Finite Group Databasebeta

← #27181077

Subgroups — #27181077

Order 12928.

410 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 121 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
1C22⋊Q8×D101C211yesmaximal, characteristic
2C2×#15692213C211yesmaximal, characteristic
3#20265520C212yesmaximal, complemented
4#20265516C212yesmaximal, complemented
5#20265512C211yesmaximal, characteristic, complemented
6#20265508C211yesmaximal, characteristic, complemented
7#20265505C211yesmaximal, characteristic, complemented
8#20265218C212yesmaximal, complemented
9#20265184C212yesmaximal, complemented
10#20265139C211yesmaximal, characteristic
11#484×C101C211yesmaximal, characteristic, complemented
12C2×Q8×D101C2211yescharacteristic
13C22×C4×D101C2211yescharacteristic
14C2×#13108057C2212yes—
15#15692235C2211yescharacteristic
16#15692234C2211yescharacteristic
17#15692233—21no—
18#15692232—21no—
19#15692231C2211yescharacteristic
20#15692213—21no—
21#15692213—21no—
22#15692213C2212yes—
23#15692213C2212yes—
24C2×C4⋊Dic101C2211yescharacteristic
25C4⋊C4×D101—21no—
26C4⋊C4×D101C2211yescharacteristic
27C2×#13108051—21no—
28C22⋊C4×D101—21no—
29C2×C4×Dic101—21no—
30#15692188C2212yescomplemented
31#15692187C2212yescomplemented
32#15692186C2212yescomplemented
33#15692180C2212yescomplemented
34#15692179C2212yescomplemented
35#15692178C2212yescomplemented
36C22⋊#11717579C2211yescharacteristic
37#15692167C2212yes—
38C22⋊Q8×C101C2211yescharacteristic
39C2×C4⋊C4×C101C2211yescharacteristic
40#107×C101C2212yescomplemented
41#103×C101C2212yescomplemented
42C22⋊C8×C101C2211yescharacteristic
43C23×D101D411yescharacteristic
44C2×Dic202—21no—
45C2×Dic202C2311yescharacteristic
46Q8×D101—24no—
47C22×Dic101—21no—
48C22×Dic101D411yescharacteristic
49C2×C4×D101—24no—
50C2×C4×D101—21no—
51C2×C4×D101—21no—
52C2×C4×D101D411yescharacteristic
53C2×C4×D101D411yescharacteristic
54C2×C4×D101C2311yescharacteristic
55C2×C4×D101C2311yescharacteristic
56#13108057—24no—
57C2×#11717579C2312yes—
58C4⋊Dic101—21no—
59C4⋊Dic101—21no—
60C4⋊Dic101C2312yes—
61C4⋊Dic101C2311yescharacteristic
62#13108053—21no—
63#13108053—21no—
64#13108053—21no—
65C22⋊Dic101—21no—
66#13108051—21no—
67#13108051—21no—
68#13108051—22no—
69#13108051—22no—
70C4×Dic101—22no—
71C4×Dic101—22no—
72C2×Q8×C101C2311yescharacteristic
73C22×C4×C101—21no—
74C22×C4×C101C2311yescharacteristic
75C2×C8×C101C2312yes—
76C4⋊C4×C101—21no—
77C4⋊C4×C101—21no—
78C4⋊C4×C101C2312yes—
79C4⋊C4×C101C2311yescharacteristic
80C22⋊C4×C101—21no—
81C23×C101C2×D411yescharacteristic
82C22×D101—22no—
83C22×D101—22no—
84C22×D101—41no—
85C22×D101C2×D411yescharacteristic
86C22×D101C2×D411yescharacteristic
87Q8×C101—22no—
88C2×C4×C101—22no—
89C2×C4×C101—21no—
90C2×C4×C101—21no—
91C2×C4×C101—22no—
92C2×C4×C101—42no—
93C2×C4×C101C2×D411yescharacteristic
94C2×C4×C101C2411yescharacteristic
95C2×Dic101—22no—
96C2×Dic101—22no—
97C2×Dic101—21no—
98C2×Dic101—21no—
99C2×Dic101—42no—
100C2×Dic101C2×D411yescharacteristic
101C2×Dic101C2×D411yescharacteristic
102C4×D101—22no—
103C4×D101—22no—
104C4×D101—22no—
105C4×D101—42no—
106C4×D101—42no—
107C4×D101#4114yes—
108Dic202—22no—
109Dic202—42no—
110C8×C101—22no—
111#11717579—22no—
112C22×C101—21no—
113C22×C101—21no—
114C22×C101#13712yes—
115C22×C101C22×D411yescharacteristic
116C2×D101—21no—
117C2×D101—21no—
118C2×D101—21no—
119C2×D101—21no—
120C2×D101—21no—
121C2×D101—44no—
122C2×D101#12312yes—
123C4×C101—21no—
124C4×C101—41no—
125C4×C101—41no—
126C4×C101—42no—
127C4×C101C2×#4112yes—
128Dic101—21no—
129Dic101—41no—
130Dic101—42no—
131Dic101—41no—
132Dic101#12312yes—
133C2×C101—22no—
134C2×C101C2×#12311yescharacteristic
135C2×C101C2×#13711yescharacteristic
136C2×C101#57511yescharacteristic
137D101—22no—
138D101—41no—
139#3095—1011nomaximal, Sylow
140C101#309511yescharacteristic, complemented, Sylow
141C2×C22⋊Q8—1011no—
142C2×#117—1011no—
143#485—1012no—
144#485—1012no—
145#484—1011no—
146#484D10111yescharacteristic, complemented
147#483—1011no—
148#483—1011no—
149#428—1012no—
150#419—1012no—
151#407—1011no—
152C22×Q8—1011no—
153C23×C4—1011no—
154C2×M16—1012no—
155C22⋊Q8—1011no—
156C22⋊Q8—1011no—
157C22⋊Q8—1011no—
158C22⋊Q8—2021no—
159C22⋊Q8—2021no—
160C22⋊Q8C2×D10111yescharacteristic
161#117—1012no—
162#117—1012no—
163#117—2021no—
164#117—2021no—
165C2×C4⋊C4—1011no—
166C2×C4⋊C4—1011no—
167C2×C4⋊C4—2021no—
168C2×C4⋊C4C2×D10111yescharacteristic
169C2×C22⋊C4—2021no—
170C2×C22⋊C4—2021no—
171C2×C42—2021no—
172#107—1012no—
173#107C2×D10112yescomplemented
174#106—1012no—
175#106—1012no—
176#103—1012no—
177#103—1012no—
178#103—1012no—
179#103C2×D10112yescomplemented
180C22⋊C8—1011no—
181C22⋊C8—1012no—
182C22⋊C8C2×D10111yescharacteristic
183C24—1011no—
184C2×Q8—1011no—
185C2×Q8—2021no—
186C2×Q8—2024no—
187C2×Q8C22×D10111yescharacteristic
188C22×C4—21no—
189C22×C4—1011no—
190C22×C4—1011no—
191C22×C4—1011no—
192C22×C4—1011no—
193C22×C4—1011no—
194C22×C4—2024no—
195C22×C4—2021no—
196C22×C4—2021no—
197C22×C4—2021no—
198C22×C4C22×D10111yescharacteristic
199M16—2024no—
200C2×C8—1012no—
201C2×C8C22×D10112yes—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—1012no—
205C4⋊C4—1011no—
206C4⋊C4—2021no—
207C4⋊C4—2021no—
208C4⋊C4—2021no—
209C4⋊C4—2021no—
210C4⋊C4—2021no—
211C4⋊C4C22×D10112yes—
212C4⋊C4C22×D10111yescharacteristic
213C22⋊C4—21no—
214C22⋊C4—2022no—
215C22⋊C4—2022no—
216C22⋊C4—2021no—
217C22⋊C4—2021no—
218C22⋊C4—2021no—
219C42—2022no—
220C42—2022no—
221C23—1011no—
222C23—1011no—
223C23—4041no—
224C23—2022no—
225C23—2022no—
226C23D4×D10111yescharacteristic
227Q8—22no—
228Q8—4042no—
229Q8—2022no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—42no—
235C2×C4—1011no—
236C2×C4—1014no—
237C2×C4—1011no—
238C2×C4—4042no—
239C2×C4—4042no—
240C2×C4—4042no—
241C2×C4—2022no—
242C2×C4—2022no—
243C2×C4—2022no—
244C2×C4—2022no—
245C2×C4—2021no—
246C2×C4—2021no—
247C2×C4—2022no—
248C2×C4D4×D10111yescharacteristic
249C2×C4C23×D10111yescharacteristic
250C8—22no—
251C8—2022no—
252C22—21no—
253C22—21no—
254C22—1012no—
255C22—4044no—
256C22—2021no—
257C22—2021no—
258C22—2021no—
259C22—2021no—
260C22—2021no—
261C22#1569228712yes—
262C22C2×D4×D10111yescharacteristic
263C4—21no—
264C4—41no—
265C4—42no—
266C4—41no—
267C4—1012no—
268C4—4041no—
269C4—4042no—
270C4—4041no—
271C4—2021no—
272C4#41×D10112yes—
273C2—22no—
274C2—4041no—
275C2—2022no—
276C2#123×D10111yescharacteristic
277C2C2×#1569228711yescharacteristic
278C2#2026597311yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#484×C101C2
socleC22×C101C22×D4
prime radicalC23×D101D4
hypercenter#484D101
hypocenterC101#3095
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
3. #20265520C2semidirect
4. #20265516C2semidirect
5. #20265512C2semidirect
6. #20265508C2semidirect
7. #20265505C2semidirect
8. #20265218C2semidirect
9. #20265184C2semidirect
11. #484×C101C2semidirect
30. #15692188C22semidirect
31. #15692187C22semidirect
32. #15692186C22semidirect
33. #15692180C22semidirect
34. #15692179C22semidirect
35. #15692178C22semidirect
40. #107×C101C22semidirect
41. #103×C101C22semidirect
140. C101#3095semidirect
146. #484D101semidirect
173. #107C2×D101semidirect
179. #103C2×D101semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D1012
C2×#156922132
#202655202
#202655162
#202655122
#202655082
#202655052
#202652182
#202651842
#202651392
#484×C1012
#3095101