Finite Group Databasebeta

← #27169669

Subgroups — #27169669

Order 12928.

222 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 61 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#20265713C211yesmaximal, characteristic
2#20265231C211yesmaximal, characteristic, complemented
3#20265041C211yesmaximal, characteristic, complemented
4#20264914C211yesmaximal, characteristic, complemented
5#20264912C211yesmaximal, characteristic
6#20264910C211yesmaximal, characteristic
7#328×C101C211yesmaximal, characteristic, complemented
8#15692250—21no—
9#15692246C411yescharacteristic, complemented
10#15692244C411yescharacteristic, complemented
11#15692239—21no—
12#15692229C2211yescharacteristic
13Q8×Dic101C411yescharacteristic, complemented
14#15692217C411yescharacteristic, complemented
15#15692215—21no—
16#15692213—21no—
17C2×#13108053—21no—
18C2×C4×Dic101C2211yescharacteristic
19#15692184—21no—
20C22⋊#11717579C2211yescharacteristic
21#15692165—21no—
22#15692162—21no—
23#15692162C2211yescharacteristic
24C22⋊Q8×C101C2211yescharacteristic
25C22⋊C8×C101C2211yescharacteristic
26#95×C101C2211yescharacteristic
27C2×D404C2×C411yescharacteristic
28C2×C22⋊D101—21no—
29C22×Dic101—21no—
30C22×Dic101D411yescharacteristic
31C22×Dic101D411yescharacteristic
32C2×C4×D101—21no—
33C2×#11717579—21no—
34C4⋊Dic101—21no—
35C4⋊Dic101—21no—
36C4⋊Dic101C2×C411yescharacteristic
37#13108053—21no—
38#13108053—42no—
39#13108051—21no—
40#13108051—21no—
41#13108051—21no—
42#13108051—21no—
43C4×Dic101—21no—
44C4×Dic101C2×C411yescharacteristic
45C4×Dic101C2×C411yescharacteristic
46C4×Dic101D411yescharacteristic
47C4×Dic101D411yescharacteristic
48C2×Q8×C101C2×C411yescharacteristic
49C22×C4×C101—21no—
50C22×C4×C101C2311yescharacteristic
51C2×C8×C101—21no—
52C4⋊C4×C101—21no—
53C4⋊C4×C101C2×C411yescharacteristic
54C22⋊C4×C101—21no—
55C23×C101C2×D411yescharacteristic
56C22×D101—21no—
57Q8×C101—41no—
58C2×C4×C101—21no—
59C2×C4×C101—21no—
60C2×C4×C101—41no—
61C2×C4×C101—41no—
62C2×C4×C101—41no—
63C2×C4×C101—41no—
64C2×C4×C101C22×C411yescharacteristic
65C2×C4×C101C2×D411yescharacteristic
66C22⋊D101—41no—
67C22⋊D101—41no—
68C2×Dic101—21no—
69C2×Dic101—21no—
70C2×Dic101—21no—
71C2×Dic101—41no—
72C2×Dic101—41no—
73C2×Dic101—41no—
74C2×Dic101—41no—
75C2×Dic101C22⋊C411yescharacteristic
76C2×Dic101C22⋊C411yescharacteristic
77C2×Dic101C22⋊C411yescharacteristic
78C2×Dic101C22⋊C411yescharacteristic
79D404—41no—
80C4×D101—42no—
81C8×C101—41no—
82#11717579—41no—
83C22×C101—21no—
84C22×C101—21no—
85C22×C101C2×C22⋊C411yescharacteristic
86C22×C101AGL(1,ℤ/8)11yescharacteristic
87C22×C101#13711yescharacteristic
88C2×D101—41no—
89C2×D101—41no—
90C2×D101—41no—
91C4×C101—21no—
92C4×C101—21no—
93C4×C101—41no—
94C4×C101—41no—
95C4×C101—81no—
96Dic101—21no—
97Dic101—21no—
98Dic101—21no—
99Dic101—21no—
100Dic101—41no—
101Dic101—81no—
102C2×C101—21no—
103C2×C101—21no—
104C2×C101#41011yescharacteristic
105C2×C101#41711yescharacteristic
106C2×C101#42111yescharacteristic
107D101—81no—
108#1512—1011nomaximal, Sylow
109C101#151211yescharacteristic, complemented, Sylow
110#529—1011no—
111#432—1011no—
112#382—1011no—
113#328—1011no—
114#328D10111yescharacteristic, complemented
115#327—1011no—
116#327—1011no—
117#126—2021no—
118#125—1011no—
119#124—1011no—
120#123—2021no—
121C22⋊Q8C2×D10111yescharacteristic
122#121—1011no—
123C4×Q8—1011no—
124C4×D4—1011no—
125C4×D4—2021no—
126#117—2021no—
127C2×C4⋊C4—2021no—
128C2×C42—1011no—
129C4⋊C8—2021no—
130C22⋊C8—1011no—
131C22⋊C8C2×D10111yescharacteristic
132#97—2021no—
133#95—1011no—
134#95—2021no—
135#95C2×D10111yescharacteristic
136C2×Q8C4×D10111yescharacteristic
137C2×D4—1011no—
138C2×D4—2021no—
139C22×C4—21no—
140C22×C4—1011no—
141C22×C4—1011no—
142C22×C4—2021no—
143C22×C4—2021no—
144C22×C4C22×D10111yescharacteristic
145C2×C8—21no—
146C2×C8—2021no—
147C4⋊C4—21no—
148C4⋊C4—1011no—
149C4⋊C4—4042no—
150C4⋊C4—2021no—
151C4⋊C4—2021no—
152C4⋊C4—2021no—
153C4⋊C4C4×D10111yescharacteristic
154C22⋊C4—21no—
155C22⋊C4—2021no—
156C22⋊C4—2021no—
157C22⋊C4—2021no—
158C22⋊C4—2021no—
159C42—1011no—
160C42—1011no—
161C42—1011no—
162C42—1011no—
163C42—2021no—
164C23—2021no—
165C23D4×D10111yescharacteristic
166Q8—41no—
167D4—4041no—
168D4—4041no—
169D4—4041no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—1011no—
177C2×C4—1011no—
178C2×C4—1011no—
179C2×C4—1011no—
180C2×C4—4042no—
181C2×C4—4041no—
182C2×C4—4041no—
183C2×C4—4041no—
184C2×C4—4041no—
185C2×C4—2021no—
186C2×C4—2021no—
187C2×C4—2021no—
188C2×C4C2×C4×D10111yescharacteristic
189C2×C4D4×D10111yescharacteristic
190C8—41no—
191C8—4041no—
192C22—21no—
193C22—21no—
194C22—4041no—
195C22—4041no—
196C22—4041no—
197C22C22⋊C4×D10111yescharacteristic
198C22#1569228211yescharacteristic
199C22#1569228711yescharacteristic
200C4—21no—
201C4—21no—
202C4—41no—
203C4—41no—
204C4—81no—
205C4—8081no—
206C4—4041no—
207C4—2021no—
208C4—2021no—
209C4—2021no—
210C4—2021no—
211C2—21no—
212C2—21no—
213C2—8081no—
214C2#2026514911yescharacteristic
215C2#2026517711yescharacteristic
216C2#2026519211yescharacteristic
subgroupG/H
centerC22C22⋊C4×D101
commutatorC2×C4×C101C22×C4
FrattiniC22×C4C22×D101
Fitting#328×C101C2
socleC22×C101C2×C22⋊C4
prime radical#15692229C22
cosocleC22×C4×C101C23
hypercenter#328D101
hypocenterC101#1512
LCSG ▷ C2×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #20265231C2semidirect
3. #20265041C2semidirect
4. #20264914C2semidirect
7. #328×C101C2semidirect
9. #15692246C4semidirect
10. #15692244C4semidirect
13. Q8×Dic101C4semidirect
14. #15692217C4semidirect
109. C101#1512semidirect
114. #328D101semidirect

Maximal subgroups

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

subgroupindex
#202657132
#202652312
#202650412
#202649142
#202649122
#202649102
#328×C1012
#1512101