Finite Group Databasebeta

← #27170103

Subgroups — #27170103

Order 12928.

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

normal subgroups only

This group has 282 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#20265758C211yesmaximal, characteristic
2#20265318C212yesmaximal, complemented
3#20265097C211yesmaximal, characteristic, complemented
4#20264912C212yesmaximal, complemented
5#324×C101C211yesmaximal, characteristic, complemented
6C2×#13108074—21no—
7C22×Dic202C2211yescharacteristic
8C2×#13108063—22no—
9#15692235—22no—
10#15692230—22no—
11#15692229—21no—
12#15692229C2212yes—
13C2×#13108053—21no—
14#15692180—22no—
15#15692162—21no—
16#15692162C2211yescharacteristic
17C2×C4⋊C4×C101C2211yescharacteristic
18C22⋊C8×C101C2212yes—
19#13108074—42no—
20#13108074—42no—
21C2×Dic202—22no—
22C2×Dic202—21no—
23C2×Dic202—42no—
24C2×D404—21no—
25C2×D404—21no—
26C2×D404D412yes—
27C2×C22⋊D101—22no—
28C22×Dic101—21no—
29C22×Dic101—21no—
30C2×C4×D101—22no—
31#13108063—44no—
32C4⋊Dic101—21no—
33C4⋊Dic101D412yes—
34#13108053—42no—
35#13108051—22no—
36#13108051—22no—
37C22×C4×C101D411yescharacteristic
38C22×C4×C101D411yescharacteristic
39C22×C4×C101C2311yescharacteristic
40C2×C8×C101—22no—
41C4⋊C4×C101—22no—
42C23×C101C2×D411yescharacteristic
43C22×D101—22no—
44C2×C4×C101—21no—
45C2×C4×C101—21no—
46C2×C4×C101—21no—
47C2×C4×C101—41no—
48C2×C4×C101—41no—
49C2×C4×C101SD1612yescomplemented
50C2×C4×C101C2×D412yes—
51C22⋊D101—42no—
52C22⋊D101—42no—
53C2×Dic101—22no—
54C2×Dic101—22no—
55C2×Dic101—22no—
56C2×Dic101—41no—
57C2×Dic101—41no—
58C2×Dic101—41no—
59C2×Dic101—41no—
60D404—42no—
61D404—42no—
62D404—81no—
63C4×D101—44no—
64Dic202—24no—
65Dic202—42no—
66Dic202—42no—
67C8×C101—42no—
68C22×C101—21no—
69C22×C101—21no—
70C22×C101#12011yescharacteristic
71C22×C101C2×SD1611yescharacteristic
72C22×C101AGL(1,ℤ/8)11yescharacteristic
73C2×D101—42no—
74C2×D101—42no—
75C2×D101—42no—
76C4×C101—22no—
77C4×C101—22no—
78C4×C101—41no—
79Dic101—42no—
80Dic101—42no—
81Dic101—42no—
82C2×C101—21no—
83C2×C101—21no—
84C2×C101#45011yescharacteristic
85C2×C101#45611yescharacteristic
86C2×C101B(4,𝔽2)11yescharacteristic
87D101—82no—
88#1604—1011nomaximal, Sylow
89C101#160411yescharacteristic, complemented, Sylow
90#537—1011no—
91#448—1012no—
92#395—1011no—
93#327—1012no—
94#324D10111yescharacteristic, complemented
95C2×#41—2021no—
96C22×Q8—1011no—
97C2×SD16—2022no—
98C22⋊Q8—2022no—
99#121—1012no—
100#121—2022no—
101#121—2021no—
102C2×C4⋊C4—2021no—
103C2×C4⋊C4C2×D10111yescharacteristic
104#103—2022no—
105C22⋊C8C2×D10112yes—
106#95—1011no—
107#95—2021no—
108#41—4042no—
109#41—4042no—
110C2×Q8—4042no—
111C2×Q8—2021no—
112C2×Q8—2022no—
113C2×D4—1012no—
114C2×D4—2021no—
115C2×D4—2021no—
116C2×D4—2022no—
117C22×C4—2022no—
118C22×C4—2021no—
119C22×C4—2021no—
120C22×C4D40411yescharacteristic
121C22×C4D40411yescharacteristic
122C22×C4C22×D10111yescharacteristic
123SD16—4044no—
124C2×C8—22no—
125C4⋊C4—22no—
126C4⋊C4—1012no—
127C4⋊C4—4042no—
128C4⋊C4—2021no—
129C22⋊C4—2022no—
130C22⋊C4—2022no—
131C23—2022no—
132C23C2×D40411yescharacteristic
133Q8—4042no—
134Q8—4042no—
135Q8—2024no—
136D4—8081no—
137D4—4042no—
138D4—4042no—
139D4—4042no—
140D4—4042no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—4044no—
147C2×C4—4041no—
148C2×C4—4041no—
149C2×C4—4041no—
150C2×C4—4041no—
151C2×C4—2022no—
152C2×C4—2022no—
153C2×C4—2022no—
154C2×C4#1310806312yescomplemented
155C2×C4D4×D10112yes—
156C8—42no—
157C22—21no—
158C22—21no—
159C22—4042no—
160C22—4042no—
161C22—4042no—
162C22#1569222511yescharacteristic
163C22C2×#1310806311yescharacteristic
164C22#1569228311yescharacteristic
165C4—22no—
166C4—22no—
167C4—41no—
168C4—4042no—
169C4—4042no—
170C4—4042no—
171C2—21no—
172C2—21no—
173C2—8082no—
174C2#2026533011yescharacteristic
175C2#2026536211yescharacteristic
176C2#2026536511yescharacteristic
subgroupG/H
centerC22#15692225
commutator, cosocleC22×C4×C101C23
FrattiniC22×C4C22×D101
Fitting#324×C101C2
socleC22×C101#120
prime radical#20265758C2
hypercenter#324D101
hypocenterC101#1604
LCSG ▷ C22×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
2. #20265318C2semidirect
3. #20265097C2semidirect
4. #20264912C2semidirect
5. #324×C101C2semidirect
49. C2×C4×C101SD16semidirect
89. C101#1604semidirect
94. #324D101semidirect
154. C2×C4#13108063semidirect

Maximal subgroups

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

subgroupindex
#202657582
#202653182
#202650972
#202649122
#324×C1012
#1604101