Finite Group Databasebeta

← #27170091

Subgroups — #27170091

Order 12928.

256 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 53 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#20265803C211yesmaximal, characteristic
2#20265335C212yesmaximal, complemented
3#20265089C211yesmaximal, characteristic
4#20264915C212yesmaximal, complemented
5#323×C101C211yesmaximal, characteristic
6C22×Dic202C2211yescharacteristic
7C2×Dic404—22no—
8#15692234—21no—
9#15692234C2212yes—
10#15692232—22no—
11#15692232—22no—
12C2×C22⋊Dic101—21no—
13C2×C22⋊Dic101—21no—
14#15692180—22no—
15#15692162—21no—
16#15692162C2211yescharacteristic
17C2×C22⋊C4×C101C2211yescharacteristic
18C22⋊C8×C101C2212yes—
19C2×Dic202—22no—
20C2×Dic202—21no—
21C2×Dic202—42no—
22C2×Dic202D412yes—
23C22×Dic101—21no—
24C22×Dic101—21no—
25C22×Dic101—21no—
26C22×Dic101—21no—
27Dic404—44no—
28C4⋊Dic101—21no—
29C4⋊Dic101D412yes—
30#13108053—22no—
31#13108053—22no—
32C22⋊Dic101—22no—
33C22⋊Dic101—22no—
34C22⋊Dic101—42no—
35C24×C101D411yescharacteristic
36C22×C4×C101D411yescharacteristic
37C22×C4×C101C2311yescharacteristic
38C2×C8×C101—22no—
39C22⋊C4×C101—22no—
40C23×C101—21no—
41C23×C101—21no—
42C23×C101—41no—
43C23×C101—41no—
44C23×C101Q1612yescomplemented
45C23×C101C2×D411yescharacteristic
46C2×C4×C101—21no—
47C2×C4×C101—41no—
48C2×C4×C101—41no—
49C2×C4×C101C2×D412yes—
50C2×Dic101—22no—
51C2×Dic101—22no—
52C2×Dic101—22no—
53C2×Dic101—41no—
54C2×Dic101—41no—
55C2×Dic101—41no—
56C2×Dic101—41no—
57C2×Dic101—41no—
58C2×Dic101—81no—
59C2×Dic101—81no—
60Dic202—24no—
61Dic202—42no—
62Dic202—42no—
63Dic202—81no—
64C8×C101—42no—
65C22×C101—21no—
66C22×C101—21no—
67C22×C101—22no—
68C22×C101—42no—
69C22×C101—41no—
70C22×C101—41no—
71C22×C101—81no—
72C22×C101#12011yescharacteristic
73C22×C101C2×Q1611yescharacteristic
74C22×C101#13711yescharacteristic
75C4×C101—22no—
76C4×C101—41no—
77Dic101—42no—
78Dic101—42no—
79Dic101—42no—
80Dic101—81no—
81C2×C101—21no—
82C2×C101—21no—
83C2×C101—42no—
84C2×C101#45111yescharacteristic
85C2×C101#45511yescharacteristic
86C2×C101B(4,𝔽2)11yescharacteristic
87#1602—1011nomaximal, Sylow
88C101#160211yescharacteristic, complemented, Sylow
89#543—1011no—
90#451—1012no—
91#393—1011no—
92#328—1012no—
93#323D10111yescharacteristic
94C22×Q8—1011no—
95C2×Q16—2022no—
96C22⋊Q8—1012no—
97C22⋊Q8—2022no—
98C22⋊Q8—2022no—
99C22⋊Q8—2021no—
100C2×C22⋊C4—2021no—
101C2×C22⋊C4—2021no—
102C2×C22⋊C4C2×D10111yescharacteristic
103#103—2022no—
104C22⋊C8C2×D10112yes—
105#95—1011no—
106#95—2021no—
107C24D40411yescharacteristic
108C2×Q8—1012no—
109C2×Q8—4042no—
110C2×Q8—2021no—
111C2×Q8—2022no—
112C22×C4—2021no—
113C22×C4—2021no—
114C22×C4—2021no—
115C22×C4—2021no—
116C22×C4D40411yescharacteristic
117C22×C4C22×D10111yescharacteristic
118Q16—4044no—
119C2×C8—22no—
120C4⋊C4—1012no—
121C4⋊C4—2022no—
122C4⋊C4—2021no—
123C4⋊C4—2022no—
124C22⋊C4—22no—
125C22⋊C4—4042no—
126C22⋊C4—2022no—
127C22⋊C4—2022no—
128C23—21no—
129C23—21no—
130C23—41no—
131C23—41no—
132C23Dic40412yescomplemented
133C23C2×D40411yescharacteristic
134Q8—8081no—
135Q8—4042no—
136Q8—4042no—
137Q8—2024no—
138C2×C4—21no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—8081no—
142C2×C4—8081no—
143C2×C4—4041no—
144C2×C4—4041no—
145C2×C4—4041no—
146C2×C4—4041no—
147C2×C4—4041no—
148C2×C4—2022no—
149C2×C4—2022no—
150C2×C4—2022no—
151C2×C4D4×D10112yes—
152C8—42no—
153C22—21no—
154C22—21no—
155C22—22no—
156C22—42no—
157C22—41no—
158C22—41no—
159C22—81no—
160C22#1569222511yescharacteristic
161C22C2×Dic40411yescharacteristic
162C22#1569228811yescharacteristic
163C4—22no—
164C4—41no—
165C4—8081no—
166C4—4042no—
167C4—4042no—
168C4—4042no—
169C2—21no—
170C2—21no—
171C2—42no—
172C2#2026533511yescharacteristic
173C2#2026535711yescharacteristic
174C2#2026536511yescharacteristic
subgroupG/H
centerC22#15692225
commutator, cosocleC22×C4×C101C23
FrattiniC22×C4C22×D101
Fitting#323×C101C2
socleC22×C101#120
prime radicalC24×C101D4
hypercenter#323D101
hypocenterC101#1602
LCSG ▷ C22×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #20265335C2semidirect
4. #20264915C2semidirect
44. C23×C101Q16semidirect
88. C101#1602semidirect
132. C23Dic404semidirect

Maximal subgroups

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

subgroupindex
#202658032
#202653352
#202650892
#202649152
#323×C1012
#1602101