Finite Group Databasebeta

← #27186538

Subgroups — #27186538

Order 12928.

326 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 58 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#20266148C211yesmaximal, characteristic, complemented
2#20266120C211yesmaximal, characteristic, complemented
3#20266029C212yesmaximal, complemented
4#20266028C212yesmaximal, complemented
5#121×D101C211yesmaximal, characteristic
6C2×#13108084—21no—
7C2×C22⋊#87938—21no—
8#15692312C2212yescomplemented
9C22⋊#11717581—21no—
10(C2×#87938)⋊C4—21no—
11(C2×#87938)⋊C4C2212yescomplemented
12C2×D4×D101—21no—
13C2×D4×D101C411yescharacteristic, complemented
14C22×C4×D101C2211yescharacteristic
15#15692230C2211yescharacteristic
16#15692229C411yescharacteristic, complemented
17#15692228—21no—
18#15692227—21no—
19#15692226C2211yescharacteristic
20C4⋊C4×D101C411yescharacteristic, complemented
21C22⋊C4×D101—21no—
22#121×C101C411yescharacteristic, complemented
23C22×#87938—22no—
24#13108084—42no—
25C2×#11717581—22no—
26C22⋊#87938—42no—
27C23×D101—21no—
28C23×D101D411yescharacteristic
29C2×D404—21no—
30C2×D404C2×C411yescharacteristic
31C2×C22⋊D101—21no—
32C2×C22⋊D101—21no—
33C2×C22⋊D101—21no—
34D4×D101—42no—
35D4×D101—44no—
36C22×Dic101D411yescharacteristic
37C2×C4×D101—21no—
38C2×C4×D101—42no—
39C2×C4×D101C2×C411yescharacteristic
40C2×C4×D101D411yescharacteristic
41C2×C4×D101D411yescharacteristic
42C2×C4×D101C2311yescharacteristic
43C4⋊Dic101C2×C411yescharacteristic
44#13108053—21no—
45C22⋊Dic101—21no—
46#13108051—21no—
47#13108051—21no—
48C2×D4×C101—21no—
49C2×D4×C101C2×C411yescharacteristic
50C22×C4×C101C2×C411yescharacteristic
51C4⋊C4×C101C2×C411yescharacteristic
52C22⋊C4×C101—21no—
53C23×C101—21no—
54C23×C101C22⋊C411yescharacteristic
55C22×D101—22no—
56C22×D101—22no—
57C22×D101—21no—
58C22×D101—41no—
59C22×D101—42no—
60C22×D101—42no—
61C22×D101—42no—
62C22×D101C22⋊C411yescharacteristic
63C22×D101C2×D411yescharacteristic
64C2×#87938—42no—
65C2×#87938—42no—
66C2×#87938—42no—
67D4×C101—41no—
68D4×C101—42no—
69C2×C4×C101—21no—
70C2×C4×C101—41no—
71C2×C4×C101C22⋊C411yescharacteristic
72C2×C4×C101C22×C411yescharacteristic
73C22⋊D101—42no—
74C22⋊D101—42no—
75C22⋊D101—42no—
76C2×Dic101—21no—
77C2×Dic101—41no—
78C2×Dic101C22⋊C411yescharacteristic
79C2×Dic101C2×D411yescharacteristic
80D404—41no—
81D404—42no—
82C4×D101—22no—
83C4×D101—22no—
84C4×D101—41no—
85C4×D101—41no—
86C4×D101—42no—
87#11717581—42no—
88C22×C101—21no—
89C22×C101—21no—
90C22×C101—21no—
91C22×C101—41no—
92C22×C101—41no—
93C22×C101—41no—
94C22×C101C2×C22⋊C411yescharacteristic
95C2×D101—21no—
96C2×D101—21no—
97C2×D101—21no—
98C2×D101—21no—
99C2×D101—44no—
100C2×D101—41no—
101C2×D101—41no—
102C2×D101—41no—
103C2×D101—84no—
104C2×D101#9912yes—
105C2×D101AGL(1,ℤ/8)12yes—
106#87938—82no—
107C4×C101—21no—
108C4×C101—21no—
109C4×C101—41no—
110Dic101—21no—
111Dic101—21no—
112Dic101—41no—
113C2×C101—41no—
114C2×C101—81no—
115C2×C101C2×#9911yescharacteristic
116C2×C101#41811yescharacteristic
117C2×C101#42111yescharacteristic
118D101—22no—
119D101—22no—
120D101—81no—
121#1510—1011nomaximal, Sylow
122C101#151011yescharacteristic, complemented, Sylow
123C2×#121—1011no—
124#407—1011no—
125#381—1011no—
126#327—1012no—
127#327—1012no—
128C22×D4—1011no—
129C22×D4—2021no—
130C23×C4—1011no—
131C2×M16—2021no—
132#121—1011no—
133#121—1011no—
134#121—1011no—
135#121—2021no—
136#121—2021no—
137#121#8793811yescharacteristic, complemented
138C2×C4⋊C4—1011no—
139C2×C22⋊C4—2021no—
140C2×C22⋊C4—2021no—
141C22⋊C8—1012no—
142C22⋊C8—2021no—
143#95—1012no—
144#95—2021no—
145C24—1011no—
146C24—2021no—
147C2×D4—21no—
148C2×D4—1011no—
149C2×D4—4044no—
150C2×D4—4042no—
151C2×D4—2021no—
152C2×D4—2021no—
153C2×D4—2021no—
154C2×D4—2021no—
155C2×D4C2×#8793811yescharacteristic
156C22×C4—1011no—
157C22×C4—1011no—
158C22×C4—1011no—
159C22×C4—1011no—
160C22×C4—1011no—
161C22×C4—4042no—
162C22×C4—2021no—
163C22×C4—2022no—
164C22×C4C2×#8793811yescharacteristic
165M16—4042no—
166C2×C8—2022no—
167C4⋊C4—1011no—
168C4⋊C4—2021no—
169C4⋊C4C2×#8793811yescharacteristic
170C22⋊C4—21no—
171C22⋊C4—4042no—
172C22⋊C4—2021no—
173C22⋊C4—2021no—
174C22⋊C4—2021no—
175C23—21no—
176C23—1011no—
177C23—1011no—
178C23—4041no—
179C23—4042no—
180C23—4042no—
181C23—4042no—
182C23—2022no—
183C23—2022no—
184C23—2021no—
185C23C22⋊#8793811yescharacteristic
186D4—41no—
187D4—42no—
188D4—4041no—
189D4—4042no—
190D4—4042no—
191D4—4042no—
192D4—4042no—
193C2×C4—21no—
194C2×C4—41no—
195C2×C4—1011no—
196C2×C4—1011no—
197C2×C4—4041no—
198C2×C4—4041no—
199C2×C4—4042no—
200C2×C4—4042no—
201C2×C4—4042no—
202C2×C4—4042no—
203C2×C4—4041no—
204C2×C4—2022no—
205C2×C4—2022no—
206C2×C4—2021no—
207C2×C4C22⋊#8793811yescharacteristic
208C2×C4C22×#8793811yescharacteristic
209C8—4042no—
210C22—21no—
211C22—21no—
212C22—21no—
213C22—41no—
214C22—41no—
215C22—41no—
216C22—8084no—
217C22—1012no—
218C22—1012no—
219C22—4044no—
220C22—4041no—
221C22—4041no—
222C22—4041no—
223C22—2021no—
224C22—2021no—
225C22—2021no—
226C22—2021no—
227C22C2×C22⋊#8793811yescharacteristic
228C4—21no—
229C4—21no—
230C4—41no—
231C4—8082no—
232C4—4041no—
233C4—2021no—
234C4—2021no—
235C2—41no—
236C2—81no—
237C2—8081no—
238C2—2022no—
239C2—2022no—
240C2D101⋊#9911yescharacteristic
241C2#2026616911yescharacteristic
242C2#2026617411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87938
commutatorC2×C4×C101C22×C4
FrattiniC2×C4C22×#87938
Fitting#121×C101C4
socleC22×C101C2×C22⋊C4
prime radical#121×D101C2
cosocleC2×C4×D101C23
hypercenter#121#87938
hypocenterC101#1510
LCSG ▷ C2×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
1. #20266148C2semidirect
2. #20266120C2semidirect
3. #20266029C2semidirect
4. #20266028C2semidirect
8. #15692312C22semidirect
11. (C2×#87938)⋊C4C22semidirect
13. C2×D4×D101C4semidirect
16. #15692229C4semidirect
20. C4⋊C4×D101C4semidirect
22. #121×C101C4semidirect
122. C101#1510semidirect
137. #121#87938semidirect

Maximal subgroups

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

subgroupindex
#202661482
#202661202
#202660292
#202660282
#121×D1012
#1510101