no—
30C2×#11717579—21no—
31C2×#11717579—21no—
32C4⋊Dic101—21no—
33C4⋊Dic101—21no—
34C4⋊Dic101D411yescharacteristic
35#13108053—22no—
36#13108053—22no—
37#13108051—22no—
38#13108051—21no—
39C2×Q8×C101—21no—
40C2×Q8×C101D411yescharacteristic
41C2×C8×C101—21no—
42C4⋊C4×C101—21no—
43C4⋊C4×C101—21no—
44C4⋊C4×C101—21no—
45C4⋊C4×C101D411yescharacteristic
46C42×C101—21no—
47C42×C101C2311yescharacteristic
48C22×D101—21no—
49Q8×C101—21no—
50Q8×C101—21no—
51Q8×C101—42no—
52C2×C4×C101—21no—
53C2×C4×C101—21no—
54C2×C4×C101—41no—
55C2×C4×C101C2×D411yescharacteristic
56C2×C4×C101C2×D411yescharacteristic
57C2×C4×C101C2×D411yescharacteristic
58C2×Dic101—21no—
59C2×Dic101—22no—
60D404—21no—
61D404Q1612yescomplemented
62C4×D101—41no—
63C4×D101—41no—
64C4×D101—82no—
65Dic202—42no—
66C8×C101—41no—
67#11717579—22no—
68#11717579—41no—
69C22×C101#12011yescharacteristic
70C2×D101—22no—
71C2×D101—41no—
72C2×D101—41no—
73C4×C101—21no—
74C4×C101—41no—
75C4×C101—41no—
76C4×C101—81no—
77C4×C101C2×Q1611yescharacteristic
78C4×C101#13511yescharacteristic
79C4×C101#13711yescharacteristic
80C4×C101#13711yescharacteristic
81Dic101—41no—
82Dic101—42no—
83C2×C101#45111yescharacteristic
84C2×C101#45211yescharacteristic
85C2×C101#45311yescharacteristic
86D101—42no—
87#1632—1011nomaximal, Sylow
88C101#163211yescharacteristic, complemented, Sylow
89#554—1011no—
90#494—1011no—
91#477D10111yescharacteristic, complemented
92#476—1011no—
93#333—1011no—
94Q8⋊C8—1011no—
95D4⋊C8—1011no—
96C2×Q16—2022no—
97C4⋊Q8—1011no—
98C4⋊Q8C2×D10111yescharacteristic
99#125—2021no—
100C22⋊Q8—2022no—
101C22⋊Q8—2021no—
102C4×Q8C2×D10111yescharacteristic
103C4×D4—1011no—
104C4×D4—2021no—
105C2×C4⋊C4—2021no—
106#107—21no—
107#106—2021no—
108C4⋊C8—1011no—
109C4⋊C8C2×D10111yescharacteristic
110#103—21no—
111#102—2021no—
112C4×C8—1011no—
113C2×Q8—21no—
114C2×Q8—1012no—
115C2×Q8C22⋊D10111yescharacteristic
116C2×D4—1011no—
117C22×C4—2022no—
118C22×C4—2021no—
119Q16—4044no—
120C2×C8—21no—
121C2×C8—2021no—
122C2×C8—2021no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—1011no—
127C4⋊C4—2022no—
128C4⋊C4—2022no—
129C4⋊C4—2021no—
130C4⋊C4—2021no—
131C4⋊C4C22⋊D10111yescharacteristic
132C22⋊C4—2022no—
133C22⋊C4—2021no—
134C42—21no—
135C42C22×D10111yescharacteristic
136C23—2021no—
137Q8—21no—
138Q8—21no—
139Q8—42no—
140Q8—4042no—
141D4—1012no—
142D4—2021no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—41no—
146C2×C4—8082no—
147C2×C4—4041no—
148C2×C4—4041no—
149C2×C4—2021no—
150C2×C4—2022no—
151C2×C4D4×D10111yescharacteristic
152C2×C4D4×D10111yescharacteristic
153C2×C4C2×C22⋊D10111yescharacteristic
154C8—41no—
155C8—4041no—
156C8—2022no—
157C22—4041no—
158C22—4041no—
159C22—2022no—
160C22#1569222311yescharacteristic
161C4—21no—
162C4—41no—
163C4—41no—
164C4—81no—
165C4—4041no—
166C4—4042no—
167C4Q16×D10111yescharacteristic
168C4#1569227511yescharacteristic
169C4#1569228611yescharacteristic
170C4#1569228711yescharacteristic
171C2—4042no—
172C2#2026533111yescharacteristic
173C2#2026533711yescharacteristic
174C2#2026534311yescharacteristic
subgroupG/H
centerC22#15692223
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
Fitting#477×C101C2
socleC22×C101#120
prime radicalC2×D404D4
hypercenter#477D101
hypocenterC101#1632
LCSG ▷ C42×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
2. #20265556C2semidirect
4. #20264925C2semidirect
5. Q8⋊#11717579C2semidirect
7. #477×C101C2semidirect
61. D404Q16semidirect
88. C101#1632semidirect
91. #477D101semidirect

Maximal subgroups

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

subgroupindex
#202658872
#202655562
#202654642
#202649252
Q8⋊#117175792
#202649042
#477×C1012
#1632101