Finite Group Databasebeta

← #27181082

Subgroups — #27181082

Order 12928.

410 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 127 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
1C2×#15692235C211yesmaximal, characteristic
2#117×D101C211yesmaximal, characteristic
3#20265521C212yesmaximal, complemented
4#20265520C212yesmaximal, complemented
5#20265514C212yesmaximal, complemented
6#20265509C212yesmaximal, complemented
7#20265217C211yesmaximal, characteristic, complemented
8#20265187C211yesmaximal, characteristic, complemented
9#20265185C211yesmaximal, characteristic, complemented
10#20265140C211yesmaximal, characteristic
11#428×C101C211yesmaximal, characteristic, complemented
12C22×Dic202C2211yescharacteristic
13C22×C4×D101C2211yescharacteristic
14C2×#13108058C2211yescharacteristic
15#15692235—21no—
16#15692235—21no—
17#15692235C2212yes—
18#15692235C2212yes—
19#15692214C2211yescharacteristic
20#15692213C2211yescharacteristic
21#15692213C2211yescharacteristic
22#15692212—21no—
23#15692211—21no—
24C2×C4⋊Dic101—21no—
25C4⋊C4×D101C2211yescharacteristic
26C4⋊C4×D101C2211yescharacteristic
27C2×#13108051—21no—
28C22⋊C4×D101—21no—
29C42×D101—21no—
30#15692188C2212yescomplemented
31#15692186C2212yescomplemented
32#15692180C2212yescomplemented
33#15692180C2212yescomplemented
34#15692178C2212yescomplemented
35#15692178C2212yescomplemented
36#15692167C2212yes—
37#15692167C2212yes—
38C2×M16×C101C2211yescharacteristic
39#117×C101C2211yescharacteristic
40C2×C4⋊C4×C101C2211yescharacteristic
41#107×C101C2212yescomplemented
42#106×C101C2212yescomplemented
43C23×D101D411yescharacteristic
44C2×Dic202—22no—
45C2×Dic202—22no—
46C2×Dic202—21no—
47C2×Dic202C2312yes—
48C22×Dic101—21no—
49C22×Dic101D411yescharacteristic
50C2×C4×D101—22no—
51C2×C4×D101—22no—
52C2×C4×D101—21no—
53C2×C4×D101—21no—
54C2×C4×D101D411yescharacteristic
55C2×C4×D101D411yescharacteristic
56C2×C4×D101C2311yescharacteristic
57C2×C4×D101C2311yescharacteristic
58#13108058—22no—
59C2×#11717579C2312yes—
60C4⋊Dic101—22no—
61C4⋊Dic101—22no—
62C4⋊Dic101C2311yescharacteristic
63C4⋊Dic101C2311yescharacteristic
64#13108053—21no—
65#13108053—21no—
66C22⋊Dic101—21no—
67#13108051—21no—
68#13108051—21no—
69#13108051—22no—
70#13108051—22no—
71C4×Dic101—21no—
72C4×Dic101—21no—
73C4×Dic101—21no—
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—22no—
87C22×D101—41no—
88C22×D101C2×D411yescharacteristic
89C22×D101C2×D411yescharacteristic
90C2×C4×C101—21no—
91C2×C4×C101—21no—
92C2×C4×C101—22no—
93C2×C4×C101—42no—
94C2×C4×C101C2×D411yescharacteristic
95C2×C4×C101#4112yes—
96C2×C4×C101#4112yes—
97C2×C4×C101C2411yescharacteristic
98C2×Dic101—22no—
99C2×Dic101—21no—
100C2×Dic101—21no—
101C2×Dic101—22no—
102C2×Dic101—42no—
103C2×Dic101C2×D411yescharacteristic
104C2×Dic101C2×D411yescharacteristic
105C4×D101—22no—
106C4×D101—22no—
107C4×D101—21no—
108C4×D101—21no—
109C4×D101—21no—
110C4×D101—21no—
111C4×D101—42no—
112C4×D101—42no—
113Dic202—22no—
114Dic202—22no—
115Dic202—42no—
116C8×C101—22no—
117#11717579—22no—
118C22×C101—21no—
119C22×C101—21no—
120C22×C101#12312yes—
121C22×C101C22×D411yescharacteristic
122C2×D101—21no—
123C2×D101—21no—
124C2×D101—21no—
125C2×D101—21no—
126C2×D101—21no—
127C2×D101—44no—
128C2×D101#13712yes—
129C4×C101—41no—
130C4×C101—41no—
131C4×C101—42no—
132C4×C101#12312yes—
133C4×C101C2×#4111yescharacteristic
134C4×C101C2×#4111yescharacteristic
135Dic101—21no—
136Dic101—21no—
137Dic101—42no—
138Dic101—41no—
139Dic101—41no—
140C2×C101—22no—
141C2×C101C2×#12311yescharacteristic
142C2×C101C2×#13711yescharacteristic
143C2×C101#57511yescharacteristic
144D101—22no—
145D101—41no—
146#3095—1011nomaximal, Sylow
147C101#309511yescharacteristic, complemented, Sylow
148C2×C22⋊Q8—1011no—
149C2×#117—1011no—
150#485—1012no—
151#485—1012no—
152#484—1012no—
153#483—1012no—
154#428—1011no—
155#428D10111yescharacteristic, complemented
156#419—1011no—
157#419—1011no—
158#407—1011no—
159C22×Q8—1011no—
160C23×C4—1011no—
161C2×M16—1011no—
162C2×M16C2×D10111yescharacteristic
163C22⋊Q8—1012no—
164C22⋊Q8—1012no—
165C22⋊Q8—2021no—
166C22⋊Q8—2021no—
167#117—1011no—
168#117—1011no—
169#117—1011no—
170#117—2021no—
171#117—2021no—
172#117C2×D10111yescharacteristic
173C2×C4⋊C4—1011no—
174C2×C4⋊C4—1011no—
175C2×C4⋊C4—2021no—
176C2×C4⋊C4C2×D10111yescharacteristic
177C2×C22⋊C4—2021no—
178C2×C22⋊C4—2021no—
179C2×C42—2021no—
180#107—1012no—
181#107C2×D10112yescomplemented
182#106—1012no—
183#106C2×D10112yescomplemented
184#103—1012no—
185#103—1012no—
186#103—1012no—
187#103—1012no—
188C22⋊C8—1012no—
189C22⋊C8—1012no—
190C24—1011no—
191C2×Q8—1012no—
192C2×Q8—2022no—
193C2×Q8—2022no—
194C2×Q8—2021no—
195C22×C4—21no—
196C22×C4—1011no—
197C22×C4—1011no—
198C22×C4—1011no—
199C22×C4—1011no—
200C22×C4—1011no—
201C22×C4—2021no—
202C22×C4—2021no—
203C22×C4—2021no—
204C22×C4—2022no—
205C22×C4—2022no—
206C22×C4C22×D10111yescharacteristic
207M16—22no—
208M16—2022no—
209C2×C8—1012no—
210C2×C8C22×D10112yes—
211C4⋊C4—21no—
212C4⋊C4—1011no—
213C4⋊C4—1011no—
214C4⋊C4—2021no—
215C4⋊C4—2021no—
216C4⋊C4—2022no—
217C4⋊C4—2022no—
218C4⋊C4C22×D10111yescharacteristic
219C4⋊C4C22×D10111yescharacteristic
220C4⋊C4C22×D10112yes—
221C22⋊C4—21no—
222C22⋊C4—2021no—
223C22⋊C4—2021no—
224C22⋊C4—2022no—
225C22⋊C4—2022no—
226C22⋊C4—2021no—
227C42—21no—
228C42—2021no—
229C42—2021no—
230C42—2021no—
231C23—1011no—
232C23—1011no—
233C23—4041no—
234C23—2022no—
235C23—2022no—
236C23D4×D10111yescharacteristic
237Q8—4042no—
238Q8—2022no—
239Q8—2022no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—1011no—
245C2×C4—1011no—
246C2×C4—4042no—
247C2×C4—4042no—
248C2×C4—4042no—
249C2×C4—2021no—
250C2×C4—2021no—
251C2×C4—2022no—
252C2×C4—2022no—
253C2×C4—2021no—
254C2×C4—2021no—
255C2×C4—2021no—
256C2×C4—2021no—
257C2×C4—2022no—
258C2×C4—2022no—
259C2×C4D4×D10111yescharacteristic
260C2×C4#1310807412yes—
261C2×C4#1310807512yes—
262C2×C4C23×D10111yescharacteristic
263C8—22no—
264C8—2022no—
265C22—21no—
266C22—21no—
267C22—1012no—
268C22—4044no—
269C22—2021no—
270C22—2021no—
271C22—2021no—
272C22—2021no—
273C22—2021no—
274C22#1569224012yes—
275C22C2×D4×D10111yescharacteristic
276C4—41no—
277C4—42no—
278C4—41no—
279C4—4042no—
280C4—4041no—
281C4—4041no—
282C4—2021no—
283C4—2021no—
284C4#1569224012yes—
285C4C2×#1310807411yescharacteristic
286C4C2×#1310807511yescharacteristic
287C2—22no—
288C2—4041no—
289C2—2022no—
290C2C2×#1569224011yescharacteristic
291C2#137×D10111yescharacteristic
292C2#2026597311yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#428×C101C2
socleC22×C101C22×D4
prime radicalC23×D101D4
hypercenter#428D101
hypocenterC101#3095
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
3. #20265521C2semidirect
4. #20265520C2semidirect
5. #20265514C2semidirect
6. #20265509C2semidirect
7. #20265217C2semidirect
8. #20265187C2semidirect
9. #20265185C2semidirect
11. #428×C101C2semidirect
30. #15692188C22semidirect
31. #15692186C22semidirect
32. #15692180C22semidirect
33. #15692180C22semidirect
34. #15692178C22semidirect
35. #15692178C22semidirect
41. #107×C101C22semidirect
42. #106×C101C22semidirect
147. C101#3095semidirect
155. #428D101semidirect
181. #107C2×D101semidirect
183. #106C2×D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922352
#117×D1012
#202655212
#202655202
#202655142
#202655092
#202652172
#202651872
#202651852
#202651402
#428×C1012
#3095101