Finite Group Databasebeta

← #27170131

Subgroups — #27170131

Order 12928.

216 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 49 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#20265774C211yesmaximal, characteristic
2#20265508C211yesmaximal, characteristic, complemented
3#20265491C211yesmaximal, characteristic
4#20265107C211yesmaximal, characteristic, complemented
5#20264915C211yesmaximal, characteristic, complemented
6#20264912C211yesmaximal, characteristic
7#324×C101C211yesmaximal, characteristic, complemented
8#15692250—21no—
9#15692245—21no—
10#15692240—21no—
11#15692238—21no—
12#15692234C2211yescharacteristic
13#15692233—21no—
14#15692229C2211yescharacteristic
15C4×C22⋊D101—21no—
16C2×C4⋊Dic101C2211yescharacteristic
17C2×#13108053—21no—
18#15692189—21no—
19#15692187—21no—
20#15692180—21no—
21#15692177—21no—
22#15692162—21no—
23#15692162C2211yescharacteristic
24C2×C4⋊C4×C101C2211yescharacteristic
25C22⋊C8×C101C2211yescharacteristic
26C22⋊C8×C101C2211yescharacteristic
27C2×Dic202D411yescharacteristic
28C2×D404D411yescharacteristic
29C2×C22⋊D101—21no—
30C22×Dic101—21no—
31C22×Dic101—21no—
32C2×C4×D101—21no—
33C4⋊Dic101—21no—
34C4⋊Dic101—21no—
35C4⋊Dic101D411yescharacteristic
36C4⋊Dic101D411yescharacteristic
37#13108053—21no—
38#13108053—21no—
39#13108053—21no—
40#13108053—21no—
41#13108053—42no—
42C22⋊Dic101—21no—
43#13108051—21no—
44#13108051—21no—
45#13108051—21no—
46#13108051—21no—
47C4×Dic101—21no—
48C22×C4×C101D411yescharacteristic
49C22×C4×C101D411yescharacteristic
50C22×C4×C101C2311yescharacteristic
51C2×C8×C101—21no—
52C2×C8×C101—21no—
53C4⋊C4×C101—21no—
54C4⋊C4×C101—21no—
55C23×C101C2×D411yescharacteristic
56C22×D101—21no—
57C2×C4×C101—21no—
58C2×C4×C101—21no—
59C2×C4×C101—21no—
60C2×C4×C101—21no—
61C2×C4×C101—41no—
62C2×C4×C101—41no—
63C2×C4×C101C2×D411yescharacteristic
64C2×C4×C101C2×D411yescharacteristic
65C22⋊D101—41no—
66C22⋊D101—41no—
67C2×Dic101—22no—
68C2×Dic101—21no—
69C2×Dic101—21no—
70C2×Dic101—21no—
71C2×Dic101—41no—
72C2×Dic101—41no—
73C2×Dic101—41no—
74C2×Dic101—41no—
75C2×Dic101—41no—
76D404—41no—
77C4×D101—81no—
78Dic202—41no—
79C8×C101—41no—
80C8×C101—41no—
81C22×C101—21no—
82C22×C101—21no—
83C22×C101#12011yescharacteristic
84C22×C101#13511yescharacteristic
85C22×C101AGL(1,ℤ/8)11yescharacteristic
86C2×D101—41no—
87C2×D101—41no—
88C2×D101—41no—
89C4×C101—21no—
90C4×C101—21no—
91C4×C101—41no—
92C4×C101—41no—
93Dic101—42no—
94Dic101—41no—
95Dic101—41no—
96Dic101—41no—
97Dic101—81no—
98C2×C101—21no—
99C2×C101—21no—
100C2×C101#44911yescharacteristic
101C2×C101#45611yescharacteristic
102C2×C101#45811yescharacteristic
103D101—81no—
104#1609—1011nomaximal, Sylow
105C101#160911yescharacteristic, complemented, Sylow
106#539—1011no—
107#483—1011no—
108#480—1011no—
109#398—1011no—
110#328—1011no—
111#327—1011no—
112#324D10111yescharacteristic, complemented
113#126—2021no—
114#125—2021no—
115#123—2021no—
116#123—2021no—
117C22⋊Q8—1011no—
118C22⋊Q8—2021no—
119#121—1011no—
120C4×D4—2021no—
121C2×C4⋊C4—1011no—
122C2×C4⋊C4—2021no—
123C2×C4⋊C4C2×D10111yescharacteristic
124#107—2021no—
125#106—2021no—
126#103—2021no—
127#102—2021no—
128C22⋊C8C2×D10111yescharacteristic
129C22⋊C8C2×D10111yescharacteristic
130#95—1011no—
131#95—2021no—
132C2×Q8—1011no—
133C2×D4—1011no—
134C2×D4—2021no—
135C22×C4—2021no—
136C22×C4—2021no—
137C22×C4—2021no—
138C22×C4D40411yescharacteristic
139C22×C4D40411yescharacteristic
140C22×C4C22×D10111yescharacteristic
141C2×C8—21no—
142C2×C8—21no—
143C4⋊C4—21no—
144C4⋊C4—21no—
145C4⋊C4—1011no—
146C4⋊C4—1011no—
147C4⋊C4—4042no—
148C4⋊C4—2021no—
149C4⋊C4—2021no—
150C4⋊C4—2021no—
151C4⋊C4—2021no—
152C4⋊C4—2021no—
153C4⋊C4—2021no—
154C22⋊C4—2021no—
155C22⋊C4—2021no—
156C22⋊C4—2021no—
157C22⋊C4—2021no—
158C22⋊C4—2021no—
159C42—2021no—
160C23—2021no—
161C23C2×D40411yescharacteristic
162Q8—4041no—
163D4—4041no—
164D4—4041no—
165D4—4041no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—8081no—
173C2×C4—4041no—
174C2×C4—4041no—
175C2×C4—4041no—
176C2×C4—4041no—
177C2×C4—4041no—
178C2×C4—2022no—
179C2×C4—2021no—
180C2×C4—2021no—
181C2×C4—2021no—
182C2×C4D4×D10111yescharacteristic
183C2×C4D4×D10111yescharacteristic
184C8—41no—
185C8—41no—
186C22—21no—
187C22—21no—
188C22—4041no—
189C22—4041no—
190C22—4041no—
191C22#1569222511yescharacteristic
192C22#1569227711yescharacteristic
193C22#1569228311yescharacteristic
194C4—21no—
195C4—21no—
196C4—41no—
197C4—41no—
198C4—8081no—
199C4—4042no—
200C4—4041no—
201C4—4041no—
202C4—4041no—
203C2—21no—
204C2—21no—
205C2—8081no—
206C2#2026532411yescharacteristic
207C2#2026536211yescharacteristic
208C2#2026536811yescharacteristic
subgroupG/H
centerC22#15692225
commutator, cosocleC22×C4×C101C23
FrattiniC22×C4C22×D101
Fitting#324×C101C2
socleC22×C101#120
prime radical#15692229C22
hypercenter#324D101
hypocenterC101#1609
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. #20265508C2semidirect
4. #20265107C2semidirect
5. #20264915C2semidirect
7. #324×C101C2semidirect
105. C101#1609semidirect
112. #324D101semidirect

Maximal subgroups

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

subgroupindex
#202657742
#202655082
#202654912
#202651072
#202649152
#202649122
#324×C1012
#1609101