Finite Group Databasebeta

← #27170171

Subgroups — #27170171

Order 12928.

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

normal subgroups only

This group has 246 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#20265496C212yesmaximal, complemented
3#20265116C211yesmaximal, characteristic, complemented
4#20264912C212yesmaximal, complemented
5#324×C101C211yesmaximal, characteristic, complemented
6C2×#13108074—21no—
7#15692235—22no—
8#15692230—22no—
9#15692229—21no—
10#15692229C2212yes—
11C2×C4⋊Dic101C2211yescharacteristic
12#15692187—22no—
13#15692177—22no—
14#15692162—21no—
15#15692162—21no—
16#15692162C2211yescharacteristic
17C2×C4⋊C4×C101C2211yescharacteristic
18C22⋊C8×C101C2212yes—
19#13108074—42no—
20#13108074—42no—
21C2×Dic202—21no—
22C2×D404—21no—
23C2×D404—21no—
24C2×D404D412yes—
25C2×C22⋊D101—22no—
26C22×Dic101—21no—
27C22×Dic101—21no—
28C2×C4×D101—22no—
29C4⋊Dic101—21no—
30C4⋊Dic101—22no—
31C4⋊Dic101D412yes—
32#13108051—22no—
33#13108051—22no—
34C22×C4×C101D411yescharacteristic
35C22×C4×C101D411yescharacteristic
36C22×C4×C101C2311yescharacteristic
37C2×C8×C101—22no—
38C4⋊C4×C101—22no—
39C23×C101C2×D411yescharacteristic
40C22×D101—22no—
41C2×C4×C101—21no—
42C2×C4×C101—21no—
43C2×C4×C101—21no—
44C2×C4×C101—41no—
45C2×C4×C101—41no—
46C2×C4×C101SD1612yes—
47C2×C4×C101C2×D412yes—
48C22⋊D101—42no—
49C22⋊D101—42no—
50C2×Dic101—22no—
51C2×Dic101—41no—
52C2×Dic101—41no—
53C2×Dic101—42no—
54C2×Dic101—41no—
55C2×Dic101—41no—
56D404—42no—
57D404—42no—
58D404—81no—
59C4×D101—44no—
60Dic202—42no—
61C8×C101—42no—
62C22×C101—21no—
63C22×C101—21no—
64C22×C101#12011yescharacteristic
65C22×C101C2×SD1611yescharacteristic
66C22×C101AGL(1,ℤ/8)11yescharacteristic
67C2×D101—42no—
68C2×D101—42no—
69C2×D101—42no—
70C4×C101—22no—
71C4×C101—22no—
72C4×C101—41no—
73Dic101—42no—
74Dic101—81no—
75Dic101—81no—
76C2×C101—21no—
77C2×C101—21no—
78C2×C101#45011yescharacteristic
79C2×C101#45411yescharacteristic
80C2×C101#45811yescharacteristic
81D101—82no—
82#1616—1011nomaximal, Sylow
83C101#161611yescharacteristic, complemented, Sylow
84#537—1011no—
85#481—1012no—
86#400—1011no—
87#327—1012no—
88#324D10111yescharacteristic, complemented
89C2×#41—2021no—
90C22⋊Q8—2022no—
91#121—1012no—
92#121—2022no—
93#121—2021no—
94C2×C4⋊C4—1011no—
95C2×C4⋊C4C2×D10111yescharacteristic
96#106—2022no—
97#102—2022no—
98C22⋊C8C2×D10112yes—
99#95—1011no—
100#95—2021no—
101#95—2021no—
102#41—4042no—
103#41—4042no—
104C2×Q8—2021no—
105C2×D4—1012no—
106C2×D4—2021no—
107C2×D4—2021no—
108C2×D4—2022no—
109C22×C4—2022no—
110C22×C4—2021no—
111C22×C4—2021no—
112C22×C4D40411yescharacteristic
113C22×C4D40411yescharacteristic
114C22×C4C22×D10111yescharacteristic
115C2×C8—22no—
116C4⋊C4—22no—
117C4⋊C4—1012no—
118C4⋊C4—2021no—
119C4⋊C4—2022no—
120C22⋊C4—2022no—
121C22⋊C4—2022no—
122C23—2022no—
123C23C2×D40411yescharacteristic
124Q8—4042no—
125D4—8081no—
126D4—4042no—
127D4—4042no—
128D4—4042no—
129D4—4042no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—4044no—
136C2×C4—4041no—
137C2×C4—4041no—
138C2×C4—4042no—
139C2×C4—4041no—
140C2×C4—4041no—
141C2×C4—2022no—
142C2×C4#1310806312yes—
143C2×C4D4×D10112yes—
144C8—42no—
145C22—21no—
146C22—21no—
147C22—4042no—
148C22—4042no—
149C22—4042no—
150C22#1569222511yescharacteristic
151C22C2×#1310806311yescharacteristic
152C22#1569228311yescharacteristic
153C4—22no—
154C4—22no—
155C4—41no—
156C4—8081no—
157C4—8081no—
158C4—4042no—
159C2—21no—
160C2—21no—
161C2—8082no—
162C2#2026533011yescharacteristic
163C2#2026535211yescharacteristic
164C2#2026536811yescharacteristic
subgroupG/H
centerC22#15692225
commutator, cosocleC22×C4×C101C23
FrattiniC22×C4C22×D101
Fitting#324×C101C2
socleC22×C101#120
prime radical#20265758C2
hypercenter#324D101
hypocenterC101#1616
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. #20265496C2semidirect
3. #20265116C2semidirect
4. #20264912C2semidirect
5. #324×C101C2semidirect
83. C101#1616semidirect
88. #324D101semidirect

Maximal subgroups

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

subgroupindex
#202657582
#202654962
#202651162
#202649122
#324×C1012
#1616101