Finite Group Databasebeta

← #27170137

Subgroups — #27170137

Order 12928.

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

normal subgroups only

This group has 254 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#20265491C212yesmaximal, complemented
3#20265107C211yesmaximal, 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
12C2×#13108053—21no—
13#15692189—22no—
14#15692177—22no—
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#13108053—42no—
33#13108051—22no—
34#13108051—22no—
35C22×C4×C101D411yescharacteristic
36C22×C4×C101D411yescharacteristic
37C22×C4×C101C2311yescharacteristic
38C2×C8×C101—22no—
39C4⋊C4×C101—22no—
40C23×C101C2×D411yescharacteristic
41C22×D101—22no—
42C2×C4×C101—21no—
43C2×C4×C101—21no—
44C2×C4×C101—21no—
45C2×C4×C101—41no—
46C2×C4×C101—41no—
47C2×C4×C101D812yes—
48C2×C4×C101C2×D412yes—
49C22⋊D101—42no—
50C22⋊D101—42no—
51C2×Dic101—22no—
52C2×Dic101—22no—
53C2×Dic101—41no—
54C2×Dic101—41no—
55C2×Dic101—41no—
56C2×Dic101—41no—
57C2×Dic101—41no—
58D404—42no—
59D404—42no—
60D404—81no—
61C4×D101—44no—
62Dic202—42no—
63C8×C101—42no—
64C22×C101—21no—
65C22×C101—21no—
66C22×C101#12011yescharacteristic
67C22×C101C2×D811yescharacteristic
68C22×C101AGL(1,ℤ/8)11yescharacteristic
69C2×D101—42no—
70C2×D101—42no—
71C2×D101—42no—
72C4×C101—22no—
73C4×C101—22no—
74C4×C101—41no—
75Dic101—42no—
76Dic101—42no—
77Dic101—81no—
78C2×C101—21no—
79C2×C101—21no—
80C2×C101#44711yescharacteristic
81C2×C101#45611yescharacteristic
82C2×C101#45811yescharacteristic
83D101—82no—
84#1610—1011nomaximal, Sylow
85C101#161011yescharacteristic, complemented, Sylow
86#537—1011no—
87#480—1012no—
88#398—1011no—
89#327—1012no—
90#324D10111yescharacteristic, complemented
91C2×#41—2021no—
92C22⋊Q8—2022no—
93#121—1012no—
94#121—2022no—
95#121—2021no—
96C2×C4⋊C4—1011no—
97C2×C4⋊C4—2021no—
98C2×C4⋊C4C2×D10111yescharacteristic
99#107—2022no—
100#102—2022no—
101C22⋊C8C2×D10112yes—
102#95—1011no—
103#95—2021no—
104#41—4042no—
105#41—4042no—
106C2×Q8—2021no—
107C2×D4—1012no—
108C2×D4—2021no—
109C2×D4—2021no—
110C2×D4—2022no—
111C22×C4—2022no—
112C22×C4—2021no—
113C22×C4—2021no—
114C22×C4D40411yescharacteristic
115C22×C4D40411yescharacteristic
116C22×C4C22×D10111yescharacteristic
117C2×C8—22no—
118C4⋊C4—22no—
119C4⋊C4—1012no—
120C4⋊C4—4042no—
121C4⋊C4—2021no—
122C4⋊C4—2022no—
123C22⋊C4—2022no—
124C22⋊C4—2022no—
125C23—2022no—
126C23C2×D40411yescharacteristic
127Q8—4042no—
128D4—8081no—
129D4—4042no—
130D4—4042no—
131D4—4042no—
132D4—4042no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—4044no—
139C2×C4—4041no—
140C2×C4—4041no—
141C2×C4—4041no—
142C2×C4—4041no—
143C2×C4—4041no—
144C2×C4—2022no—
145C2×C4—2022no—
146C2×C4D80812yes—
147C2×C4D4×D10112yes—
148C8—42no—
149C22—21no—
150C22—21no—
151C22—4042no—
152C22—4042no—
153C22—4042no—
154C22#1569222511yescharacteristic
155C22C2×D80811yescharacteristic
156C22#1569228311yescharacteristic
157C4—22no—
158C4—22no—
159C4—41no—
160C4—8081no—
161C4—4042no—
162C4—4042no—
163C2—21no—
164C2—21no—
165C2—8082no—
166C2#2026531311yescharacteristic
167C2#2026536211yescharacteristic
168C2#2026536811yescharacteristic
subgroupG/H
centerC22#15692225
commutator, cosocleC22×C4×C101C23
FrattiniC22×C4C22×D101
Fitting#324×C101C2
socleC22×C101#120
prime radical#20265758C2
hypercenter#324D101
hypocenterC101#1610
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. #20265491C2semidirect
3. #20265107C2semidirect
4. #20264912C2semidirect
5. #324×C101C2semidirect
85. C101#1610semidirect
90. #324D101semidirect

Maximal subgroups

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

subgroupindex
#202657582
#202654912
#202651072
#202649122
#324×C1012
#1610101