Finite Group Databasebeta

← #27186408

Subgroups — #27186408

Order 12928.

322 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 64 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#20266147C211yesmaximal, characteristic, complemented
2C2×C22⋊#11717581C211yesmaximal, characteristic, complemented
3#20266021C212yesmaximal, complemented
4#20266020C212yesmaximal, complemented
5C2×#15692227C211yesmaximal, characteristic
6C2×#13108083—21no—
7C22×#11717581—21no—
8C22⋊#11717581—21no—
9C22⋊#11717581—21no—
10C22⋊#11717581C2212yes—
11C22⋊#11717581C2212yes—
12C22×C22⋊D101C411yescharacteristic
13C22×C22⋊D101C411yescharacteristic
14C23×Dic101C2211yescharacteristic
15#15692227—21no—
16#15692227—21no—
17#15692227—21no—
18#15692227—21no—
19C2×#13108053C411yescharacteristic
20C2×C22⋊Dic101C2211yescharacteristic
21C2×#13108051C2211yescharacteristic
22C22×D4×C101C411yescharacteristic
23#13108083—42no—
24C2×#11717581—22no—
25C2×#11717581—22no—
26C2×#11717581—42no—
27C23×D101C2×C411yescharacteristic
28C2×C22⋊D101—21no—
29C2×C22⋊D101—21no—
30C2×C22⋊D101—21no—
31C2×C22⋊D101—21no—
32C2×C22⋊D101—42no—
33C2×C22⋊D101—42no—
34C22×Dic101—42no—
35C22×Dic101C2×C411yescharacteristic
36C22×Dic101C2×C411yescharacteristic
37C22×Dic101D411yescharacteristic
38C22×Dic101D411yescharacteristic
39C22×Dic101D411yescharacteristic
40C22×Dic101D411yescharacteristic
41C22×Dic101C2311yescharacteristic
42#13108053—21no—
43#13108053—21no—
44C22⋊Dic101—21no—
45C22⋊Dic101—21no—
46#13108051—21no—
47#13108051—21no—
48C24×C101C2×C411yescharacteristic
49C24×C101C2×C411yescharacteristic
50C2×D4×C101—21no—
51C2×D4×C101—21no—
52C2×D4×C101—42no—
53C22×C4×C101C2×C411yescharacteristic
54C23×C101—22no—
55C23×C101—22no—
56C23×C101—21no—
57C23×C101—22no—
58C23×C101—41no—
59C23×C101—41no—
60C23×C101—41no—
61C23×C101C22⋊C411yescharacteristic
62C23×C101C22⋊C411yescharacteristic
63C23×C101C22×C411yescharacteristic
64C22×D101—21no—
65C22×D101—22no—
66C22×D101—41no—
67C22×D101—41no—
68D4×C101—42no—
69D4×C101—42no—
70C2×C4×C101—21no—
71C2×C4×C101—41no—
72C22⋊D101—42no—
73C22⋊D101—42no—
74C22⋊D101—42no—
75C22⋊D101—42no—
76C2×Dic101—22no—
77C2×Dic101—22no—
78C2×Dic101—22no—
79C2×Dic101—21no—
80C2×Dic101—41no—
81C2×Dic101—41no—
82C2×Dic101—41no—
83C2×Dic101—41no—
84C2×Dic101C22⋊C411yescharacteristic
85C2×Dic101C22⋊C411yescharacteristic
86C2×Dic101C2×D411yescharacteristic
87C2×Dic101C2×D411yescharacteristic
88#11717581—42no—
89#11717581—42no—
90C22×C101—21no—
91C22×C101—21no—
92C22×C101—21no—
93C22×C101—21no—
94C22×C101—21no—
95C22×C101—21no—
96C22×C101—21no—
97C22×C101—44no—
98C22×C101—41no—
99C22×C101—41no—
100C22×C101—41no—
101C22×C101—82no—
102C22×C101#10012yes—
103C22×C101C2×C22⋊C411yescharacteristic
104C22×C101#13112yes—
105C2×D101—41no—
106C2×D101—41no—
107C2×D101—41no—
108C2×D101—82no—
109C4×C101—41no—
110Dic101—21no—
111Dic101—21no—
112Dic101—21no—
113Dic101—41no—
114Dic101#9912yes—
115C2×C101—22no—
116C2×C101—22no—
117C2×C101—41no—
118C2×C101—81no—
119C2×C101#40811yescharacteristic
120C2×C101C2×#9911yescharacteristic
121C2×C101C2×#10011yescharacteristic
122D101—81no—
123#1471—1011nomaximal, Sylow
124C101#147111yescharacteristic, complemented, Sylow
125C2×#121—1011no—
126#407—1011no—
127C2×C22⋊C8—1011no—
128#323—1012no—
129#323—1012no—
130C22×D4—1011no—
131C22×D4—1011no—
132C22×D4#8793811yescharacteristic
133C23×C4—1011no—
134C2×M16—2021no—
135C22×C8—2021no—
136#121—2021no—
137#121—2021no—
138#121—2021no—
139#121—2021no—
140C2×C4⋊C4—1011no—
141C2×C22⋊C4—1011no—
142C2×C22⋊C4—1011no—
143C22⋊C8—1012no—
144C22⋊C8—1012no—
145C22⋊C8—2021no—
146C22⋊C8—2021no—
147C24—1011no—
148C24C2×#8793811yescharacteristic
149C24C2×#8793811yescharacteristic
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—42no—
153C2×D4—4042no—
154C2×D4—4042no—
155C2×D4—2021no—
156C2×D4—2021no—
157C2×D4—2021no—
158C2×D4—2021no—
159C22×C4—1011no—
160C22×C4—1011no—
161C22×C4—1011no—
162C22×C4—1011no—
163C22×C4—1011no—
164C22×C4—1011no—
165C22×C4—1011no—
166C22×C4—4042no—
167C22×C4C2×#8793811yescharacteristic
168M16—4042no—
169C2×C8—4042no—
170C2×C8—2022no—
171C2×C8—2022no—
172C4⋊C4—2021no—
173C4⋊C4—2021no—
174C22⋊C4—2021no—
175C22⋊C4—2021no—
176C22⋊C4—2021no—
177C22⋊C4—2021no—
178C23—22no—
179C23—22no—
180C23—21no—
181C23—22no—
182C23—41no—
183C23—41no—
184C23—41no—
185C23—4041no—
186C23—4041no—
187C23—2022no—
188C23—2021no—
189C23C22⋊#8793811yescharacteristic
190C23C22⋊#8793811yescharacteristic
191C23C22×#8793811yescharacteristic
192D4—42no—
193D4—42no—
194D4—4042no—
195D4—4042no—
196D4—4042no—
197D4—4042no—
198C2×C4—21no—
199C2×C4—41no—
200C2×C4—1011no—
201C2×C4—1011no—
202C2×C4—1011no—
203C2×C4—1011no—
204C2×C4—4041no—
205C2×C4—4041no—
206C2×C4—4041no—
207C2×C4—4041no—
208C2×C4—2022no—
209C2×C4—2022no—
210C2×C4—2022no—
211C2×C4—2021no—
212C8—4042no—
213C8—4042no—
214C22—21no—
215C22—21no—
216C22—21no—
217C22—21no—
218C22—21no—
219C22—21no—
220C22—21no—
221C22—44no—
222C22—41no—
223C22—41no—
224C22—41no—
225C22—82no—
226C22—8082no—
227C22—4041no—
228C22—4041no—
229C22—4041no—
230C22#1569231512yes—
231C22C2×C22⋊#8793811yescharacteristic
232C22#1569234212yes—
233C4—41no—
234C4—1012no—
235C4—4041no—
236C4—2021no—
237C4—2021no—
238C4—2021no—
239C2—22no—
240C2—22no—
241C2—41no—
242C2—81no—
243C2—8081no—
244C2#2026615011yescharacteristic
245C2D101⋊#9911yescharacteristic
246C2C2×#1569231511yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87938
commutatorC23×C101C22×C4
FrattiniC23C22×#87938
FittingC22×D4×C101C4
socleC22×C101C2×C22⋊C4
prime radicalC2×#15692227C2
cosocleC22×Dic101C23
hypercenterC22×D4#87938
hypocenterC101#1471
LCSG ▷ C23×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C24 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. #20266147C2semidirect
2. C2×C22⋊#11717581C2semidirect
3. #20266021C2semidirect
4. #20266020C2semidirect
124. C101#1471semidirect

Maximal subgroups

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

subgroupindex
#202661472
C2×C22⋊#117175812
#202660212
#202660202
C2×#156922272
#1471101