Finite Group Databasebeta

← #27181517

Subgroups — #27181517

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#20265732C211yesmaximal, characteristic, complemented
2#124×D101C211yesmaximal, characteristic
3#20265539C211yesmaximal, characteristic, complemented
4#20265537C211yesmaximal, characteristic, complemented
5#20265536C211yesmaximal, characteristic, complemented
6#20265533C212yesmaximal, complemented
7#20265530C212yesmaximal, complemented
8#20265184C212yesmaximal, complemented
9#20265180C212yesmaximal
10#20265129C211yesmaximal, characteristic, complemented
11#489×C101C211yesmaximal, characteristic, complemented
12C2×Q8×D101C2211yescharacteristic
13C2×D4×D101C2211yescharacteristic
14C2×#13108057C2212yes—
15#15692255C2211yescharacteristic, complemented
16#15692246C2212yescomplemented
17#15692244C2211yescharacteristic
18#15692243C2211yescharacteristic
19#15692242C2211yescharacteristic
20#15692241—21no—
21#15692241—21no—
22#15692235—21no—
23#15692231—21no—
24Q8×Dic101—21no—
25#15692220—21no—
26#15692213C2212yes—
27C22⋊C4×D101—21no—
28C22⋊C4×D101—21no—
29C42×D101C2211yescharacteristic
30#15692180C2212yescomplemented
31#15692179C2212yescomplemented
32#15692178C2212yescomplemented
33#15692177C2212yes—
34#15692176C2212yes—
35#15692175C2212yes—
36#15692166C2211yescharacteristic, complemented
37C8×Dic101C2212yescomplemented
38C4⋊Q8×C101C2211yescharacteristic, complemented
39#124×C101C2211yescharacteristic
40#103×C101C2212yescomplemented
41#102×C101C2212yes—
42#97×C101C2211yescharacteristic, complemented
43C23×D101—21no—
44C2×Dic202—21no—
45C2×Dic202—21no—
46C2×Dic202C2311yescharacteristic
47Q8×D101—24no—
48C2×D404C2311yescharacteristic
49C2×C22⋊D101—21no—
50D4×D101—24no—
51C2×C4×D101—21no—
52C2×C4×D101D411yescharacteristic
53C2×C4×D101D411yescharacteristic
54C2×C4×D101C2311yescharacteristic
55#13108057—24no—
56C2×#11717579C2312yes—
57C4⋊Dic101—21no—
58C4⋊Dic101—22no—
59C4⋊Dic101C2312yes—
60#13108053—22no—
61C22⋊Dic101—21no—
62C22⋊Dic101—21no—
63#13108051—22no—
64#13108051—21no—
65#13108051—21no—
66#13108051—21no—
67#13108051—21no—
68C4×Dic101—22no—
69C4×Dic101D411yescharacteristic
70C4×Dic101D411yescharacteristic
71C4×Dic101C2311yescharacteristic
72C2×Q8×C101—21no—
73C2×Q8×C101C2311yescharacteristic
74C2×D4×C101C2311yescharacteristic
75C2×C8×C101C2312yes—
76C4⋊C4×C101—21no—
77C4⋊C4×C101C2312yes—
78C22⋊C4×C101—21no—
79C22⋊C4×C101—21no—
80C42×C101C2311yescharacteristic
81C23×C101—21no—
82C22×D101—21no—
83C22×D101—42no—
84C22×D101—42no—
85C22×D101—42no—
86C22×D101C2×D411yescharacteristic
87Q8×C101—22no—
88Q8×C101—42no—
89D4×C101—22no—
90C2×C4×C101—21no—
91C2×C4×C101—22no—
92C2×C4×C101C2×D411yescharacteristic
93C2×C4×C101C2×D411yescharacteristic
94C2×C4×C101C2411yescharacteristic
95C22⋊D101—42no—
96C2×Dic101—22no—
97C2×Dic101—21no—
98C2×Dic101C2×D411yescharacteristic
99C2×Dic101C2×D411yescharacteristic
100C2×Dic101C2×D411yescharacteristic
101D404—22no—
102C4×D101—41no—
103C4×D101—41no—
104C4×D101—42no—
105C4×D101#4114yes—
106Dic202—22no—
107Dic202—42no—
108Dic202—42no—
109C8×C101—22no—
110#11717579—22no—
111C22×C101—41no—
112C22×C101—41no—
113C22×C101—41no—
114C22×C101C22×D411yescharacteristic
115C2×D101—21no—
116C2×D101—21no—
117C2×D101—41no—
118C2×D101—41no—
119C2×D101—41no—
120C2×D101—84no—
121C2×D101#12412yes—
122C4×C101—21no—
123C4×C101—21no—
124C4×C101—41no—
125C4×C101—42no—
126C4×C101C2×#4112yes—
127Dic101—21no—
128Dic101—21no—
129Dic101—41no—
130Dic101—42no—
131Dic101#12412yes—
132C2×C101—81no—
133C2×C101C2×#12411yescharacteristic
134C2×C101#57511yescharacteristic
135C2×C101#57511yescharacteristic
136D101—22no—
137D101—81no—
138#3140—1011nomaximal, Sylow
139C101#314011yescharacteristic, complemented, Sylow
140#533—1011no—
141C2×#124—1011no—
142#489—1011no—
143#489—1011no—
144#489—1011no—
145#489D10111yescharacteristic, complemented
146#488—1012no—
147#488—1012no—
148#419—1012no—
149#418—1012no—
150C4×M16—1011no—
151C22×Q8—1011no—
152C22×D4—1011no—
153C2×M16—1012no—
154C4⋊Q8—1011no—
155C4⋊Q8C2×D10111yescharacteristic, complemented
156#125—1012no—
157#124—1011no—
158#124—1011no—
159#124—1011no—
160#124—2021no—
161#124—2021no—
162#124C2×D10111yescharacteristic
163C22⋊Q8—2021no—
164C22⋊Q8—2021no—
165C4×Q8—2021no—
166C4×Q8—2021no—
167#117—1012no—
168C2×C22⋊C4—2021no—
169C2×C22⋊C4—2021no—
170C2×C42—1011no—
171#103—1012no—
172#103—1012no—
173#103—1012no—
174#103C2×D10112yescomplemented
175#102—1012no—
176#102—1012no—
177#102—1012no—
178#102C2×D10112yes—
179#97—1011no—
180#97C2×D10111yescharacteristic, complemented
181C4×C8—1012no—
182C24—2021no—
183C2×Q8—21no—
184C2×Q8—1011no—
185C2×Q8—2021no—
186C2×Q8—2024no—
187C2×Q8—2021no—
188C2×Q8C22×D10111yescharacteristic
189C2×D4—1011no—
190C2×D4—2024no—
191C2×D4—2021no—
192C2×D4C22×D10111yescharacteristic
193C22×C4—1011no—
194C22×C4—1011no—
195C22×C4—1011no—
196C22×C4—2021no—
197M16—2024no—
198C2×C8—1012no—
199C2×C8C22×D10112yes—
200C4⋊C4—21no—
201C4⋊C4—1012no—
202C4⋊C4—2022no—
203C4⋊C4—2022no—
204C4⋊C4—2021no—
205C4⋊C4C22×D10112yes—
206C22⋊C4—21no—
207C22⋊C4—21no—
208C22⋊C4—2021no—
209C22⋊C4—2021no—
210C22⋊C4—2022no—
211C22⋊C4—2021no—
212C22⋊C4—2021no—
213C22⋊C4—2021no—
214C22⋊C4—2021no—
215C42—1011no—
216C42—1011no—
217C42—1011no—
218C42—2022no—
219C42C22×D10111yescharacteristic
220C23—21no—
221C23—1011no—
222C23—4042no—
223C23—4042no—
224C23—4042no—
225C23—2021no—
226Q8—22no—
227Q8—42no—
228Q8—4042no—
229Q8—4042no—
230Q8—2022no—
231D4—22no—
232D4—4042no—
233D4—2022no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—1011no—
237C2×C4—1014no—
238C2×C4—1011no—
239C2×C4—1011no—
240C2×C4—4041no—
241C2×C4—4041no—
242C2×C4—4042no—
243C2×C4—2022no—
244C2×C4—2021no—
245C2×C4D4×D10111yescharacteristic
246C2×C4D4×D10111yescharacteristic
247C2×C4C23×D10111yescharacteristic
248C8—22no—
249C8—2022no—
250C22—41no—
251C22—41no—
252C22—41no—
253C22—8084no—
254C22—1012no—
255C22—4041no—
256C22—4041no—
257C22—4041no—
258C22—2021no—
259C22—2021no—
260C22C2×D4×D10111yescharacteristic
261C4—21no—
262C4—21no—
263C4—41no—
264C4—42no—
265C4—1012no—
266C4—4041no—
267C4—4042no—
268C4—2021no—
269C4—2021no—
270C4#41×D10112yes—
271C2—81no—
272C2—8081no—
273C2—2022no—
274C2#124×D10111yescharacteristic
275C2#2026597311yescharacteristic
276C2#2026597611yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#489×C101C2
socleC22×C101C22×D4
prime radicalC2×D4×D101C22
hypercenter#489D101
hypocenterC101#3140
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. #20265732C2semidirect
3. #20265539C2semidirect
4. #20265537C2semidirect
5. #20265536C2semidirect
6. #20265533C2semidirect
7. #20265530C2semidirect
8. #20265184C2semidirect
10. #20265129C2semidirect
11. #489×C101C2semidirect
15. #15692255C22semidirect
16. #15692246C22semidirect
30. #15692180C22semidirect
31. #15692179C22semidirect
32. #15692178C22semidirect
36. #15692166C22semidirect
37. C8×Dic101C22semidirect
38. C4⋊Q8×C101C22semidirect
40. #103×C101C22semidirect
42. #97×C101C22semidirect
139. C101#3140semidirect
145. #489D101semidirect
155. C4⋊Q8C2×D101semidirect
174. #103C2×D101semidirect
180. #97C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202657322
#124×D1012
#202655392
#202655372
#202655362
#202655332
#202655302
#202651842
#202651802
#202651292
#489×C1012
#3140101