Finite Group Databasebeta

← #27181051

Subgroups — #27181051

Order 12928.

458 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 123 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#20265687C211yesmaximal, characteristic
2C2×C4⋊C4×D101C211yesmaximal, characteristic, complemented
3#20265514C212yesmaximal, complemented
4#20265508C211yesmaximal, characteristic, complemented
5#20265507C211yesmaximal, characteristic, complemented
6#20265494C212yesmaximal, complemented
7#20265488C211yesmaximal, characteristic, complemented
8#20265218C212yesmaximal, complemented
9#20265175C212yesmaximal, complemented
10#20265139C211yesmaximal, characteristic, complemented
11#480×C101C211yesmaximal, characteristic, complemented
12C2×#13108073C2211yescharacteristic
13C22×C4×D101—21no—
14C22×C4×D101C2211yescharacteristic
15C2×#13108057C2212yes—
16#15692237—21no—
17#15692235C2211yescharacteristic
18#15692234C2211yescharacteristic
19#15692226C2211yescharacteristic
20#15692223—21no—
21D4×Dic101—21no—
22#15692215—21no—
23#15692213C2211yescharacteristic
24C2×C4⋊Dic101C2211yescharacteristic
25C2×#13108053—21no—
26C4⋊C4×D101—21no—
27C4⋊C4×D101—21no—
28C4⋊C4×D101C2212yes—
29C4⋊C4×D101C2212yes—
30#15692188C2212yescomplemented
31#15692187C2212yescomplemented
32#15692186C2212yescomplemented
33#15692180C2212yescomplemented
34#15692178C2212yescomplemented
35#15692176C2212yescomplemented
36C22⋊#11717579C2211yescharacteristic
37#15692167C2212yes—
38#121×C101C2211yescharacteristic
39C2×C4⋊C4×C101C2211yescharacteristic
40#107×C101C2212yescomplemented
41#102×C101C2212yescomplemented
42C22⋊C8×C101C2211yescharacteristic
43C23×D101D411yescharacteristic
44#13108073—24no—
45C2×Dic202C2311yescharacteristic
46C2×C22⋊D101—21no—
47C2×C22⋊D101—21no—
48C22×Dic101—21no—
49C22×Dic101—21no—
50C22×Dic101D411yescharacteristic
51C2×C4×D101—24no—
52C2×C4×D101—22no—
53C2×C4×D101—22no—
54C2×C4×D101—44no—
55C2×C4×D101D411yescharacteristic
56C2×C4×D101D411yescharacteristic
57C2×C4×D101C2311yescharacteristic
58C2×C4×D101C2311yescharacteristic
59#13108057—24no—
60C2×#11717579C2312yes—
61C4⋊Dic101—21no—
62C4⋊Dic101—21no—
63C4⋊Dic101C2312yes—
64C4⋊Dic101C2311yescharacteristic
65#13108053—21no—
66#13108053—22no—
67#13108053—22no—
68C22⋊Dic101—21no—
69C22⋊Dic101—21no—
70#13108051—21no—
71#13108051—21no—
72C4×Dic101—21no—
73C2×D4×C101—21no—
74C2×D4×C101C2311yescharacteristic
75C22×C4×C101—21no—
76C22×C4×C101C2311yescharacteristic
77C2×C8×C101C2312yes—
78C4⋊C4×C101—21no—
79C4⋊C4×C101C2312yes—
80C4⋊C4×C101C2311yescharacteristic
81C22⋊C4×C101—21no—
82C23×C101—21no—
83C23×C101C2×D411yescharacteristic
84C22×D101—22no—
85C22×D101—22no—
86C22×D101—22no—
87C22×D101C2×D411yescharacteristic
88C22×D101C2×D411yescharacteristic
89D4×C101—22no—
90D4×C101—42no—
91C2×C4×C101—22no—
92C2×C4×C101—21no—
93C2×C4×C101—22no—
94C2×C4×C101—42no—
95C2×C4×C101C2×D411yescharacteristic
96C2×C4×C101C2411yescharacteristic
97C22⋊D101—42no—
98C22⋊D101—42no—
99C2×Dic101—22no—
100C2×Dic101—21no—
101C2×Dic101—21no—
102C2×Dic101—22no—
103C2×Dic101—42no—
104C2×Dic101—42no—
105C2×Dic101C2×D411yescharacteristic
106C2×Dic101C2×D411yescharacteristic
107C4×D101—22no—
108C4×D101—22no—
109C4×D101—22no—
110C4×D101—44no—
111C4×D101—44no—
112C4×D101#4114yes—
113Dic202—22no—
114C8×C101—22no—
115#11717579—22no—
116C22×C101—21no—
117C22×C101—21no—
118C22×C101—41no—
119C22×C101—41no—
120C22×C101—41no—
121C22×C101AGL(1,ℤ/8)12yes—
122C22×C101C22×D411yescharacteristic
123C2×D101—28no—
124C2×D101—21no—
125C2×D101—21no—
126C2×D101—21no—
127C2×D101—21no—
128C2×D101#12312yes—
129C2×D101#13712yes—
130C4×C101—21no—
131C4×C101—42no—
132C4×C101—41no—
133C4×C101C2×#4112yes—
134Dic101—21no—
135Dic101—41no—
136Dic101—42no—
137Dic101—41no—
138Dic101#12312yes—
139C2×C101—22no—
140C2×C101—81no—
141C2×C101C2×#12311yescharacteristic
142C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
143C2×C101C2×#13711yescharacteristic
144D101—22no—
145D101—22no—
146#3092—1011nomaximal, Sylow
147C101#309211yescharacteristic, complemented, Sylow
148#525—1011no—
149C22×C4⋊C4—1011no—
150#484—1012no—
151#483—1011no—
152#483—1011no—
153#481—1012no—
154#480—1011no—
155#480D10111yescharacteristic, complemented
156#428—1012no—
157#417—1012no—
158#407—1011no—
159C2×#41—1011no—
160C23×C4—1011no—
161C23×C4—2021no—
162C2×M16—1012no—
163#123—2021no—
164C22⋊Q8—1011no—
165C22⋊Q8—1011no—
166#121—1011no—
167#121C2×D10111yescharacteristic
168#120—2021no—
169C4×D4—2021no—
170C4×D4—2021no—
171#117—1011no—
172C2×C4⋊C4—1011no—
173C2×C4⋊C4—1012no—
174C2×C4⋊C4—1012no—
175C2×C4⋊C4—2021no—
176C2×C4⋊C4—2021no—
177C2×C4⋊C4—2021no—
178C2×C4⋊C4C2×D10111yescharacteristic
179#107—1012no—
180#107C2×D10112yescomplemented
181#106—1012no—
182#106—1012no—
183#103—1012no—
184#103—1012no—
185#102—1012no—
186#102C2×D10112yescomplemented
187C22⋊C8—1011no—
188C22⋊C8—1012no—
189C22⋊C8C2×D10111yescharacteristic
190C24—1011no—
191#41—2024no—
192C2×Q8—1011no—
193C2×D4—21no—
194C2×D4—2021no—
195C2×D4—2021no—
196C2×D4C22×D10111yescharacteristic
197C22×C4—21no—
198C22×C4—1011no—
199C22×C4—1011no—
200C22×C4—1011no—
201C22×C4—1011no—
202C22×C4—1011no—
203C22×C4—4044no—
204C22×C4—2024no—
205C22×C4—2022no—
206C22×C4—2022no—
207C22×C4—2021no—
208C22×C4—2021no—
209C22×C4C22×D10111yescharacteristic
210M16—2024no—
211C2×C8—1012no—
212C2×C8C22×D10112yes—
213C4⋊C4—21no—
214C4⋊C4—1012no—
215C4⋊C4—1011no—
216C4⋊C4—2022no—
217C4⋊C4—2022no—
218C4⋊C4—2021no—
219C4⋊C4—2021no—
220C4⋊C4—2021no—
221C4⋊C4C22×D10112yes—
222C4⋊C4C22×D10111yescharacteristic
223C22⋊C4—21no—
224C22⋊C4—2021no—
225C22⋊C4—2021no—
226C22⋊C4—2021no—
227C22⋊C4—2021no—
228C42—2021no—
229C23—21no—
230C23—1011no—
231C23—1011no—
232C23—2022no—
233C23—2022no—
234C23—2022no—
235C23D4×D10111yescharacteristic
236Q8—2022no—
237D4—22no—
238D4—42no—
239D4—4042no—
240D4—4042no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—42no—
245C2×C4—1011no—
246C2×C4—1014no—
247C2×C4—1011no—
248C2×C4—4044no—
249C2×C4—4044no—
250C2×C4—4042no—
251C2×C4—4042no—
252C2×C4—2022no—
253C2×C4—2022no—
254C2×C4—2022no—
255C2×C4—2021no—
256C2×C4—2021no—
257C2×C4—2022no—
258C2×C4—2022no—
259C2×C4D4×D10111yescharacteristic
260C2×C4C23×D10111yescharacteristic
261C8—22no—
262C8—2022no—
263C22—21no—
264C22—21no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—1012no—
269C22—1012no—
270C22—2028no—
271C22—2021no—
272C22—2021no—
273C22—2021no—
274C22—2021no—
275C22#1569227912yes—
276C22C2×D4×D10111yescharacteristic
277C4—21no—
278C4—42no—
279C4—41no—
280C4—1012no—
281C4—4041no—
282C4—4042no—
283C4—4041no—
284C4—2021no—
285C4#41×D10112yes—
286C2—22no—
287C2—81no—
288C2—2022no—
289C2—2022no—
290C2#123×D10111yescharacteristic
291C2C2×#1569227911yescharacteristic
292C2#137×D10111yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#480×C101C2
socleC22×C101C22×D4
prime radical#20265687C2
hypercenter#480D101
hypocenterC101#3092
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D101C2semidirect
3. #20265514C2semidirect
4. #20265508C2semidirect
5. #20265507C2semidirect
6. #20265494C2semidirect
7. #20265488C2semidirect
8. #20265218C2semidirect
9. #20265175C2semidirect
10. #20265139C2semidirect
11. #480×C101C2semidirect
30. #15692188C22semidirect
31. #15692187C22semidirect
32. #15692186C22semidirect
33. #15692180C22semidirect
34. #15692178C22semidirect
35. #15692176C22semidirect
40. #107×C101C22semidirect
41. #102×C101C22semidirect
147. C101#3092semidirect
155. #480D101semidirect
180. #107C2×D101semidirect
186. #102C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202656872
C2×C4⋊C4×D1012
#202655142
#202655082
#202655072
#202654942
#202654882
#202652182
#202651752
#202651392
#480×C1012
#3092101