Finite Group Databasebeta

← #27180928

Subgroups — #27180928

Order 12928.

440 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 135 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#20265730C211yesmaximal, characteristic, complemented
2C2×#15692217C211yesmaximal, characteristic
3#20265480C212yesmaximal, complemented
4#20265476C212yesmaximal, complemented
5#20265466C212yesmaximal, complemented
6#20265456C212yesmaximal, complemented
7#20265217C211yesmaximal, characteristic, complemented
8#20265200C211yesmaximal, characteristic, complemented
9#20265183C211yesmaximal, characteristic
10#20265181C211yesmaximal, characteristic
11#428×C101C211yesmaximal, characteristic, complemented
12C22×D404C2211yescharacteristic
13C22×C4×D101—21no—
14C2×#13108058C2211yescharacteristic
15#15692254C2211yescharacteristic, complemented
16#15692254C2211yescharacteristic, complemented
17#15692246C2211yescharacteristic, complemented
18#15692246C2211yescharacteristic, complemented
19#15692232—21no—
20#15692232—21no—
21C4×Dic202—21no—
22C4×Dic202—21no—
23#15692217—21no—
24#15692217—21no—
25#15692217C2212yes—
26#15692217C2212yes—
27#15692214C2211yescharacteristic
28C2×#13108051—21no—
29C2×C4×Dic101C2211yescharacteristic
30#15692188C2212yescomplemented
31#15692186C2212yescomplemented
32#15692184C2212yescomplemented
33#15692184C2212yescomplemented
34#15692177C2212yes—
35#15692177C2212yes—
36#15692175C2212yes—
37#15692175C2212yes—
38C2×M16×C101C2211yescharacteristic
39#117×C101C2211yescharacteristic
40C2×C4⋊C4×C101C2211yescharacteristic
41#107×C101C2212yescomplemented
42#106×C101C2212yescomplemented
43C23×D101—21no—
44C2×Dic202—21no—
45C2×Dic202—21no—
46C2×D404—21no—
47C2×D404—22no—
48C2×D404Q814yescomplemented
49C2×D404C2312yes—
50C22×Dic101D411yescharacteristic
51C22×Dic101D411yescharacteristic
52C2×C4×D101—22no—
53C2×C4×D101—22no—
54C2×C4×D101—44no—
55#13108058—22no—
56C2×#11717579C2312yes—
57C4⋊Dic101—21no—
58C4⋊Dic101—21no—
59C4⋊Dic101C2311yescharacteristic
60C4⋊Dic101C2311yescharacteristic
61#13108053—21no—
62#13108053—21no—
63#13108053—21no—
64#13108053—21no—
65C22⋊Dic101—21no—
66#13108051—22no—
67#13108051—22no—
68C4×Dic101—21no—
69C4×Dic101D411yescharacteristic
70C4×Dic101D411yescharacteristic
71C4×Dic101C2311yescharacteristic
72C4×Dic101C2311yescharacteristic
73C22×C4×C101—21no—
74C22×C4×C101C2311yescharacteristic
75M16×C101—22no—
76C2×C8×C101C2312yes—
77C4⋊C4×C101—21no—
78C4⋊C4×C101C2311yescharacteristic
79C4⋊C4×C101C2312yes—
80C4⋊C4×C101C2311yescharacteristic
81C22⋊C4×C101—21no—
82C42×C101—21no—
83C23×C101C2×D411yescharacteristic
84C22×D101—24no—
85C22×D101—22no—
86C22×D101—42no—
87C22×D101—42no—
88C2×C4×C101—21no—
89C2×C4×C101—21no—
90C2×C4×C101—22no—
91C2×C4×C101—42no—
92C2×C4×C101C2×D411yescharacteristic
93C2×C4×C101C2×Q812yes—
94C2×C4×C101#4112yes—
95C2×C4×C101C2411yescharacteristic
96C2×Dic101—22no—
97C2×Dic101—21no—
98C2×Dic101—21no—
99C2×Dic101—41no—
100C2×Dic101C2×D411yescharacteristic
101C2×Dic101C2×D411yescharacteristic
102C2×Dic101C2×D411yescharacteristic
103C2×Dic101C2×D411yescharacteristic
104D404—24no—
105D404—22no—
106D404C2×Q814yes—
107C4×D101—44no—
108C4×D101—44no—
109Dic202—42no—
110Dic202—42no—
111C8×C101—22no—
112#11717579—22no—
113C22×C101—21no—
114C22×C101—21no—
115C22×C101C22⋊Q812yes—
116C22×C101C22×D411yescharacteristic
117C2×D101—24no—
118C2×D101—48no—
119C2×D101—42no—
120C2×D101—42no—
121C4×C101—41no—
122C4×C101—42no—
123C4×C101—41no—
124C4×C101C22⋊Q812yes—
125C4×C101C22×Q811yescharacteristic
126C4×C101C2×#4111yescharacteristic
127Dic101—21no—
128Dic101—21no—
129Dic101—21no—
130Dic101—41no—
131Dic101—41no—
132Dic101AGL(1,ℤ/8)12yes—
133C2×C101—22no—
134C2×C101C2×C22⋊Q811yescharacteristic
135C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
136C2×C101#57511yescharacteristic
137D101—44no—
138#3077—1011nomaximal, Sylow
139C101#307711yescharacteristic, complemented, Sylow
140#533—1011no—
141C2×C4×D4—1011no—
142#478—1012no—
143#478—1012no—
144#476—1012no—
145#474—1012no—
146#428—1011no—
147#428D10111yescharacteristic, complemented
148#423—1011no—
149#418—1011no—
150#418—1011no—
151C22×D4—1011no—
152C23×C4—2021no—
153C2×M16—1011no—
154C2×M16C2×D10111yescharacteristic
155C4⋊Q8—1011no—
156C4⋊Q8—1011no—
157#125—1011no—
158#125—1011no—
159C22⋊Q8—2021no—
160C22⋊Q8—2021no—
161C4×Q8—2021no—
162C4×Q8—2021no—
163C4×D4—1012no—
164C4×D4—1012no—
165C4×D4—2021no—
166C4×D4—2021no—
167#117—1011no—
168#117C2×D10111yescharacteristic
169C2×C4⋊C4C2×D10111yescharacteristic
170C2×C22⋊C4—2021no—
171C2×C42—1011no—
172#107—1012no—
173#107C2×D10112yescomplemented
174#106—1012no—
175#106C2×D10112yescomplemented
176C4⋊C8—1012no—
177C4⋊C8—1012no—
178#102—1012no—
179#102—1012no—
180#102—1012no—
181#102—1012no—
182C24—2021no—
183C2×Q8—2021no—
184C2×Q8—2021no—
185C2×D4—1014no—
186C2×D4—1012no—
187C2×D4—2021no—
188C2×D4—2022no—
189C22×C4—21no—
190C22×C4—1011no—
191C22×C4—1011no—
192C22×C4—4044no—
193C22×C4—2022no—
194C22×C4—2022no—
195C22×C4C22×D10111yescharacteristic
196M16—22no—
197M16—2022no—
198C2×C8—1012no—
199C2×C8C22×D10112yes—
200C4⋊C4—21no—
201C4⋊C4—1011no—
202C4⋊C4—1011no—
203C4⋊C4—2021no—
204C4⋊C4—2021no—
205C4⋊C4—2021no—
206C4⋊C4—2021no—
207C4⋊C4—2021no—
208C4⋊C4—2021no—
209C4⋊C4C22×D10111yescharacteristic
210C4⋊C4C22×D10111yescharacteristic
211C4⋊C4C22×D10112yes—
212C22⋊C4—21no—
213C22⋊C4—2022no—
214C22⋊C4—2022no—
215C22⋊C4—2021no—
216C42—21no—
217C42—1011no—
218C42—1011no—
219C42—1011no—
220C42—1011no—
221C42—2021no—
222C23—4042no—
223C23—4042no—
224C23—2022no—
225C23—2024no—
226C23D4×D10111yescharacteristic
227Q8—4042no—
228Q8—4042no—
229D4—1014no—
230D4—2022no—
231D4—2024no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—42no—
236C2×C4—1011no—
237C2×C4—1011no—
238C2×C4—1011no—
239C2×C4—1011no—
240C2×C4—4044no—
241C2×C4—4044no—
242C2×C4—4041no—
243C2×C4—2022no—
244C2×C4—2021no—
245C2×C4—2021no—
246C2×C4D4×D10111yescharacteristic
247C2×C4Q8×D10112yes—
248C2×C4#1310807412yes—
249C2×C4C23×D10111yescharacteristic
250C8—22no—
251C8—2022no—
252C22—21no—
253C22—21no—
254C22—4048no—
255C22—4042no—
256C22—4042no—
257C22—2024no—
258C22#1569223312yes—
259C22C2×D4×D10111yescharacteristic
260C4—42no—
261C4—41no—
262C4—41no—
263C4—1012no—
264C4—4041no—
265C4—4041no—
266C4—2021no—
267C4—2021no—
268C4—2021no—
269C4#1569223312yes—
270C4C2×Q8×D10111yescharacteristic
271C4C2×#1310807411yescharacteristic
272C2—22no—
273C2—4044no—
274C2C2×#1569223311yescharacteristic
275C2AGL(1,ℤ/8)×D10111yescharacteristic
276C2#2026597611yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#428×C101C2
socleC22×C101C22×D4
prime radicalC22×D404C22
hypercenter#428D101
hypocenterC101#3077
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
1. #20265730C2semidirect
3. #20265480C2semidirect
4. #20265476C2semidirect
5. #20265466C2semidirect
6. #20265456C2semidirect
7. #20265217C2semidirect
8. #20265200C2semidirect
11. #428×C101C2semidirect
15. #15692254C22semidirect
16. #15692254C22semidirect
17. #15692246C22semidirect
18. #15692246C22semidirect
30. #15692188C22semidirect
31. #15692186C22semidirect
32. #15692184C22semidirect
33. #15692184C22semidirect
41. #107×C101C22semidirect
42. #106×C101C22semidirect
48. C2×D404Q8semidirect
139. C101#3077semidirect
147. #428D101semidirect
173. #107C2×D101semidirect
175. #106C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202657302
C2×#156922172
#202654802
#202654762
#202654662
#202654562
#202652172
#202652002
#202651832
#202651812
#428×C1012
#3077101