Finite Group Databasebeta

← #27170154

Subgroups — #27170154

Order 12928.

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

normal subgroups only

This group has 288 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#20265739C211yesmaximal, characteristic
2#20265496C212yesmaximal, complemented
3#20265113C211yesmaximal, characteristic, complemented
4#20264912C212yesmaximal, complemented
5#323×C101C211yesmaximal, characteristic, complemented
6C22×C22⋊D101—21no—
7#15692239—22no—
8#15692229—21no—
9#15692229C2212yes—
10#15692225—22no—
11C2×C4⋊Dic101C2211yescharacteristic, complemented
12C2×#13108053—21no—
13C2×C22⋊Dic101—21no—
14#15692187—22no—
15#15692177—22no—
16#15692162C2211yescharacteristic, complemented
17C2×C22⋊C4×C101C2211yescharacteristic
18C22⋊C8×C101C2212yescomplemented
19C23×D101—21no—
20C2×D404—21no—
21C2×D404D412yes—
22C2×C22⋊D101—22no—
23C2×C22⋊D101—22no—
24C2×C22⋊D101—42no—
25C2×C22⋊D101—42no—
26C22×Dic101—21no—
27C22×Dic101—21no—
28C22×Dic101—21no—
29C4⋊Dic101—21no—
30C4⋊Dic101—22no—
31C4⋊Dic101D412yes—
32#13108053—42no—
33C22⋊Dic101—42no—
34#13108051—22no—
35#13108051—22no—
36C24×C101D411yescharacteristic
37C22×C4×C101D411yescharacteristic
38C22×C4×C101C2311yescharacteristic
39C2×C8×C101—22no—
40C22⋊C4×C101—22no—
41C23×C101—21no—
42C23×C101—21no—
43C23×C101—41no—
44C23×C101—41no—
45C23×C101SD1612yes—
46C23×C101C2×D411yescharacteristic
47C22×D101—22no—
48C22×D101—42no—
49C22×D101—81no—
50C22×D101—81no—
51C2×C4×C101—21no—
52C2×C4×C101—41no—
53C2×C4×C101—41no—
54C2×C4×C101C2×D412yes—
55C22⋊D101—44no—
56C22⋊D101—42no—
57C22⋊D101—42no—
58C2×Dic101—22no—
59C2×Dic101—22no—
60C2×Dic101—41no—
61C2×Dic101—42no—
62C2×Dic101—41no—
63C2×Dic101—41no—
64C2×Dic101—41no—
65C2×Dic101—41no—
66D404—42no—
67D404—81no—
68C8×C101—42no—
69C22×C101—21no—
70C22×C101—21no—
71C22×C101—22no—
72C22×C101—42no—
73C22×C101—41no—
74C22×C101—41no—
75C22×C101—81no—
76C22×C101#12011yescharacteristic
77C22×C101C2×SD1611yescharacteristic
78C22×C101AGL(1,ℤ/8)11yescharacteristic
79C2×D101—42no—
80C2×D101—42no—
81C2×D101—42no—
82C2×D101—84no—
83C4×C101—22no—
84C4×C101—41no—
85Dic101—42no—
86Dic101—42no—
87Dic101—81no—
88C2×C101—21no—
89C2×C101—21no—
90C2×C101—42no—
91C2×C101#45011yescharacteristic
92C2×C101#45311yescharacteristic
93C2×C101#45811yescharacteristic
94D101—82no—
95#1613—1011nomaximal, Sylow
96C101#161311yescharacteristic, complemented, Sylow
97#534—1011no—
98#481—1012no—
99#399—1011no—
100#327—1012no—
101#323D10111yescharacteristic, complemented
102C22×D4—2021no—
103#123—2022no—
104#121—1012no—
105#121—2021no—
106#120—2022no—
107C2×C4⋊C4—1011no—
108C2×C4⋊C4—2021no—
109C2×C22⋊C4—2021no—
110C2×C22⋊C4C2×D10111yescharacteristic
111#106—2022no—
112#102—2022no—
113C22⋊C8C2×D10112yescomplemented
114#95—1011no—
115C24—2021no—
116C24D40411yescharacteristic
117C2×D4—1012no—
118C2×D4—4042no—
119C2×D4—4042no—
120C2×D4—2021no—
121C2×D4—2022no—
122C2×D4—2022no—
123C22×C4—2021no—
124C22×C4—2021no—
125C22×C4—2021no—
126C22×C4D40411yescharacteristic
127C22×C4C22×D10111yescharacteristic
128C2×C8—22no—
129C4⋊C4—1012no—
130C4⋊C4—4042no—
131C4⋊C4—2021no—
132C4⋊C4—2022no—
133C22⋊C4—22no—
134C22⋊C4—4042no—
135C22⋊C4—2022no—
136C22⋊C4—2022no—
137C23—21no—
138C23—21no—
139C23—41no—
140C23—41no—
141C23—8081no—
142C23—8081no—
143C23—4042no—
144C23—2022no—
145C23#1310806312yes—
146C23C2×D40411yescharacteristic
147D4—8081no—
148D4—4042no—
149D4—4044no—
150D4—4042no—
151D4—4042no—
152C2×C4—21no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—4041no—
156C2×C4—4041no—
157C2×C4—4041no—
158C2×C4—4042no—
159C2×C4—4041no—
160C2×C4—4041no—
161C2×C4—2022no—
162C2×C4—2022no—
163C2×C4D4×D10112yes—
164C8—42no—
165C22—21no—
166C22—21no—
167C22—22no—
168C22—42no—
169C22—41no—
170C22—41no—
171C22—81no—
172C22—8084no—
173C22—4042no—
174C22—4042no—
175C22—4042no—
176C22#1569222511yescharacteristic
177C22C2×#1310806311yescharacteristic
178C22#1569228311yescharacteristic
179C4—22no—
180C4—41no—
181C4—8081no—
182C4—4042no—
183C4—4042no—
184C2—21no—
185C2—21no—
186C2—42no—
187C2—8082no—
188C2#2026533011yescharacteristic
189C2#2026534711yescharacteristic
190C2#2026536811yescharacteristic
subgroupG/H
centerC22#15692225
commutator, cosocleC22×C4×C101C23
FrattiniC22×C4C22×D101
Fitting#323×C101C2
socleC22×C101#120
prime radical#20265739C2
hypercenter#323D101
hypocenterC101#1613
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. #20265496C2semidirect
3. #20265113C2semidirect
4. #20264912C2semidirect
5. #323×C101C2semidirect
11. C2×C4⋊Dic101C22semidirect
16. #15692162C22semidirect
18. C22⋊C8×C101C22semidirect
96. C101#1613semidirect
101. #323D101semidirect
113. C22⋊C8C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202657392
#202654962
#202651132
#202649122
#323×C1012
#1613101