Finite Group Databasebeta

← #27169465

Subgroups — #27169465

Order 12928.

252 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 67 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
1C2×#15692213C211yesmaximal, characteristic
2#20265231C212yesmaximal, complemented
3#20264903C211yesmaximal, characteristic, complemented
4#20264900C212yesmaximal
5#324×C101C211yesmaximal, characteristic, complemented
6C22×C4×D101C411yescharacteristic
7#15692213—22no—
8#15692213—22no—
9C2×C4⋊Dic101C411yescharacteristic
10C2×#13108051C411yescharacteristic
11C2×#13108051C2211yescharacteristic
12C2×C4×Dic101C411yescharacteristic
13C2×C4×Dic101C2211yescharacteristic
14#15692184—22no—
15C22⋊#11717579C2212yes—
16#15692165—22no—
17C2×C4⋊C4×C101C2211yescharacteristic
18C22⋊C8×C101C2212yes—
19C23×D101C2×C411yescharacteristic
20C22×Dic101C2×C411yescharacteristic
21C22×Dic101C2×C411yescharacteristic
22C22×Dic101C2×C411yescharacteristic
23C2×C4×D101—22no—
24C2×C4×D101—22no—
25C2×C4×D101—21no—
26C2×C4×D101—21no—
27C2×#11717579—22no—
28C4⋊Dic101—22no—
29#13108051—22no—
30#13108051—22no—
31C4×Dic101—21no—
32C4×Dic101—21no—
33C4×Dic101D412yes—
34C4×Dic101D412yes—
35C22×C4×C101C2×C411yescharacteristic
36C22×C4×C101C2×C411yescharacteristic
37C22×C4×C101C2311yescharacteristic
38C2×C8×C101—22no—
39C4⋊C4×C101—22no—
40C23×C101C22×C411yescharacteristic
41C22×D101—21no—
42C22×D101—21no—
43C22×D101—41no—
44C22×D101—41no—
45C22×D101M1612yes—
46C2×C4×C101—21no—
47C2×C4×C101—21no—
48C2×C4×C101—21no—
49C2×C4×C101—41no—
50C2×C4×C101—41no—
51C2×C4×C101M1612yes—
52C2×C4×C101C2×D412yes—
53C2×Dic101—21no—
54C2×Dic101—21no—
55C2×Dic101—21no—
56C2×Dic101—41no—
57C2×Dic101—41no—
58C2×Dic101C22⋊C411yescharacteristic
59C2×Dic101C22⋊C411yescharacteristic
60C2×Dic101C22⋊C412yes—
61C4×D101—24no—
62C4×D101—24no—
63C8×C101—42no—
64#11717579—42no—
65C22×C101—21no—
66C22×C101—21no—
67C22×C101C2×C22⋊C411yescharacteristic
68C22×C101C2×M1611yescharacteristic
69C22×C101C2×M1611yescharacteristic
70C2×D101—22no—
71C2×D101—42no—
72C2×D101—41no—
73C2×D101—41no—
74C2×D101—42no—
75C4×C101—22no—
76C4×C101—22no—
77C4×C101—41no—
78Dic101—21no—
79Dic101—21no—
80Dic101—22no—
81Dic101—41no—
82C2×C101—21no—
83C2×C101—21no—
84C2×C101#40711yescharacteristic
85C2×C101#41011yescharacteristic
86C2×C101#41311yescharacteristic
87D101—42no—
88#1464—1011nomaximal, Sylow
89C101#146411yescharacteristic, complemented, Sylow
90C2×#117—1011no—
91#432—1012no—
92#324—1011no—
93#324D10111yescharacteristic, complemented
94#323—1012no—
95C23×C4—1011no—
96#117—2022no—
97#117—2022no—
98C2×C4⋊C4—1011no—
99C2×C4⋊C4C2×D10111yescharacteristic
100C2×C22⋊C4—1011no—
101C2×C22⋊C4—1011no—
102C2×C42—1011no—
103C2×C42—1011no—
104C4⋊C8—2022no—
105C22⋊C8—1012no—
106C22⋊C8C2×D10112yes—
107#97—2022no—
108C24—1011no—
109C22×C4—1011no—
110C22×C4—1011no—
111C22×C4—1011no—
112C22×C4—2021no—
113C22×C4—2021no—
114C22×C4—2022no—
115C22×C4—2022no—
116C22×C4C4×D10111yescharacteristic
117C22×C4C4×D10111yescharacteristic
118C22×C4C22×D10111yescharacteristic
119C2×C8—22no—
120C2×C8—2022no—
121C4⋊C4—22no—
122C4⋊C4—2022no—
123C22⋊C4—2022no—
124C22⋊C4—2022no—
125C42—1012no—
126C42—1012no—
127C42—2021no—
128C42—2021no—
129C23—1012no—
130C23—4041no—
131C23—4041no—
132C23—2021no—
133C23—2021no—
134C23C2×C4×D10111yescharacteristic
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—1011no—
141C2×C4—1011no—
142C2×C4—1012no—
143C2×C4—4041no—
144C2×C4—4041no—
145C2×C4—2024no—
146C2×C4—2024no—
147C2×C4—2021no—
148C2×C4—2021no—
149C2×C4—2021no—
150C2×C4#1310805712yes—
151C2×C4D4×D10112yes—
152C8—42no—
153C8—4042no—
154C22—21no—
155C22—21no—
156C22—4042no—
157C22—4042no—
158C22—4041no—
159C22—4041no—
160C22—2022no—
161C22C22⋊C4×D10111yescharacteristic
162C22M16×D10111yescharacteristic
163C22C2×#1310805711yescharacteristic
164C4—22no—
165C4—22no—
166C4—41no—
167C4—4041no—
168C4—2021no—
169C4—2021no—
170C4—2022no—
171C2—21no—
172C2—21no—
173C2—4042no—
174C2#2026513911yescharacteristic
175C2#2026514911yescharacteristic
176C2#2026516011yescharacteristic
subgroupG/H
centerC22C22⋊C4×D101
commutatorC23×C101C22×C4
FrattiniC22×C4C22×D101
Fitting#324×C101C2
socleC22×C101C2×C22⋊C4
prime radicalC23×D101C2×C4
cosocleC22×C4×C101C23
hypercenter#324D101
hypocenterC101#1464
LCSG ▷ C23×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
2. #20265231C2semidirect
3. #20264903C2semidirect
5. #324×C101C2semidirect
89. C101#1464semidirect
93. #324D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922132
#202652312
#202649032
#202649002
#324×C1012
#1464101