Finite Group Databasebeta

← #27169656

Subgroups — #27169656

Order 12928.

242 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 63 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#20265028C211yesmaximal, 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×Dic101—21no—
19C2×C4×Dic101C2211yescharacteristic
20#15692184—21no—
21C22⋊#11717579C2211yescharacteristic
22#15692165—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—24no—
38#13108053—21no—
39#13108051—21no—
40#13108051—21no—
41#13108051—21no—
42#13108051—21no—
43C4×Dic101—21no—
44C4×Dic101—42no—
45C4×Dic101C2×C411yescharacteristic
46C4×Dic101C2×C411yescharacteristic
47C4×Dic101D411yescharacteristic
48C4×Dic101D411yescharacteristic
49C2×Q8×C101C2×C411yescharacteristic
50C22×C4×C101—21no—
51C22×C4×C101C2311yescharacteristic
52C2×C8×C101—21no—
53C4⋊C4×C101—21no—
54C4⋊C4×C101C2×C411yescharacteristic
55C22⋊C4×C101—21no—
56C23×C101C2×D411yescharacteristic
57C22×D101—21no—
58Q8×C101—41no—
59C2×C4×C101—22no—
60C2×C4×C101—21no—
61C2×C4×C101—21no—
62C2×C4×C101—41no—
63C2×C4×C101—41no—
64C2×C4×C101—41no—
65C2×C4×C101C22×C411yescharacteristic
66C2×C4×C101C2×D411yescharacteristic
67C22⋊D101—41no—
68C22⋊D101—41no—
69C2×Dic101—21no—
70C2×Dic101—21no—
71C2×Dic101—22no—
72C2×Dic101—22no—
73C2×Dic101—21no—
74C2×Dic101—41no—
75C2×Dic101—41no—
76C2×Dic101C22⋊C411yescharacteristic
77C2×Dic101C22⋊C411yescharacteristic
78C2×Dic101C22⋊C411yescharacteristic
79C2×Dic101C22⋊C411yescharacteristic
80D404—41no—
81C4×D101—42no—
82C8×C101—41no—
83#11717579—41no—
84C22×C101—21no—
85C22×C101—21no—
86C22×C101C2×C22⋊C411yescharacteristic
87C22×C101AGL(1,ℤ/8)11yescharacteristic
88C22×C101#13711yescharacteristic
89C2×D101—41no—
90C2×D101—41no—
91C2×D101—41no—
92C4×C101—21no—
93C4×C101—21no—
94C4×C101—42no—
95C4×C101—41no—
96C4×C101—41no—
97Dic101—21no—
98Dic101—21no—
99Dic101—21no—
100Dic101—42no—
101Dic101—41no—
102Dic101#10412yes—
103C2×C101—21no—
104C2×C101—21no—
105C2×C101#41011yescharacteristic
106C2×C101#41711yescharacteristic
107C2×C101C2×#10411yescharacteristic
108D101—81no—
109#1509—1011nomaximal, Sylow
110C101#150911yescharacteristic, complemented, Sylow
111#529—1011no—
112#432—1011no—
113C4×C4⋊C4—1011no—
114#328—1011no—
115#328D10111yescharacteristic, complemented
116#327—1011no—
117#327—1011no—
118#126—2021no—
119#125—1011no—
120#124—1011no—
121#123—2021no—
122C22⋊Q8C2×D10111yescharacteristic
123#121—1011no—
124C4×Q8—1011no—
125C4×D4—1011no—
126C4×D4—2021no—
127#117—2021no—
128C2×C4⋊C4—2021no—
129C2×C42—1011no—
130C2×C42—2021no—
131C4⋊C8—2021no—
132C22⋊C8—1011no—
133C22⋊C8C2×D10111yescharacteristic
134#97—2021no—
135#95—1011no—
136#95C2×D10111yescharacteristic
137C2×Q8C4×D10111yescharacteristic
138C2×D4—1011no—
139C2×D4—2021no—
140C22×C4—21no—
141C22×C4—1011no—
142C22×C4—1011no—
143C22×C4—2021no—
144C22×C4—2021no—
145C22×C4C22×D10111yescharacteristic
146C2×C8—21no—
147C2×C8—2021no—
148C4⋊C4—21no—
149C4⋊C4—1011no—
150C4⋊C4—2021no—
151C4⋊C4—2021no—
152C4⋊C4—2021no—
153C4⋊C4—2024no—
154C4⋊C4C4×D10111yescharacteristic
155C22⋊C4—21no—
156C22⋊C4—2021no—
157C22⋊C4—2021no—
158C22⋊C4—2021no—
159C22⋊C4—2021no—
160C42—1011no—
161C42—1011no—
162C42—1011no—
163C42—1011no—
164C42—4042no—
165C42—2021no—
166C23—2021no—
167C23D4×D10111yescharacteristic
168Q8—41no—
169D4—4041no—
170D4—4041no—
171D4—4041no—
172C2×C4—22no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—1011no—
179C2×C4—1011no—
180C2×C4—1011no—
181C2×C4—1011no—
182C2×C4—4042no—
183C2×C4—4041no—
184C2×C4—4041no—
185C2×C4—2022no—
186C2×C4—2022no—
187C2×C4—2021no—
188C2×C4—2021no—
189C2×C4—2021no—
190C2×C4C2×C4×D10111yescharacteristic
191C2×C4D4×D10111yescharacteristic
192C8—41no—
193C8—4041no—
194C22—21no—
195C22—21no—
196C22—4041no—
197C22—4041no—
198C22—4041no—
199C22C22⋊C4×D10111yescharacteristic
200C22#1569228211yescharacteristic
201C22#1569228711yescharacteristic
202C4—21no—
203C4—21no—
204C4—42no—
205C4—41no—
206C4—41no—
207C4—1012no—
208C4—4042no—
209C4—4041no—
210C4—2021no—
211C4—2021no—
212C4—2021no—
213C2—21no—
214C2—21no—
215C2—8081no—
216C2#2026514911yescharacteristic
217C2#2026517711yescharacteristic
218C2#104×D10111yescharacteristic
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#1509
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. #20265028C2semidirect
4. #20264914C2semidirect
7. #328×C101C2semidirect
9. #15692246C4semidirect
10. #15692244C4semidirect
13. Q8×Dic101C4semidirect
14. #15692217C4semidirect
110. C101#1509semidirect
115. #328D101semidirect

Maximal subgroups

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

subgroupindex
#202657132
#202652312
#202650282
#202649142
#202649122
#202649102
#328×C1012
#1509101