Finite Group Databasebeta

← #27170266

Subgroups — #27170266

Order 12928.

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

normal subgroups only

This group has 214 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#20265556C211yesmaximal, characteristic, complemented
2#20265472C211yesmaximal, characteristic, complemented
3#20265467C211yesmaximal, characteristic
4#20264926C211yesmaximal, characteristic, complemented
5Q8⋊#11717579C211yesmaximal, characteristic, complemented
6D4⋊#11717579C211yesmaximal, characteristic
7#554×C101C211yesmaximal, characteristic
8C2×#13108064—22no—
9#15692255C2211yescharacteristic
10#15692188—21no—
11#15692186—21no—
12C4⋊#11717579C2211yescharacteristic
13C4⋊#11717579C2211yescharacteristic
14#15692179—21no—
15#15692176—21no—
16C4×#11717579C2211yescharacteristic
17C4⋊Q8×C101C2211yescharacteristic
18#125×C101—21no—
19C22⋊Q8×C101—21no—
20C22⋊Q8×C101—22no—
21C4×Q8×C101C2211yescharacteristic
22C4×D4×C101—21no—
23C4×D4×C101C2211yescharacteristic
24C2×C4⋊C4×C101—21no—
25C2×Dic202—21no—
26#13108064—44no—
27C2×#11717579—21no—
28C2×#11717579—21no—
29C2×#11717579—21no—
30C4⋊Dic101—21no—
31C4⋊Dic101—21no—
32C2×Q8×C101D411yescharacteristic
33C2×Q8×C101D412yes—
34C2×D4×C101D411yescharacteristic
35C22×C4×C101—22no—
36C22×C4×C101—21no—
37C4⋊C4×C101—22no—
38C4⋊C4×C101—22no—
39C4⋊C4×C101—21no—
40C4⋊C4×C101—21no—
41C4⋊C4×C101—21no—
42C4⋊C4×C101D411yescharacteristic
43C4⋊C4×C101D411yescharacteristic
44C22⋊C4×C101—22no—
45C22⋊C4×C101—21no—
46C42×C101—21no—
47C42×C101C2311yescharacteristic
48C23×C101—21no—
49Q8×C101—21no—
50Q8×C101—21no—
51Q8×C101—42no—
52D4×C101—21no—
53D4×C101Q1612yescomplemented
54C2×C4×C101—21no—
55C2×C4×C101—21no—
56C2×C4×C101—21no—
57C2×C4×C101—22no—
58C2×C4×C101—41no—
59C2×C4×C101—41no—
60C2×C4×C101—82no—
61C2×C4×C101C2×D411yescharacteristic
62C2×C4×C101C2×D411yescharacteristic
63C2×C4×C101C2×D411yescharacteristic
64C2×Dic101—41no—
65Dic202—42no—
66#11717579—22no—
67#11717579—41no—
68#11717579—41no—
69C22×C101—22no—
70C22×C101—41no—
71C22×C101—41no—
72C22×C101#12011yescharacteristic
73C4×C101—21no—
74C4×C101—41no—
75C4×C101—41no—
76C4×C101—41no—
77C4×C101—42no—
78C4×C101C2×Q1611yescharacteristic
79C4×C101#13511yescharacteristic
80C4×C101#13711yescharacteristic
81C4×C101#13711yescharacteristic
82Dic101—81no—
83C2×C101—42no—
84C2×C101#45111yescharacteristic
85C2×C101#45211yescharacteristic
86C2×C101#45311yescharacteristic
87#1632—1011nomaximal, Sylow
88C101#163211yescharacteristic, complemented, Sylow
89#554D10111yescharacteristic
90#494—1011no—
91#477—1011no—
92#476—1011no—
93#333—1011no—
94Q8⋊C8—1011no—
95D4⋊C8—1011no—
96C2×Q16—2022no—
97C4⋊Q8—1011no—
98C4⋊Q8C2×D10111yescharacteristic
99#125—21no—
100C22⋊Q8—21no—
101C22⋊Q8—22no—
102C4×Q8C2×D10111yescharacteristic
103C4×D4—21no—
104C4×D4C2×D10111yescharacteristic
105C2×C4⋊C4—21no—
106#107—2021no—
107#106—2021no—
108C4⋊C8—1011no—
109C4⋊C8—1011no—
110#103—2021no—
111#102—2021no—
112C4×C8—1011no—
113C2×Q8—2021no—
114C2×Q8C22⋊D10111yescharacteristic
115C2×Q8C22⋊D10112yes—
116C2×D4C22⋊D10111yescharacteristic
117C22×C4—22no—
118C22×C4—21no—
119Q16—4044no—
120C2×C8—2021no—
121C2×C8—2021no—
122C2×C8—2021no—
123C4⋊C4—22no—
124C4⋊C4—22no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—2021no—
129C4⋊C4—2021no—
130C4⋊C4C22⋊D10111yescharacteristic
131C4⋊C4C22⋊D10111yescharacteristic
132C22⋊C4—22no—
133C22⋊C4—21no—
134C42—21no—
135C42C22×D10111yescharacteristic
136C23—21no—
137Q8—21no—
138Q8—21no—
139Q8—42no—
140Q8—4042no—
141D4—21no—
142D4#1310806412yescomplemented
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—22no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—82no—
150C2×C4—4041no—
151C2×C4C2×C22⋊D10111yescharacteristic
152C2×C4C2×C22⋊D10111yescharacteristic
153C2×C4C2×C22⋊D10111yescharacteristic
154C8—4041no—
155C8—4041no—
156C8—2022no—
157C22—22no—
158C22—41no—
159C22—41no—
160C22#1569222411yescharacteristic
161C4—21no—
162C4—41no—
163C4—41no—
164C4—41no—
165C4—42no—
166C4—8081no—
167C4C2×#1310806411yescharacteristic
168C4#1569227511yescharacteristic
169C4#1569228511yescharacteristic
170C4#1569228611yescharacteristic
171C2—42no—
172C2#2026533311yescharacteristic
173C2#2026533911yescharacteristic
174C2#2026534511yescharacteristic
subgroupG/H
centerC22#15692224
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
Fitting#554×C101C2
socleC22×C101#120
prime radicalC2×D4×C101D4
hypercenter#554D101
hypocenterC101#1632
LCSG ▷ C42×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #554

Complements

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

NH ≅ G/Ntype
1. #20265556C2semidirect
2. #20265472C2semidirect
4. #20264926C2semidirect
5. Q8⋊#11717579C2semidirect
53. D4×C101Q16semidirect
88. C101#1632semidirect
142. D4#13108064semidirect

Maximal subgroups

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

subgroupindex
#202655562
#202654722
#202654672
#202649262
Q8⋊#117175792
D4⋊#117175792
#554×C1012
#1632101