Finite Group Databasebeta

← #27170086

Subgroups — #27170086

Order 12928.

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

normal subgroups only

This group has 308 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#20265318C212yesmaximal, complemented
3#20265089C211yesmaximal, characteristic, complemented
4#20264912C212yesmaximal, complemented
5#323×C101C211yesmaximal, characteristic, complemented
6C22×Dic202C2211yescharacteristic, complemented
7C22×C22⋊D101—21no—
8C2×#13108063—22no—
9#15692239—22no—
10#15692229—21no—
11#15692229C2212yes—
12#15692225—22no—
13C2×C22⋊Dic101—21no—
14#15692180—22no—
15#15692162—21no—
16#15692162C2211yescharacteristic, complemented
17C2×C22⋊C4×C101C2211yescharacteristic
18C22⋊C8×C101C2212yescomplemented
19C23×D101—21no—
20C2×Dic202—22no—
21C2×Dic202—42no—
22C2×D404—21no—
23C2×D404D412yes—
24C2×C22⋊D101—22no—
25C2×C22⋊D101—22no—
26C2×C22⋊D101—42no—
27C2×C22⋊D101—42no—
28C22×Dic101—21no—
29C22×Dic101—21no—
30C22×Dic101—21no—
31#13108063—44no—
32C4⋊Dic101—21no—
33C4⋊Dic101D412yes—
34C22⋊Dic101—42no—
35#13108051—22no—
36#13108051—22no—
37C24×C101D411yescharacteristic
38C22×C4×C101D411yescharacteristic
39C22×C4×C101C2311yescharacteristic
40C2×C8×C101—22no—
41C22⋊C4×C101—22no—
42C23×C101—21no—
43C23×C101—21no—
44C23×C101—41no—
45C23×C101—41no—
46C23×C101SD1612yescomplemented
47C23×C101C2×D411yescharacteristic
48C22×D101—22no—
49C22×D101—42no—
50C22×D101—81no—
51C22×D101—81no—
52C2×C4×C101—21no—
53C2×C4×C101—41no—
54C2×C4×C101—41no—
55C2×C4×C101C2×D412yes—
56C22⋊D101—44no—
57C22⋊D101—42no—
58C22⋊D101—42no—
59C2×Dic101—22no—
60C2×Dic101—22no—
61C2×Dic101—42no—
62C2×Dic101—41no—
63C2×Dic101—41no—
64C2×Dic101—41no—
65C2×Dic101—41no—
66C2×Dic101—41no—
67D404—42no—
68D404—81no—
69Dic202—24no—
70Dic202—42no—
71C8×C101—42no—
72C22×C101—21no—
73C22×C101—21no—
74C22×C101—22no—
75C22×C101—42no—
76C22×C101—41no—
77C22×C101—41no—
78C22×C101—81no—
79C22×C101#12011yescharacteristic
80C22×C101C2×SD1611yescharacteristic
81C22×C101AGL(1,ℤ/8)11yescharacteristic
82C2×D101—42no—
83C2×D101—42no—
84C2×D101—42no—
85C2×D101—84no—
86C4×C101—22no—
87C4×C101—41no—
88Dic101—42no—
89Dic101—42no—
90Dic101—81no—
91C2×C101—21no—
92C2×C101—21no—
93C2×C101—42no—
94C2×C101#45011yescharacteristic
95C2×C101#45511yescharacteristic
96C2×C101B(4,𝔽2)11yescharacteristic
97D101—82no—
98#1601—1011nomaximal, Sylow
99C101#160111yescharacteristic, complemented, Sylow
100#534—1011no—
101#448—1012no—
102#393—1011no—
103#327—1012no—
104#323D10111yescharacteristic, complemented
105C22×Q8—1011no—
106C22×D4—2021no—
107C2×SD16—2022no—
108#123—2022no—
109#121—1012no—
110#121—2021no—
111#120—2022no—
112C2×C22⋊C4—2021no—
113C2×C22⋊C4C2×D10111yescharacteristic
114#103—2022no—
115C22⋊C8C2×D10112yescomplemented
116#95—1011no—
117#95—2021no—
118C24—2021no—
119C24D40411yescharacteristic
120C2×Q8—4042no—
121C2×Q8—2022no—
122C2×D4—1012no—
123C2×D4—4042no—
124C2×D4—4042no—
125C2×D4—2021no—
126C2×D4—2022no—
127C2×D4—2022no—
128C22×C4—2021no—
129C22×C4—2021no—
130C22×C4—2021no—
131C22×C4D40411yescharacteristic
132C22×C4C22×D10111yescharacteristic
133SD16—4044no—
134C2×C8—22no—
135C4⋊C4—1012no—
136C4⋊C4—2021no—
137C22⋊C4—22no—
138C22⋊C4—4042no—
139C22⋊C4—2022no—
140C22⋊C4—2022no—
141C23—21no—
142C23—21no—
143C23—41no—
144C23—41no—
145C23—8081no—
146C23—8081no—
147C23—4042no—
148C23—2022no—
149C23#1310806312yescomplemented
150C23C2×D40411yescharacteristic
151Q8—4042no—
152Q8—2024no—
153D4—8081no—
154D4—4042no—
155D4—4044no—
156D4—4042no—
157D4—4042no—
158C2×C4—21no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—4041no—
162C2×C4—4041no—
163C2×C4—4041no—
164C2×C4—4042no—
165C2×C4—4041no—
166C2×C4—4041no—
167C2×C4—2022no—
168C2×C4—2022no—
169C2×C4D4×D10112yes—
170C8—42no—
171C22—21no—
172C22—21no—
173C22—22no—
174C22—42no—
175C22—41no—
176C22—41no—
177C22—81no—
178C22—8084no—
179C22—4042no—
180C22—4042no—
181C22—4042no—
182C22#1569222511yescharacteristic
183C22C2×#1310806311yescharacteristic
184C22#1569228311yescharacteristic
185C4—22no—
186C4—41no—
187C4—8081no—
188C4—4042no—
189C4—4042no—
190C2—21no—
191C2—21no—
192C2—42no—
193C2—8082no—
194C2#2026533011yescharacteristic
195C2#2026535711yescharacteristic
196C2#2026536511yescharacteristic
subgroupG/H
centerC22#15692225
commutator, cosocleC22×C4×C101C23
FrattiniC22×C4C22×D101
Fitting#323×C101C2
socleC22×C101#120
prime radical#20265739C2
hypercenter#323D101
hypocenterC101#1601
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. #20265318C2semidirect
3. #20265089C2semidirect
4. #20264912C2semidirect
5. #323×C101C2semidirect
6. C22×Dic202C22semidirect
16. #15692162C22semidirect
18. C22⋊C8×C101C22semidirect
46. C23×C101SD16semidirect
99. C101#1601semidirect
104. #323D101semidirect
115. C22⋊C8C2×D101semidirect
149. C23#13108063semidirect

Maximal subgroups

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

subgroupindex
#202657392
#202653182
#202650892
#202649122
#323×C1012
#1601101