Finite Group Databasebeta

← #27170150

Subgroups — #27170150

Order 12928.

288 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 51 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#20265735C211yesmaximal, characteristic
2#20265495C211yesmaximal, characteristic, complemented
3#20265112C211yesmaximal, characteristic, complemented
4#20264912C211yesmaximal, characteristic, complemented
5#20264911C211yesmaximal, characteristic, complemented
6#20264900C211yesmaximal, characteristic, complemented
7#481×C101C211yesmaximal, characteristic, complemented
8C2×D4×D101—21no—
9#15692240—21no—
10#15692237—21no—
11#15692229C2211yescharacteristic
12#15692227—21no—
13#15692225—21no—
14#15692223—21no—
15C2×#13108053—21no—
16C2×#13108051—21no—
17C2×#13108051C2211yescharacteristic
18#15692186—21no—
19#15692175—21no—
20C22⋊#11717579C2211yescharacteristic, complemented
21#15692162C2211yescharacteristic, complemented
22#121×C101C2211yescharacteristic
23C2×C4⋊C4×C101C2211yescharacteristic, complemented
24#106×C101—21no—
25#102×C101—21no—
26C22⋊C8×C101C2211yescharacteristic, complemented
27C23×D101—21no—
28C23×D101D411yescharacteristic
29C2×D404—21no—
30C2×D404D411yescharacteristic
31C2×C22⋊D101—21no—
32C2×C22⋊D101—21no—
33C2×C22⋊D101—21no—
34D4×D101—42no—
35D4×D101—42no—
36C22×Dic101—21no—
37C22×Dic101D411yescharacteristic
38C2×C4×D101—21no—
39C2×#11717579—21no—
40C4⋊Dic101D411yescharacteristic
41#13108053—21no—
42#13108053—42no—
43C22⋊Dic101—21no—
44#13108051—21no—
45#13108051—21no—
46#13108051—21no—
47#13108051—21no—
48#13108051—42no—
49C2×D4×C101—21no—
50C2×D4×C101D411yescharacteristic
51C22×C4×C101—21no—
52C22×C4×C101C2311yescharacteristic
53C2×C8×C101—21no—
54C4⋊C4×C101—21no—
55C4⋊C4×C101—21no—
56C4⋊C4×C101D411yescharacteristic
57C22⋊C4×C101—21no—
58C23×C101—21no—
59C23×C101C2×D411yescharacteristic
60C22×D101—21no—
61C22×D101—21no—
62C22×D101—21no—
63C22×D101—42no—
64C22×D101—41no—
65C22×D101—41no—
66C22×D101—81no—
67C22×D101—81no—
68C22×D101SD1612yes—
69D4×C101—41no—
70D4×C101—41no—
71D4×C101—41no—
72C2×C4×C101—21no—
73C2×C4×C101—41no—
74C2×C4×C101—41no—
75C2×C4×C101—41no—
76C2×C4×C101—41no—
77C2×C4×C101C2×D411yescharacteristic
78C2×C4×C101C2×D411yescharacteristic
79C22⋊D101—42no—
80C22⋊D101—41no—
81C22⋊D101—41no—
82C22⋊D101—81no—
83C2×Dic101—22no—
84C2×Dic101—21no—
85C2×Dic101—21no—
86C2×Dic101—41no—
87C2×Dic101—41no—
88C2×Dic101—41no—
89D404—41no—
90D404—42no—
91C4×D101—42no—
92C8×C101—41no—
93#11717579—41no—
94C22×C101—21no—
95C22×C101—21no—
96C22×C101—41no—
97C22×C101—41no—
98C22×C101—41no—
99C22×C101#12011yescharacteristic
100C22×C101C2×SD1611yescharacteristic
101C22×C101AGL(1,ℤ/8)11yescharacteristic
102C2×D101—22no—
103C2×D101—41no—
104C2×D101—41no—
105C2×D101—42no—
106C2×D101—41no—
107C2×D101—41no—
108C2×D101—41no—
109C2×D101—81no—
110C2×D101—84no—
111C4×C101—21no—
112C4×C101—21no—
113C4×C101—41no—
114C4×C101—81no—
115Dic101—42no—
116Dic101—41no—
117Dic101—41no—
118C2×C101—21no—
119C2×C101—21no—
120C2×C101—81no—
121C2×C101#45011yescharacteristic
122C2×C101#45311yescharacteristic
123C2×C101#45811yescharacteristic
124D101—42no—
125D101—81no—
126#1613—1011nomaximal, Sylow
127C101#161311yescharacteristic, complemented, Sylow
128#534—1011no—
129#481—1011no—
130#481D10111yescharacteristic, complemented
131#399—1011no—
132#327—1011no—
133#327—1011no—
134#323—1011no—
135C22×D4—2021no—
136#123—2021no—
137#123—2021no—
138#121—1011no—
139#121—2021no—
140#121C2×D10111yescharacteristic
141#120—2021no—
142#120—2021no—
143C2×C4⋊C4—2021no—
144C2×C4⋊C4C2×D10111yescharacteristic, complemented
145C2×C22⋊C4—1011no—
146C2×C22⋊C4—2021no—
147#106—21no—
148#106—2021no—
149#102—21no—
150#102—2021no—
151C22⋊C8—1011no—
152C22⋊C8C2×D10111yescharacteristic, complemented
153#95—1011no—
154C24—1011no—
155C24—2021no—
156C2×D4—21no—
157C2×D4—1011no—
158C2×D4—4042no—
159C2×D4—4042no—
160C2×D4—2021no—
161C2×D4—2021no—
162C2×D4—2021no—
163C2×D4—2021no—
164C2×D4C22⋊D10111yescharacteristic
165C22×C4—21no—
166C22×C4—1011no—
167C22×C4—2021no—
168C22×C4—2021no—
169C22×C4C22×D10111yescharacteristic
170C2×C8—21no—
171C2×C8—2021no—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—1011no—
175C4⋊C4—4042no—
176C4⋊C4—2021no—
177C4⋊C4C22⋊D10111yescharacteristic
178C22⋊C4—21no—
179C22⋊C4—4042no—
180C22⋊C4—2021no—
181C22⋊C4—2021no—
182C22⋊C4—2021no—
183C22⋊C4—2021no—
184C22⋊C4—2021no—
185C23—21no—
186C23—8081no—
187C23—8081no—
188C23—1012no—
189C23—4041no—
190C23—4041no—
191C23—4042no—
192C23—2021no—
193C23—2021no—
194C23—2021no—
195C23D4×D10111yescharacteristic
196D4—41no—
197D4—41no—
198D4—41no—
199D4—8081no—
200D4—4041no—
201D4—4042no—
202D4—4042no—
203D4—4041no—
204D4—4041no—
205C2×C4—21no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—4042no—
211C2×C4—4041no—
212C2×C4—4041no—
213C2×C4—4041no—
214C2×C4—2022no—
215C2×C4—2021no—
216C2×C4—2021no—
217C2×C4D4×D10111yescharacteristic
218C2×C4C2×C22⋊D10111yescharacteristic
219C8—41no—
220C8—4041no—
221C22—21no—
222C22—21no—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—8081no—
227C22—8084no—
228C22—4042no—
229C22—4041no—
230C22—4041no—
231C22—4041no—
232C22—4041no—
233C22—4041no—
234C22—2022no—
235C22#1569222311yescharacteristic
236C22SD16×D10111yescharacteristic
237C22#1569228211yescharacteristic
238C4—21no—
239C4—21no—
240C4—41no—
241C4—81no—
242C4—4042no—
243C4—4041no—
244C4—4041no—
245C2—21no—
246C2—21no—
247C2—81no—
248C2—8081no—
249C2—4042no—
250C2#2026532611yescharacteristic
251C2#2026534311yescharacteristic
252C2#2026536611yescharacteristic
subgroupG/H
centerC22#15692223
commutator, cosocleC22×C4×C101C23
FrattiniC22×C4C22×D101
Fitting#481×C101C2
socleC22×C101#120
prime radical#20265735C2
hypercenter#481D101
hypocenterC101#1613
LCSG ▷ C22×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
2. #20265495C2semidirect
3. #20265112C2semidirect
4. #20264912C2semidirect
5. #20264911C2semidirect
6. #20264900C2semidirect
7. #481×C101C2semidirect
20. C22⋊#11717579C22semidirect
21. #15692162C22semidirect
23. C2×C4⋊C4×C101C22semidirect
26. C22⋊C8×C101C22semidirect
127. C101#1613semidirect
130. #481D101semidirect
144. C2×C4⋊C4C2×D101semidirect
152. C22⋊C8C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202657352
#202654952
#202651122
#202649122
#202649112
#202649002
#481×C1012
#1613101