Finite Group Databasebeta

← #27181047

Subgroups — #27181047

Order 12928.

420 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 137 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
2C2×#15692214C211yesmaximal, characteristic, complemented
3#20265519C212yesmaximal, complemented
4#20265513C211yesmaximal, characteristic, complemented
5#20265508C211yesmaximal, characteristic, complemented
6#20265502C212yesmaximal, complemented
7#20265496C211yesmaximal, characteristic, complemented
8#20265491C211yesmaximal, characteristic, complemented
9#20265215C212yesmaximal, complemented
10#20265174C212yesmaximal, complemented
11C2×C22⋊C8×C101C211yesmaximal, characteristic, complemented
12C2×#13108074C2211yescharacteristic
13#15692238—21no—
14#15692234C2211yescharacteristic
15#15692234C2211yescharacteristic
16#15692229C2211yescharacteristic
17#15692229C2211yescharacteristic
18#15692224—21no—
19C4×C22⋊D101—21no—
20C4×C22⋊D101—21no—
21#15692214—21no—
22#15692214—21no—
23#15692214C2212yes—
24#15692214C2212yes—
25#15692214C2211yescharacteristic
26C2×C4⋊Dic101C2211yescharacteristic
27C2×C4⋊Dic101C2211yescharacteristic
28C2×C22⋊Dic101—21no—
29C2×C4×Dic101—21no—
30#15692189C2212yescomplemented
31#15692189C2212yescomplemented
32#15692187C2212yescomplemented
33#15692187C2212yescomplemented
34#15692180C2212yescomplemented
35#15692180C2212yescomplemented
36#15692177C2212yescomplemented
37#15692177C2212yescomplemented
38C23×C4×C101C2211yescharacteristic
39C22×C8×C101C2212yes—
40C22⋊C8×C101C2211yescharacteristic
41C22⋊C8×C101C2211yescharacteristic
42C22⋊C8×C101C2212yes—
43#13108074—24no—
44C2×Dic202C2311yescharacteristic
45C2×D404C2311yescharacteristic
46C2×C22⋊D101—21no—
47C2×C22⋊D101—21no—
48C2×C22⋊D101—21no—
49C22×Dic101—21no—
50C22×Dic101—21no—
51C2×C4×D101—21no—
52C4⋊Dic101—21no—
53C4⋊Dic101—21no—
54C4⋊Dic101C2312yes—
55C4⋊Dic101C2312yes—
56C4⋊Dic101C2311yescharacteristic
57C4⋊Dic101C2311yescharacteristic
58#13108053—21no—
59#13108053—21no—
60C22⋊Dic101—22no—
61C22⋊Dic101—22no—
62C22⋊Dic101—21no—
63C22⋊Dic101—21no—
64C22⋊Dic101—21no—
65#13108051—21no—
66#13108051—21no—
67C4×Dic101—22no—
68C4×Dic101—22no—
69C4×Dic101—21no—
70C24×C101D411yescharacteristic
71C22×C4×C101—24no—
72C22×C4×C101D411yescharacteristic
73C22×C4×C101D411yescharacteristic
74C22×C4×C101D411yescharacteristic
75C22×C4×C101C2311yescharacteristic
76C22×C4×C101C2311yescharacteristic
77C22×C4×C101C2311yescharacteristic
78C2×C8×C101—24no—
79C2×C8×C101C2312yes—
80C2×C8×C101C2312yes—
81C23×C101—22no—
82C23×C101—22no—
83C23×C101—41no—
84C23×C101C2×D411yescharacteristic
85C23×C101C2×D411yescharacteristic
86C23×C101C2×D411yescharacteristic
87C22×D101—21no—
88C2×C4×C101—22no—
89C2×C4×C101—22no—
90C2×C4×C101—22no—
91C2×C4×C101—22no—
92C2×C4×C101—22no—
93C2×C4×C101C2×D411yescharacteristic
94C2×C4×C101C2×D411yescharacteristic
95C2×C4×C101C2×D411yescharacteristic
96C2×C4×C101#4114yes—
97C2×C4×C101C2411yescharacteristic
98C22⋊D101—42no—
99C22⋊D101—42no—
100C22⋊D101—42no—
101C2×Dic101—22no—
102C2×Dic101—22no—
103C2×Dic101—21no—
104C2×Dic101—21no—
105C2×Dic101—21no—
106C2×Dic101—42no—
107C2×Dic101—42no—
108D404—22no—
109C4×D101—42no—
110Dic202—22no—
111C8×C101—22no—
112C8×C101—22no—
113C22×C101—21no—
114C22×C101—21no—
115C22×C101—21no—
116C22×C101—21no—
117C22×C101—21no—
118C22×C101—21no—
119C22×C101—21no—
120C22×C101—44no—
121C22×C101#12312yes—
122C22×C101#13512yes—
123C22×C101C22×D411yescharacteristic
124C2×D101—41no—
125C2×D101—41no—
126C2×D101—41no—
127C4×C101—21no—
128C4×C101—21no—
129C4×C101#12312yes—
130C4×C101C2×#4112yes—
131Dic101—41no—
132Dic101—42no—
133Dic101—41no—
134Dic101—42no—
135Dic101—41no—
136C2×C101—22no—
137C2×C101—22no—
138C2×C101—41no—
139C2×C101C2×#12311yescharacteristic
140C2×C101C2×#13511yescharacteristic
141C2×C101#57511yescharacteristic
142D101—81no—
143#3091—1011nomaximal, Sylow
144C101#309111yescharacteristic, complemented, Sylow
145#525—1011no—
146C2×#117—1011no—
147#485—1012no—
148#484—1011no—
149#483—1011no—
150#482—1012no—
151#481—1011no—
152#480—1011no—
153#427—1012no—
154#416—1012no—
155C2×C22⋊C8D10111yescharacteristic, complemented
156C2×#41—1011no—
157C23×C4C2×D10111yescharacteristic
158C22×C8C2×D10112yes—
159#123—2021no—
160C22⋊Q8—1011no—
161C22⋊Q8—1011no—
162#121—1011no—
163#121—1011no—
164#120—2021no—
165C4×D4—2021no—
166C4×D4—2021no—
167#117—1012no—
168#117—1012no—
169#117—1011no—
170#117—2021no—
171#117—2021no—
172C2×C4⋊C4—1011no—
173C2×C4⋊C4—1011no—
174C2×C22⋊C4—2021no—
175C2×C42—2021no—
176#107—1012no—
177#107—1012no—
178#106—1012no—
179#106—1012no—
180#103—1012no—
181#103—1012no—
182#102—1012no—
183#102—1012no—
184C22⋊C8C2×D10111yescharacteristic
185C22⋊C8C2×D10111yescharacteristic
186C22⋊C8C2×D10112yes—
187C24D40411yescharacteristic
188#41—2024no—
189C2×Q8—1011no—
190C2×D4—1011no—
191C2×D4—2021no—
192C2×D4—2021no—
193C2×D4—2021no—
194C22×C4—24no—
195C22×C4—2021no—
196C22×C4—2021no—
197C22×C4—2021no—
198C22×C4D40411yescharacteristic
199C22×C4D40411yescharacteristic
200C22×C4D40411yescharacteristic
201C22×C4C22×D10111yescharacteristic
202C22×C4C22×D10111yescharacteristic
203C22×C4C22×D10111yescharacteristic
204C2×C8—24no—
205C2×C8C22×D10112yes—
206C2×C8C22×D10112yes—
207C4⋊C4—1012no—
208C4⋊C4—1012no—
209C4⋊C4—1011no—
210C4⋊C4—1011no—
211C4⋊C4—2021no—
212C4⋊C4—2021no—
213C4⋊C4—2021no—
214C4⋊C4—2021no—
215C22⋊C4—2022no—
216C22⋊C4—2022no—
217C22⋊C4—2021no—
218C22⋊C4—2021no—
219C22⋊C4—2021no—
220C22⋊C4—2021no—
221C22⋊C4—2021no—
222C42—2022no—
223C42—2022no—
224C42—2021no—
225C23—22no—
226C23—22no—
227C23—41no—
228C23—2021no—
229C23C2×D40411yescharacteristic
230C23C2×D40411yescharacteristic
231C23C2×D40411yescharacteristic
232Q8—2022no—
233D4—4042no—
234D4—4042no—
235D4—4042no—
236D4—2022no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—4042no—
243C2×C4—4042no—
244C2×C4—4042no—
245C2×C4—2022no—
246C2×C4—2022no—
247C2×C4—2021no—
248C2×C4—2021no—
249C2×C4—2021no—
250C2×C4C2×D40411yescharacteristic
251C2×C4C2×D40411yescharacteristic
252C2×C4C2×D40411yescharacteristic
253C2×C4#1310807314yes—
254C2×C4C23×D10111yescharacteristic
255C8—22no—
256C8—22no—
257C22—21no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—21no—
262C22—21no—
263C22—21no—
264C22—44no—
265C22—4041no—
266C22—4041no—
267C22—4041no—
268C22#1569223912yes—
269C22#1569227712yes—
270C22C22×D40411yescharacteristic
271C4—21no—
272C4—21no—
273C4—4041no—
274C4—4042no—
275C4—4041no—
276C4—4042no—
277C4—4041no—
278C4#1569223912yes—
279C4C2×#1310807312yes—
280C2—22no—
281C2—22no—
282C2—41no—
283C2—8081no—
284C2C2×#1569223911yescharacteristic
285C2C2×#1569227711yescharacteristic
286C2#2026597711yescharacteristic
subgroupG/H
centerC2×C4C2×D404
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC2×C22⋊C8×C101C2
socleC22×C101C22×D4
prime radical#20265688C2
hypercenterC2×C22⋊C8D101
hypocenterC101#3091
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×#15692214C2semidirect
3. #20265519C2semidirect
4. #20265513C2semidirect
5. #20265508C2semidirect
6. #20265502C2semidirect
7. #20265496C2semidirect
8. #20265491C2semidirect
9. #20265215C2semidirect
10. #20265174C2semidirect
11. C2×C22⋊C8×C101C2semidirect
30. #15692189C22semidirect
31. #15692189C22semidirect
32. #15692187C22semidirect
33. #15692187C22semidirect
34. #15692180C22semidirect
35. #15692180C22semidirect
36. #15692177C22semidirect
37. #15692177C22semidirect
144. C101#3091semidirect
155. C2×C22⋊C8D101semidirect

Maximal subgroups

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

subgroupindex
#202656882
C2×#156922142
#202655192
#202655132
#202655082
#202655022
#202654962
#202654912
#202652152
#202651742
C2×C22⋊C8×C1012
#3091101