Finite Group Databasebeta

← #27170268

Subgroups — #27170268

Order 12928.

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

normal subgroups only

This group has 214 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#20265884C211yesmaximal, characteristic
2#20265557C211yesmaximal, characteristic, complemented
3#20265473C211yesmaximal, characteristic, complemented
4#20265467C211yesmaximal, characteristic
5#20264925C211yesmaximal, characteristic, complemented
6#20264909C211yesmaximal, characteristic, complemented
7D4⋊C8×C101C211yesmaximal, characteristic
8C2×Dic404—22no—
9#15692255C2211yescharacteristic
10#15692255C2211yescharacteristic
11#15692246—21no—
12#15692234—21no—
13#15692232—22no—
14C4×Dic202C2211yescharacteristic
15D4×Dic101—21no—
16C2×C4⋊Dic101—21no—
17#15692189—21no—
18#15692186—21no—
19C4⋊#11717579C2211yescharacteristic
20#15692180—21no—
21#15692176—21no—
22C4×D4×C101C2211yescharacteristic
23C4⋊C8×C101C2211yescharacteristic
24C4×C8×C101C2211yescharacteristic
25C2×Dic202—21no—
26C2×Dic202D411yescharacteristic
27C2×Dic202D412yes—
28C22×Dic101—22no—
29Dic404—44no—
30C2×#11717579—21no—
31C4⋊Dic101—22no—
32C4⋊Dic101—21no—
33C4⋊Dic101—21no—
34C4⋊Dic101—21no—
35C4⋊Dic101—21no—
36C4⋊Dic101D411yescharacteristic
37#13108053—22no—
38#13108053—21no—
39C22⋊Dic101—22no—
40C4×Dic101—21no—
41C2×D4×C101D411yescharacteristic
42C22×C4×C101—21no—
43C2×C8×C101—21no—
44C2×C8×C101—21no—
45C4⋊C4×C101D411yescharacteristic
46C22⋊C4×C101—21no—
47C42×C101C2311yescharacteristic
48C23×C101—21no—
49D4×C101—21no—
50D4×C101Q1612yescomplemented
51C2×C4×C101—21no—
52C2×C4×C101—41no—
53C2×C4×C101—41no—
54C2×C4×C101C2×D411yescharacteristic
55C2×C4×C101C2×D411yescharacteristic
56C2×C4×C101C2×D411yescharacteristic
57C2×Dic101—21no—
58C2×Dic101—21no—
59C2×Dic101—22no—
60C2×Dic101—41no—
61C2×Dic101—82no—
62Dic202—21no—
63Dic202—21no—
64Dic202—42no—
65Dic202—42no—
66C8×C101—22no—
67C8×C101—41no—
68#11717579—41no—
69C22×C101—22no—
70C22×C101—41no—
71C22×C101—41no—
72C22×C101#12011yescharacteristic
73C4×C101—21no—
74C4×C101—41no—
75C4×C101C2×Q1611yescharacteristic
76C4×C101#13511yescharacteristic
77C4×C101#13711yescharacteristic
78C4×C101#13711yescharacteristic
79Dic101—41no—
80Dic101—42no—
81Dic101—41no—
82Dic101—81no—
83C2×C101—42no—
84C2×C101#45111yescharacteristic
85C2×C101#45211yescharacteristic
86C2×C101#45311yescharacteristic
87#1632—1011nomaximal, Sylow
88C101#163211yescharacteristic, complemented, Sylow
89#554—1011no—
90#494—1011no—
91#477—1011no—
92#476—1011no—
93#333—1011no—
94Q8⋊C8—1011no—
95D4⋊C8D10111yescharacteristic
96C2×Q16—2022no—
97C4⋊Q8—1011no—
98C4⋊Q8—1011no—
99#125—2021no—
100C22⋊Q8—2022no—
101C22⋊Q8—2021no—
102C4×Q8—1011no—
103C4×D4—2021no—
104C4×D4C2×D10111yescharacteristic
105C2×C4⋊C4—2021no—
106#107—2021no—
107#106—2021no—
108C4⋊C8—1011no—
109C4⋊C8C2×D10111yescharacteristic
110#103—2021no—
111#102—2021no—
112C4×C8C2×D10111yescharacteristic
113C2×Q8—1012no—
114C2×Q8—1011no—
115C2×Q8—2021no—
116C2×D4D40411yescharacteristic
117C22×C4—21no—
118C22×C4—2022no—
119Q16—4044no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—2021no—
123C4⋊C4—1011no—
124C4⋊C4—2022no—
125C4⋊C4—2022no—
126C4⋊C4—2021no—
127C4⋊C4—2021no—
128C4⋊C4—2021no—
129C4⋊C4—2021no—
130C4⋊C4—2021no—
131C4⋊C4D40411yescharacteristic
132C22⋊C4—21no—
133C22⋊C4—2022no—
134C42—2021no—
135C42C22×D10111yescharacteristic
136C23—21no—
137Q8—4042no—
138Q8—4042no—
139Q8—2021no—
140Q8—2021no—
141D4—21no—
142D4Dic40412yescomplemented
143C2×C4—21no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—8082no—
147C2×C4—4041no—
148C2×C4—2021no—
149C2×C4—2021no—
150C2×C4—2022no—
151C2×C4D4×D10111yescharacteristic
152C2×C4D4×D10111yescharacteristic
153C2×C4C2×D40411yescharacteristic
154C8—22no—
155C8—41no—
156C8—4041no—
157C22—22no—
158C22—41no—
159C22—41no—
160C22#1569222511yescharacteristic
161C4—21no—
162C4—41no—
163C4—8081no—
164C4—4041no—
165C4—4041no—
166C4—4042no—
167C4C2×Dic40411yescharacteristic
168C4#1569227411yescharacteristic
169C4#1569228411yescharacteristic
170C4#1569228811yescharacteristic
171C2—42no—
172C2#2026533511yescharacteristic
173C2#2026534011yescharacteristic
174C2#2026534611yescharacteristic
subgroupG/H
centerC22#15692225
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
FittingD4⋊C8×C101C2
socleC22×C101#120
prime radicalC2×D4×C101D4
hypercenterD4⋊C8D101
hypocenterC101#1632
LCSG ▷ C42×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #20265557C2semidirect
3. #20265473C2semidirect
5. #20264925C2semidirect
6. #20264909C2semidirect
50. D4×C101Q16semidirect
88. C101#1632semidirect
142. D4Dic404semidirect

Maximal subgroups

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

subgroupindex
#202658842
#202655572
#202654732
#202654672
#202649252
#202649092
D4⋊C8×C1012
#1632101