Finite Group Databasebeta

← #27181515

Subgroups — #27181515

Order 12928.

400 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 117 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#20265733C211yesmaximal, characteristic, complemented
2#124×D101C211yesmaximal, characteristic
3#20265538C212yesmaximal, complemented
4#20265535C212yesmaximal, complemented
5#20265534C211yesmaximal, characteristic, complemented
6#20265532C211yesmaximal, characteristic, complemented
7#20265531C211yesmaximal, characteristic, complemented
8#20265184C212yesmaximal, complemented
9#20265180C212yesmaximal
10C4×#13108057C211yesmaximal, characteristic, complemented
11#488×C101C211yesmaximal, characteristic, complemented
12C2×Q8×D101C2211yescharacteristic
13C2×D4×D101C2211yescharacteristic
14C2×#13108057C2212yes—
15#15692254C2212yescomplemented
16#15692247C2211yescharacteristic, complemented
17#15692244C2211yescharacteristic
18#15692243C2211yescharacteristic
19#15692242C2211yescharacteristic
20#15692241—22no—
21#15692233—22no—
22#15692220—22no—
23#15692213C2212yes—
24C22⋊C4×D101—22no—
25C42×D101C2211yescharacteristic
26#15692180C2212yescomplemented
27#15692179C2212yescomplemented
28#15692178C2212yescomplemented
29#15692177C2212yes—
30#15692176C2212yes—
31#15692175C2212yes—
32#15692165C2212yescomplemented
33C4×#11717579C2211yescharacteristic, complemented
34#125×C101C2211yescharacteristic, complemented
35#124×C101C2211yescharacteristic
36#103×C101C2212yescomplemented
37#102×C101C2212yes—
38C4×C8×C101C2211yescharacteristic, complemented
39C23×D101—21no—
40C2×Dic202—21no—
41C2×Dic202—22no—
42C2×Dic202C2311yescharacteristic
43Q8×D101—24no—
44C2×D404C2311yescharacteristic
45C2×C22⋊D101—21no—
46D4×D101—24no—
47C2×C4×D101—21no—
48C2×C4×D101D412yes—
49C2×C4×D101C2311yescharacteristic
50#13108057—24no—
51C2×#11717579C2312yes—
52C4⋊Dic101C2312yes—
53#13108053—22no—
54#13108053—22no—
55C22⋊Dic101—22no—
56#13108051—22no—
57#13108051—22no—
58#13108051—22no—
59C4×Dic101—22no—
60C4×Dic101D412yes—
61C4×Dic101C2311yescharacteristic
62C2×Q8×C101C2311yescharacteristic
63C2×D4×C101C2311yescharacteristic
64C2×C8×C101C2312yes—
65C4⋊C4×C101—22no—
66C4⋊C4×C101C2312yes—
67C22⋊C4×C101—22no—
68C42×C101C2311yescharacteristic
69C23×C101—21no—
70C22×D101—21no—
71C22×D101—44no—
72C22×D101—42no—
73C22×D101C2×D411yescharacteristic
74Q8×C101—22no—
75D4×C101—22no—
76C2×C4×C101—21no—
77C2×C4×C101—22no—
78C2×C4×C101C2×D412yes—
79C2×C4×C101C2411yescharacteristic
80C22⋊D101—42no—
81C2×Dic101—22no—
82C2×Dic101—21no—
83C2×Dic101C2×D412yes—
84C2×Dic101C2×D411yescharacteristic
85D404—22no—
86C4×D101—42no—
87C4×D101—42no—
88C4×D101#4114yes—
89Dic202—22no—
90Dic202—44no—
91Dic202—42no—
92C8×C101—22no—
93#11717579—22no—
94C22×C101—41no—
95C22×C101—42no—
96C22×C101C22×D411yescharacteristic
97C2×D101—22no—
98C2×D101—41no—
99C2×D101—42no—
100C2×D101—84no—
101C2×D101#12412yes—
102C4×C101—22no—
103C4×C101—41no—
104C4×C101—42no—
105C4×C101C2×#4112yes—
106Dic101—22no—
107Dic101—41no—
108Dic101—42no—
109Dic101#12412yes—
110C2×C101—81no—
111C2×C101C2×#12411yescharacteristic
112C2×C101#57512yes—
113D101—22no—
114D101—81no—
115#3140—1011nomaximal, Sylow
116C101#314011yescharacteristic, complemented, Sylow
117#533—1011no—
118C2×#124—1011no—
119#489—1012no—
120#489—1012no—
121#488—1011no—
122#488—1011no—
123#488—1011no—
124#488D10111yescharacteristic, complemented
125#419—1012no—
126#418—1012no—
127C4×M16—1011no—
128C22×Q8—1011no—
129C22×D4—1011no—
130C2×M16—1012no—
131C4⋊Q8—1012no—
132#125—1011no—
133#125C2×D10111yescharacteristic, complemented
134#124—1011no—
135#124—1011no—
136#124—1011no—
137#124—2022no—
138#124C2×D10111yescharacteristic
139C22⋊Q8—2022no—
140C4×Q8—2022no—
141#117—1012no—
142C2×C22⋊C4—2022no—
143C2×C42—1011no—
144#103—1012no—
145#103—1012no—
146#103—1012no—
147#103C2×D10112yescomplemented
148#102—1012no—
149#102—1012no—
150#102—1012no—
151#102C2×D10112yes—
152#97—1012no—
153C4×C8—1011no—
154C4×C8C2×D10111yescharacteristic, complemented
155C24—2021no—
156C2×Q8—1011no—
157C2×Q8—2022no—
158C2×Q8—2021no—
159C2×Q8—2024no—
160C2×Q8C22×D10111yescharacteristic
161C2×D4—1011no—
162C2×D4—2024no—
163C2×D4—2021no—
164C2×D4C22×D10111yescharacteristic
165C22×C4—1011no—
166C22×C4—1012no—
167C22×C4—2021no—
168M16—2024no—
169C2×C8—1012no—
170C2×C8C22×D10112yes—
171C4⋊C4—22no—
172C4⋊C4—1012no—
173C4⋊C4—2022no—
174C4⋊C4—2022no—
175C4⋊C4C22×D10112yes—
176C22⋊C4—22no—
177C22⋊C4—2022no—
178C22⋊C4—2022no—
179C22⋊C4—2022no—
180C22⋊C4—2022no—
181C42—1012no—
182C42—1011no—
183C42—2022no—
184C42C22×D10111yescharacteristic
185C23—21no—
186C23—1011no—
187C23—4042no—
188C23—4044no—
189C23—2021no—
190Q8—22no—
191Q8—4042no—
192Q8—4044no—
193Q8—2022no—
194D4—22no—
195D4—4042no—
196D4—2022no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—1011no—
200C2×C4—1014no—
201C2×C4—1012no—
202C2×C4—4042no—
203C2×C4—4042no—
204C2×C4—2022no—
205C2×C4—2021no—
206C2×C4D4×D10112yes—
207C2×C4C23×D10111yescharacteristic
208C8—22no—
209C8—2022no—
210C22—41no—
211C22—42no—
212C22—8084no—
213C22—1012no—
214C22—4041no—
215C22—4042no—
216C22—2022no—
217C22C2×D4×D10111yescharacteristic
218C4—22no—
219C4—41no—
220C4—42no—
221C4—1012no—
222C4—4041no—
223C4—4042no—
224C4—2022no—
225C4#41×D10112yes—
226C2—81no—
227C2—8081no—
228C2—2022no—
229C2#124×D10111yescharacteristic
230C2#2026597412yes—
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#488×C101C2
socleC22×C101C22×D4
prime radicalC2×D4×D101C22
hypercenter#488D101
hypocenterC101#3140
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #488

Complements

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

NH ≅ G/Ntype
1. #20265733C2semidirect
3. #20265538C2semidirect
4. #20265535C2semidirect
5. #20265534C2semidirect
6. #20265532C2semidirect
7. #20265531C2semidirect
8. #20265184C2semidirect
10. C4×#13108057C2semidirect
11. #488×C101C2semidirect
15. #15692254C22semidirect
16. #15692247C22semidirect
26. #15692180C22semidirect
27. #15692179C22semidirect
28. #15692178C22semidirect
32. #15692165C22semidirect
33. C4×#11717579C22semidirect
34. #125×C101C22semidirect
36. #103×C101C22semidirect
38. C4×C8×C101C22semidirect
116. C101#3140semidirect
124. #488D101semidirect
133. #125C2×D101semidirect
147. #103C2×D101semidirect
154. C4×C8C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202657332
#124×D1012
#202655382
#202655352
#202655342
#202655322
#202655312
#202651842
#202651802
C4×#131080572
#488×C1012
#3140101