Finite Group Databasebeta

← #27187389

Subgroups — #27187389

Order 12928.

280 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 54 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
1C2×#15692315C211yesmaximal, characteristic, complemented
2C2×#15692313C211yesmaximal, characteristic, complemented
3#20266080C212yesmaximal, complemented
4#20266078C212yesmaximal, complemented
5#20265739C211yesmaximal, characteristic
6C2×#13108083—21no—
7C2×C22⋊#87938—21no—
8#15692315—21no—
9#15692315C2212yes—
10#15692313—21no—
11#15692313C2212yes—
12C22×C22⋊D101C411yescharacteristic, complemented
13C22×C22⋊D101C2211yescharacteristic
14#15692239—21no—
15#15692239C411yescharacteristic, complemented
16#15692239C2211yescharacteristic
17#15692229—21no—
18#15692229—21no—
19#15692225—21no—
20#15692225C411yescharacteristic, complemented
21#15692225C2211yescharacteristic
22C2×C22⋊C4×C101C411yescharacteristic, complemented
23C22×#87938—41no—
24#13108083—22no—
25#13108083—41no—
26C2×#11717581—41no—
27C22⋊#87938—22no—
28C22⋊#87938—41no—
29C23×D101C2×C411yescharacteristic, complemented
30C23×D101D411yescharacteristic
31C2×D404—21no—
32C2×D404—21no—
33C2×C22⋊D101—21no—
34C2×C22⋊D101—21no—
35C2×C22⋊D101—41no—
36C2×C22⋊D101—41no—
37C2×C22⋊D101—41no—
38C2×C22⋊D101—41no—
39C2×C22⋊D101C2×C411yescharacteristic
40C2×C22⋊D101D411yescharacteristic
41C2×C22⋊D101D411yescharacteristic
42C2×C22⋊D101C2311yescharacteristic
43C22×Dic101C2×C411yescharacteristic, complemented
44C22×Dic101D411yescharacteristic
45C4⋊Dic101—21no—
46C4⋊Dic101—21no—
47#13108051—21no—
48#13108051—21no—
49#13108051—21no—
50#13108051—21no—
51C24×C101C2×C411yescharacteristic
52C22×C4×C101—21no—
53C22⋊C4×C101—21no—
54C22⋊C4×C101C2×C411yescharacteristic, complemented
55C22⋊C4×C101C2×C411yescharacteristic, complemented
56C23×C101—21no—
57C23×C101—21no—
58C23×C101—81no—
59C23×C101C22⋊C411yescharacteristic
60C23×C101C22⋊C411yescharacteristic
61C23×C101C22×C411yescharacteristic
62C22×D101—21no—
63C22×D101—21no—
64C22×D101—21no—
65C22×D101—41no—
66C22×D101—81no—
67C22×D101—81no—
68C22×D101C22⋊C411yescharacteristic
69C22×D101C2×D411yescharacteristic
70C2×#87938—42no—
71C2×#87938—82no—
72C2×C4×C101—21no—
73C2×C4×C101—21no—
74C2×C4×C101—81no—
75C22⋊D101—41no—
76C22⋊D101—41no—
77C22⋊D101—41no—
78C22⋊D101—41no—
79C22⋊D101—42no—
80C22⋊D101—41no—
81C22⋊D101—41no—
82C2×Dic101—21no—
83C2×Dic101—21no—
84C2×Dic101—21no—
85C2×Dic101—41no—
86C2×Dic101C22⋊C411yescharacteristic
87C2×Dic101C2×D411yescharacteristic
88D404—81no—
89D404—81no—
90#11717581—42no—
91C22×C101—21no—
92C22×C101—21no—
93C22×C101—41no—
94C22×C101—41no—
95C22×C101—41no—
96C22×C101—81no—
97C22×C101—81no—
98C22×C101#9911yescharacteristic
99C22×C101#9911yescharacteristic
100C22×C101C2×C22⋊C411yescharacteristic
101C2×D101—21no—
102C2×D101—21no—
103C2×D101—42no—
104C2×D101—41no—
105C2×D101—41no—
106C2×D101—41no—
107C2×D101—41no—
108C2×D101—41no—
109C2×D101—81no—
110C2×D101—82no—
111#87938—82no—
112C4×C101—41no—
113C4×C101—41no—
114Dic101—21no—
115Dic101—21no—
116Dic101—41no—
117C2×C101—21no—
118C2×C101—21no—
119C2×C101—21no—
120C2×C101—41no—
121C2×C101—41no—
122C2×C101C2×#9911yescharacteristic
123D101—41no—
124D101—41no—
125D101—81no—
126#2121—1011nomaximal, Sylow
127C101#212111yescharacteristic, complemented, Sylow
128#534—1011no—
129C2×#100—1011no—
130C2×#99—1011no—
131#352—1012no—
132AΣL(1,𝔽16)—1012no—
133C22×D4—1011no—
134C22×D4—1011no—
135C2×M16—2021no—
136#123—1011no—
137#123—1011no—
138#123—2021no—
139#121—2021no—
140#121—2021no—
141#120—1011no—
142#120—1011no—
143#120—2021no—
144C2×C22⋊C4—2021no—
145C2×C22⋊C4#8793811yescharacteristic, complemented
146#100—1012no—
147#100—2021no—
148#99—1012no—
149#99—2021no—
150C24—1011no—
151C24—1011no—
152C24C2×#8793811yescharacteristic
153C2×D4—1011no—
154C2×D4—1011no—
155C2×D4—1011no—
156C2×D4—1011no—
157C2×D4—4041no—
158C2×D4—4041no—
159C2×D4—4041no—
160C2×D4—4041no—
161C2×D4—2021no—
162C2×D4—2021no—
163C2×D4—2021no—
164C2×D4—2021no—
165C22×C4—21no—
166C22×C4—1011no—
167C22×C4—1011no—
168C22×C4—4041no—
169M16—4041no—
170M16—2022no—
171C2×C8—4041no—
172C4⋊C4—2021no—
173C4⋊C4—2021no—
174C22⋊C4—21no—
175C22⋊C4—4041no—
176C22⋊C4—2021no—
177C22⋊C4—2021no—
178C22⋊C4—2021no—
179C22⋊C4—2021no—
180C22⋊C4—2022no—
181C22⋊C4C2×#8793811yescharacteristic, complemented
182C22⋊C4C2×#8793811yescharacteristic, complemented
183C23—21no—
184C23—21no—
185C23—81no—
186C23—8081no—
187C23—8081no—
188C23—1011no—
189C23—1011no—
190C23—4041no—
191C23—2021no—
192C23—2021no—
193C23—2021no—
194C23C22⋊#8793811yescharacteristic
195C23C22⋊#8793811yescharacteristic
196C23C22×#8793811yescharacteristic
197D4—8081no—
198D4—8081no—
199D4—4042no—
200D4—4041no—
201D4—4041no—
202D4—4041no—
203D4—4041no—
204D4—4041no—
205D4—4041no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—81no—
209C2×C4—8082no—
210C2×C4—1011no—
211C2×C4—1011no—
212C2×C4—4042no—
213C2×C4—4041no—
214C2×C4—2021no—
215C2×C4—2021no—
216C2×C4—2021no—
217C8—4042no—
218C22—21no—
219C22—21no—
220C22—41no—
221C22—41no—
222C22—41no—
223C22—81no—
224C22—81no—
225C22—8081no—
226C22—8082no—
227C22—4041no—
228C22—4041no—
229C22—4041no—
230C22—4041no—
231C22—4042no—
232C22—4041no—
233C22—2021no—
234C22—2021no—
235C22#1569231411yescharacteristic
236C22#1569231411yescharacteristic
237C22C2×C22⋊#8793811yescharacteristic
238C4—41no—
239C4—41no—
240C4—8082no—
241C4—4041no—
242C4—2021no—
243C4—2021no—
244C2—21no—
245C2—21no—
246C2—21no—
247C2—41no—
248C2—41no—
249C2—8081no—
250C2—4041no—
251C2—4041no—
252C2C2×#1569231411yescharacteristic
subgroupG/H
centerC2C2×#15692314
commutatorC23×C101C22×C4
FrattiniC23C22×#87938
FittingC2×C22⋊C4×C101C4
socleC2×C101C2×#99
prime radical#20265739C2
cosocleC2×C22⋊D101C23
hypercenterC2×C22⋊C4#87938
hypocenterC101#2121
LCSG ▷ C23×C101 ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C23 ◁ C24 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#15692315C2semidirect
2. C2×#15692313C2semidirect
3. #20266080C2semidirect
4. #20266078C2semidirect
12. C22×C22⋊D101C4semidirect
15. #15692239C4semidirect
20. #15692225C4semidirect
22. C2×C22⋊C4×C101C4semidirect
29. C23×D101C2×C4semidirect
43. C22×Dic101C2×C4semidirect
54. C22⋊C4×C101C2×C4semidirect
55. C22⋊C4×C101C2×C4semidirect
127. C101#2121semidirect
145. C2×C22⋊C4#87938semidirect
181. C22⋊C4C2×#87938semidirect
182. C22⋊C4C2×#87938semidirect

Maximal subgroups

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

subgroupindex
C2×#156923152
C2×#156923132
#202660802
#202660782
#202657392
#2121101