Finite Group Databasebeta

← #27181066

Subgroups — #27181066

Order 12928.

414 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 125 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#20265689C211yesmaximal, characteristic
2C2×#15692213C211yesmaximal, characteristic, complemented
3#20265521C212yesmaximal, complemented
4#20265520C212yesmaximal, complemented
5#20265504C212yesmaximal, complemented
6#20265503C212yesmaximal, complemented
7#20265220C211yesmaximal, characteristic, complemented
8#20265179C211yesmaximal, characteristic, complemented
9#20265176C211yesmaximal, characteristic, complemented
10#20265140C211yesmaximal, characteristic, complemented
11#428×C101C211yesmaximal, characteristic, complemented
12C2×#13108074C2211yescharacteristic
13C22×C4×D101C2211yescharacteristic
14C2×#13108058C2211yescharacteristic
15#15692239—21no—
16#15692235C2211yescharacteristic
17#15692235C2211yescharacteristic
18#15692230C2211yescharacteristic
19#15692230C2211yescharacteristic
20#15692225—21no—
21C4×D404—21no—
22C4×D404—21no—
23#15692213—21no—
24#15692213—21no—
25#15692213C2212yes—
26#15692213C2212yes—
27C2×C4⋊Dic101C2211yescharacteristic
28C2×#13108051—21no—
29C2×C4×Dic101—21no—
30#15692188C2212yescomplemented
31#15692186C2212yescomplemented
32#15692180C2212yescomplemented
33#15692178C2212yescomplemented
34#15692177C2212yescomplemented
35#15692175C2212yescomplemented
36#15692167C2212yes—
37#15692167C2212yes—
38C2×M16×C101C2211yescharacteristic
39#117×C101C2211yescharacteristic
40C2×C4⋊C4×C101C2211yescharacteristic
41#107×C101C2212yescomplemented
42#106×C101C2212yescomplemented
43C23×D101D411yescharacteristic
44#13108074—22no—
45#13108074—22no—
46C2×Dic202C2311yescharacteristic
47C2×D404—21no—
48C2×D404—21no—
49C2×D404C2311yescharacteristic
50C2×C22⋊D101—21no—
51C22×Dic101—21no—
52C22×Dic101D411yescharacteristic
53C2×C4×D101—22no—
54C2×C4×D101—22no—
55C2×C4×D101—21no—
56C2×C4×D101D411yescharacteristic
57C2×C4×D101D411yescharacteristic
58C2×C4×D101C2311yescharacteristic
59C2×C4×D101C2311yescharacteristic
60#13108058—22no—
61C2×#11717579C2312yes—
62C4⋊Dic101—21no—
63C4⋊Dic101—21no—
64C4⋊Dic101—21no—
65C4⋊Dic101C2312yes—
66#13108051—22no—
67#13108051—22no—
68#13108051—21no—
69#13108051—21no—
70#13108051—21no—
71#13108051—21no—
72C4×Dic101—22no—
73C4×Dic101—22no—
74C22×C4×C101—21no—
75C22×C4×C101C2311yescharacteristic
76M16×C101—22no—
77C2×C8×C101C2312yes—
78C4⋊C4×C101—21no—
79C4⋊C4×C101C2311yescharacteristic
80C4⋊C4×C101C2312yes—
81C4⋊C4×C101C2311yescharacteristic
82C22⋊C4×C101—21no—
83C42×C101—21no—
84C23×C101C2×D411yescharacteristic
85C22×D101—22no—
86C22×D101—21no—
87C22×D101—41no—
88C22×D101—41no—
89C22×D101C2×D411yescharacteristic
90C22×D101C2×D411yescharacteristic
91C2×C4×C101—21no—
92C2×C4×C101—21no—
93C2×C4×C101—22no—
94C2×C4×C101—42no—
95C2×C4×C101C2×D411yescharacteristic
96C2×C4×C101#4112yes—
97C2×C4×C101#4112yes—
98C2×C4×C101C2411yescharacteristic
99C22⋊D101—42no—
100C2×Dic101—22no—
101C2×Dic101—22no—
102C2×Dic101—21no—
103C2×Dic101—42no—
104C2×Dic101C2×D411yescharacteristic
105C2×Dic101C2×D411yescharacteristic
106D404—21no—
107D404—21no—
108D404—42no—
109D404—42no—
110C4×D101—22no—
111C4×D101—22no—
112C4×D101—21no—
113C4×D101—21no—
114C4×D101—21no—
115C4×D101—21no—
116C4×D101—42no—
117Dic202—21no—
118Dic202—21no—
119C8×C101—22no—
120#11717579—22no—
121C22×C101—21no—
122C22×C101—21no—
123C22×C101#12312yes—
124C22×C101C22×D411yescharacteristic
125C2×D101—21no—
126C2×D101—21no—
127C2×D101—21no—
128C2×D101—21no—
129C2×D101—21no—
130C2×D101—21no—
131C2×D101—44no—
132C2×D101—41no—
133C2×D101—41no—
134C2×D101—41no—
135C4×C101—41no—
136C4×C101—41no—
137C4×C101—42no—
138C4×C101#12312yes—
139C4×C101C2×#4111yescharacteristic
140C4×C101C2×#4111yescharacteristic
141Dic101—21no—
142Dic101—21no—
143Dic101—41no—
144Dic101—42no—
145C2×C101—22no—
146C2×C101C2×#12311yescharacteristic
147C2×C101#57511yescharacteristic
148C2×C101#57511yescharacteristic
149D101—41no—
150D101—41no—
151D101—81no—
152#3093—1011nomaximal, Sylow
153C101#309311yescharacteristic, complemented, Sylow
154#525—1011no—
155C2×#117—1011no—
156#485—1012no—
157#485—1012no—
158#482—1012no—
159#482—1012no—
160#428—1011no—
161#428D10111yescharacteristic, complemented
162#417—1011no—
163#417—1011no—
164#407—1011no—
165C2×#41—1011no—
166C23×C4—1011no—
167C2×M16—1011no—
168C2×M16C2×D10111yescharacteristic
169#123—2021no—
170C22⋊Q8—1011no—
171C22⋊Q8—1011no—
172#121—1011no—
173#121—1011no—
174#120—2021no—
175C4×D4—2021no—
176C4×D4—2021no—
177#117—1012no—
178#117—1012no—
179#117—2021no—
180#117—2021no—
181#117C2×D10111yescharacteristic
182C2×C4⋊C4—1011no—
183C2×C4⋊C4C2×D10111yescharacteristic
184C2×C22⋊C4—2021no—
185C2×C42—2021no—
186#107—1012no—
187#107C2×D10112yescomplemented
188#106—1012no—
189#106C2×D10112yescomplemented
190#103—1012no—
191#103—1012no—
192#102—1012no—
193#102—1012no—
194C22⋊C8—1012no—
195C22⋊C8—1012no—
196C24—1011no—
197#41—2022no—
198#41—2022no—
199C2×Q8—1011no—
200C2×D4—1011no—
201C2×D4—2021no—
202C2×D4—2021no—
203C2×D4—2021no—
204C22×C4—21no—
205C22×C4—1011no—
206C22×C4—1011no—
207C22×C4—1011no—
208C22×C4—1011no—
209C22×C4—1011no—
210C22×C4—2021no—
211C22×C4—2021no—
212C22×C4—2022no—
213C22×C4—2022no—
214C22×C4C22×D10111yescharacteristic
215M16—22no—
216M16—2022no—
217C2×C8—1012no—
218C2×C8C22×D10112yes—
219C4⋊C4—21no—
220C4⋊C4—1012no—
221C4⋊C4—2021no—
222C4⋊C4—2021no—
223C4⋊C4—2021no—
224C4⋊C4C22×D10112yes—
225C4⋊C4C22×D10111yescharacteristic
226C4⋊C4C22×D10111yescharacteristic
227C22⋊C4—21no—
228C22⋊C4—2021no—
229C22⋊C4—2021no—
230C22⋊C4—2022no—
231C22⋊C4—2022no—
232C22⋊C4—2021no—
233C22⋊C4—2021no—
234C42—21no—
235C42—2022no—
236C42—2022no—
237C23—1011no—
238C23—1011no—
239C23—4041no—
240C23—4041no—
241C23—2021no—
242C23—2022no—
243C23D4×D10111yescharacteristic
244Q8—2021no—
245Q8—2021no—
246D4—4042no—
247D4—4042no—
248D4—4042no—
249D4—2021no—
250D4—2021no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—42no—
255C2×C4—1011no—
256C2×C4—1011no—
257C2×C4—4042no—
258C2×C4—4042no—
259C2×C4—2022no—
260C2×C4—2021no—
261C2×C4—2022no—
262C2×C4—2021no—
263C2×C4—2021no—
264C2×C4—2021no—
265C2×C4—2021no—
266C2×C4—2022no—
267C2×C4—2022no—
268C2×C4D4×D10111yescharacteristic
269C2×C4#1310807412yes—
270C2×C4#1310807512yes—
271C2×C4C23×D10111yescharacteristic
272C8—22no—
273C8—2022no—
274C22—21no—
275C22—21no—
276C22—4044no—
277C22—4041no—
278C22—4041no—
279C22—4041no—
280C22—2021no—
281C22—2021no—
282C22—2021no—
283C22—2021no—
284C22—2021no—
285C22—2021no—
286C22#1569224012yes—
287C22C2×D4×D10111yescharacteristic
288C4—42no—
289C4—41no—
290C4—41no—
291C4—4041no—
292C4—4042no—
293C4—2021no—
294C4—2021no—
295C4#1569224012yes—
296C4C2×#1310807411yescharacteristic
297C4C2×#1310807511yescharacteristic
298C2—22no—
299C2—8081no—
300C2—4041no—
301C2—4041no—
302C2C2×#1569224011yescharacteristic
303C2#2026597311yescharacteristic
304C2#2026597611yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#428×C101C2
socleC22×C101C22×D4
prime radical#20265689C2
hypercenter#428D101
hypocenterC101#3093
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
2. C2×#15692213C2semidirect
3. #20265521C2semidirect
4. #20265520C2semidirect
5. #20265504C2semidirect
6. #20265503C2semidirect
7. #20265220C2semidirect
8. #20265179C2semidirect
9. #20265176C2semidirect
10. #20265140C2semidirect
11. #428×C101C2semidirect
30. #15692188C22semidirect
31. #15692186C22semidirect
32. #15692180C22semidirect
33. #15692178C22semidirect
34. #15692177C22semidirect
35. #15692175C22semidirect
41. #107×C101C22semidirect
42. #106×C101C22semidirect
153. C101#3093semidirect
161. #428D101semidirect
187. #107C2×D101semidirect
189. #106C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202656892
C2×#156922132
#202655212
#202655202
#202655042
#202655032
#202652202
#202651792
#202651762
#202651402
#428×C1012
#3093101