Finite Group Databasebeta

← #27181055

Subgroups — #27181055

Order 12928.

458 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 141 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#20265688C211yesmaximal, characteristic
2#20265512C212yesmaximal, complemented
3#20265507C212yesmaximal, complemented
4#20265495C212yesmaximal, complemented
5#20265490C212yesmaximal, complemented
6#20265217C212yesmaximal, complemented
7#20265176C212yesmaximal, complemented
8#20265141C211yesmaximal, characteristic, complemented
9C22×C4⋊C4×C101C211yesmaximal, characteristic, complemented
10C2×#13108074C2211yescharacteristic
11C2×#13108058C2212yes—
12#15692238—21no—
13#15692234C2211yescharacteristic
14#15692234C2211yescharacteristic
15#15692229C2211yescharacteristic
16#15692229C2211yescharacteristic
17#15692224—21no—
18C4×C22⋊D101—21no—
19C4×C22⋊D101—21no—
20#15692214C2211yescharacteristic
21#15692188C2214yescomplemented
22#15692186C2214yescomplemented
23#15692178C2214yescomplemented
24#15692175C2214yescomplemented
25C22⋊#11717579C2212yes—
26C22⋊#11717579C2212yes—
27C23×C4×C101—21no—
28C23×C4×C101C2211yescharacteristic
29C2×C4⋊C4×C101—21no—
30C2×C4⋊C4×C101—21no—
31C2×C4⋊C4×C101—21no—
32C2×C4⋊C4×C101C2212yes—
33C2×C4⋊C4×C101C2212yes—
34C2×C4⋊C4×C101C2212yes—
35#13108074—24no—
36C2×Dic202C2311yescharacteristic
37C2×D404C2311yescharacteristic
38C2×C22⋊D101—21no—
39C2×C22⋊D101—21no—
40C2×C22⋊D101—21no—
41C2×C4×D101—21no—
42#13108058—24no—
43C2×#11717579C2314yes—
44C4⋊Dic101C2311yescharacteristic
45C4⋊Dic101C2311yescharacteristic
46#13108053—21no—
47#13108053—21no—
48C22⋊Dic101—21no—
49C22⋊Dic101—21no—
50C22⋊Dic101—21no—
51#13108051—21no—
52#13108051—21no—
53C4×Dic101—21no—
54C24×C101D411yescharacteristic
55C22×C4×C101—24no—
56C22×C4×C101—22no—
57C22×C4×C101—22no—
58C22×C4×C101—22no—
59C22×C4×C101—44no—
60C22×C4×C101D411yescharacteristic
61C22×C4×C101D411yescharacteristic
62C22×C4×C101D411yescharacteristic
63C22×C4×C101C2311yescharacteristic
64C22×C4×C101C2311yescharacteristic
65C22×C4×C101C2311yescharacteristic
66C4⋊C4×C101—22no—
67C4⋊C4×C101—22no—
68C4⋊C4×C101—22no—
69C4⋊C4×C101C2314yes—
70C23×C101—22no—
71C23×C101—22no—
72C23×C101—22no—
73C23×C101C2×D411yescharacteristic
74C23×C101C2×D411yescharacteristic
75C23×C101C2×D411yescharacteristic
76C22×D101—21no—
77C2×C4×C101—22no—
78C2×C4×C101—22no—
79C2×C4×C101—22no—
80C2×C4×C101—22no—
81C2×C4×C101—22no—
82C2×C4×C101—24no—
83C2×C4×C101—44no—
84C2×C4×C101—44no—
85C2×C4×C101—44no—
86C2×C4×C101C2×D411yescharacteristic
87C2×C4×C101C2×D411yescharacteristic
88C2×C4×C101C2×D411yescharacteristic
89C2×C4×C101#4114yes—
90C2×C4×C101C2411yescharacteristic
91C22⋊D101—42no—
92C22⋊D101—42no—
93C22⋊D101—42no—
94C2×Dic101—21no—
95C2×Dic101—21no—
96C2×Dic101—21no—
97D404—22no—
98C4×D101—42no—
99Dic202—22no—
100#11717579—24no—
101C22×C101—28no—
102C22×C101—21no—
103C22×C101—21no—
104C22×C101—21no—
105C22×C101—21no—
106C22×C101—21no—
107C22×C101—21no—
108C22×C101#12312yes—
109C22×C101AGL(1,ℤ/8)12yes—
110C22×C101#13712yes—
111C22×C101C22×D411yescharacteristic
112C2×D101—41no—
113C2×D101—41no—
114C2×D101—41no—
115C4×C101—21no—
116C4×C101—21no—
117C4×C101—44no—
118C4×C101#12312yes—
119C4×C101C2×#4112yes—
120Dic101—41no—
121Dic101—41no—
122Dic101—41no—
123C2×C101—22no—
124C2×C101—22no—
125C2×C101—22no—
126C2×C101C2×#12311yescharacteristic
127C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
128C2×C101C2×#13711yescharacteristic
129D101—81no—
130#3092—1011nomaximal, Sylow
131C101#309211yescharacteristic, complemented, Sylow
132#525—1011no—
133C22×C4⋊C4D10111yescharacteristic, complemented
134#484—1012no—
135#483—1012no—
136#481—1012no—
137#480—1012no—
138#428—1012no—
139#417—1012no—
140#407—1011no—
141C2×#41—1011no—
142C23×C4—21no—
143C23×C4C2×D10111yescharacteristic
144C2×M16—1012no—
145#123—2021no—
146C22⋊Q8—1011no—
147C22⋊Q8—1011no—
148#121—1011no—
149#121—1011no—
150#120—2021no—
151C4×D4—2021no—
152C4×D4—2021no—
153#117—1011no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4C2×D10112yes—
158C2×C4⋊C4C2×D10112yes—
159C2×C4⋊C4C2×D10112yes—
160#107—1014no—
161#106—1014no—
162#103—1014no—
163#102—1014no—
164C22⋊C8—1012no—
165C22⋊C8—1012no—
166C24C22⋊D10111yescharacteristic
167#41—2024no—
168C2×Q8—1011no—
169C2×D4—1011no—
170C2×D4—2021no—
171C2×D4—2021no—
172C2×D4—2021no—
173C22×C4—24no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—44no—
178C22×C4—2021no—
179C22×C4C22⋊D10111yescharacteristic
180C22×C4C22⋊D10111yescharacteristic
181C22×C4C22⋊D10111yescharacteristic
182C22×C4C22×D10111yescharacteristic
183C22×C4C22×D10111yescharacteristic
184C22×C4C22×D10111yescharacteristic
185M16—2024no—
186C2×C8—1014no—
187C4⋊C4—22no—
188C4⋊C4—22no—
189C4⋊C4—22no—
190C4⋊C4—1011no—
191C4⋊C4—1011no—
192C4⋊C4—2021no—
193C4⋊C4—2021no—
194C4⋊C4C22×D10114yes—
195C22⋊C4—2021no—
196C22⋊C4—2021no—
197C22⋊C4—2021no—
198C22⋊C4—2021no—
199C22⋊C4—2021no—
200C42—2021no—
201C23—22no—
202C23—22no—
203C23—22no—
204C23—2021no—
205C23C2×C22⋊D10111yescharacteristic
206C23C2×C22⋊D10111yescharacteristic
207C23C2×C22⋊D10111yescharacteristic
208Q8—2022no—
209D4—4042no—
210D4—4042no—
211D4—4042no—
212D4—2022no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—24no—
219C2×C4—44no—
220C2×C4—44no—
221C2×C4—44no—
222C2×C4—4042no—
223C2×C4—2021no—
224C2×C4—2021no—
225C2×C4—2021no—
226C2×C4C2×C22⋊D10111yescharacteristic
227C2×C4C2×C22⋊D10111yescharacteristic
228C2×C4C2×C22⋊D10111yescharacteristic
229C2×C4#1310807414yes—
230C2×C4C23×D10111yescharacteristic
231C8—2024no—
232C22—28no—
233C22—21no—
234C22—21no—
235C22—21no—
236C22—21no—
237C22—21no—
238C22—21no—
239C22—4041no—
240C22—4041no—
241C22—4041no—
242C22#1569223812yes—
243C22#1569228112yes—
244C22#1569228612yes—
245C22C22×C22⋊D10111yescharacteristic
246C4—21no—
247C4—21no—
248C4—44no—
249C4—4041no—
250C4—4041no—
251C4—4041no—
252C4#1569223812yes—
253C4C2×#1310807412yes—
254C2—22no—
255C2—22no—
256C2—22no—
257C2—8081no—
258C2C2×#1569223811yescharacteristic
259C2C2×#1569228111yescharacteristic
260C2C2×#1569228611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC22×C4⋊C4×C101C2
socleC22×C101C22×D4
prime radical#20265688C2
hypercenterC22×C4⋊C4D101
hypocenterC101#3092
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #20265512C2semidirect
3. #20265507C2semidirect
4. #20265495C2semidirect
5. #20265490C2semidirect
6. #20265217C2semidirect
7. #20265176C2semidirect
8. #20265141C2semidirect
9. C22×C4⋊C4×C101C2semidirect
21. #15692188C22semidirect
22. #15692186C22semidirect
23. #15692178C22semidirect
24. #15692175C22semidirect
131. C101#3092semidirect
133. C22×C4⋊C4D101semidirect

Maximal subgroups

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

subgroupindex
#202656882
#202655122
#202655072
#202654952
#202654902
#202652172
#202651762
#202651412
C22×C4⋊C4×C1012
#3092101