Finite Group Databasebeta

← #27181053

Subgroups — #27181053

Order 12928.

458 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 129 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#20265684C211yesmaximal, characteristic
2C2×C4⋊C4×D101C211yesmaximal, characteristic, complemented
3#20265511C212yesmaximal, complemented
4#20265506C212yesmaximal, complemented
5#20265494C212yesmaximal, complemented
6#20265489C212yesmaximal, complemented
7#20265219C211yesmaximal, characteristic, complemented
8#20265217C211yesmaximal, characteristic, complemented
9#20265178C211yesmaximal, characteristic, complemented
10#20265140C211yesmaximal, characteristic, complemented
11#417×C101C211yesmaximal, characteristic, complemented
12C22×C4×D101—21no—
13C22×C4×D101C2211yescharacteristic
14C2×#13108058C2211yescharacteristic
15#15692236—21no—
16#15692231C2211yescharacteristic
17#15692231C2211yescharacteristic
18#15692226C2211yescharacteristic
19#15692226C2211yescharacteristic
20#15692223—21no—
21C4×C22⋊D101—21no—
22C4×C22⋊D101—21no—
23#15692214C2211yescharacteristic
24C2×C4⋊Dic101C2211yescharacteristic
25C2×#13108053—21no—
26C4⋊C4×D101—21no—
27C4⋊C4×D101—21no—
28C4⋊C4×D101C2212yes—
29C4⋊C4×D101C2212yes—
30#15692189C2212yescomplemented
31#15692188C2212yescomplemented
32#15692187C2212yescomplemented
33#15692186C2212yescomplemented
34#15692179C2212yescomplemented
35#15692176C2212yescomplemented
36#15692167C2212yes—
37#15692167C2212yes—
38C2×#41×C101C2211yescharacteristic
39C2×M16×C101C2211yescharacteristic
40C2×C4⋊C4×C101C2211yescharacteristic
41#103×C101C2212yescomplemented
42#102×C101C2212yescomplemented
43C23×D101D411yescharacteristic
44C2×C22⋊D101—21no—
45C2×C22⋊D101—21no—
46C22×Dic101—21no—
47C22×Dic101D411yescharacteristic
48C2×C4×D101—22no—
49C2×C4×D101—22no—
50C2×C4×D101—22no—
51C2×C4×D101—22no—
52C2×C4×D101—44no—
53C2×C4×D101D411yescharacteristic
54C2×C4×D101D411yescharacteristic
55C2×C4×D101C2311yescharacteristic
56C2×C4×D101C2311yescharacteristic
57#13108058—22no—
58C2×#11717579C2312yes—
59C4⋊Dic101—21no—
60C4⋊Dic101C2312yes—
61C4⋊Dic101C2311yescharacteristic
62C4⋊Dic101C2311yescharacteristic
63#13108053—22no—
64#13108053—22no—
65#13108053—21no—
66#13108053—21no—
67C22⋊Dic101—21no—
68C22⋊Dic101—21no—
69C22⋊Dic101—21no—
70#13108051—21no—
71#13108051—21no—
72C4×Dic101—21no—
73#41×C101—22no—
74#41×C101—22no—
75C2×Q8×C101C2311yescharacteristic
76C2×D4×C101—21no—
77C2×D4×C101C2311yescharacteristic
78C22×C4×C101—21no—
79C22×C4×C101—21no—
80C22×C4×C101C2311yescharacteristic
81M16×C101—22no—
82C2×C8×C101C2312yes—
83C4⋊C4×C101—21no—
84C4⋊C4×C101C2312yes—
85C23×C101—21no—
86C23×C101C2×D411yescharacteristic
87C22×D101—22no—
88C22×D101—22no—
89C22×D101—22no—
90C22×D101C2×D411yescharacteristic
91C22×D101C2×D411yescharacteristic
92Q8×C101—21no—
93Q8×C101—21no—
94D4×C101—21no—
95D4×C101—21no—
96D4×C101—42no—
97C2×C4×C101—22no—
98C2×C4×C101—21no—
99C2×C4×C101—42no—
100C2×C4×C101—42no—
101C2×C4×C101C2×D411yescharacteristic
102C2×C4×C101#4112yes—
103C2×C4×C101#4112yes—
104C2×C4×C101C2411yescharacteristic
105C22⋊D101—42no—
106C22⋊D101—42no—
107C2×Dic101—22no—
108C2×Dic101—21no—
109C2×Dic101—21no—
110C2×Dic101—22no—
111C2×Dic101—42no—
112C2×Dic101C2×D411yescharacteristic
113C2×Dic101C2×D411yescharacteristic
114C4×D101—22no—
115C4×D101—22no—
116C4×D101—21no—
117C4×D101—21no—
118C4×D101—21no—
119C4×D101—21no—
120C4×D101—44no—
121C4×D101—44no—
122C8×C101—22no—
123#11717579—22no—
124C22×C101—21no—
125C22×C101—21no—
126C22×C101—41no—
127C22×C101—41no—
128C22×C101—41no—
129C22×C101#12312yes—
130C22×C101C22×D411yescharacteristic
131C2×D101—28no—
132C2×D101—21no—
133C2×D101—21no—
134C2×D101—21no—
135C2×D101—21no—
136C2×D101AGL(1,ℤ/8)12yes—
137C2×D101#13712yes—
138C4×C101—41no—
139C4×C101—42no—
140C4×C101#12312yes—
141C4×C101C2×#4111yescharacteristic
142C4×C101C2×#4111yescharacteristic
143Dic101—21no—
144Dic101—21no—
145Dic101—42no—
146Dic101—41no—
147Dic101—41no—
148C2×C101—22no—
149C2×C101—81no—
150C2×C101C2×#12311yescharacteristic
151C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
152C2×C101C2×#13711yescharacteristic
153D101—22no—
154D101—22no—
155#3092—1011nomaximal, Sylow
156C101#309211yescharacteristic, complemented, Sylow
157#525—1011no—
158C22×C4⋊C4—1011no—
159#484—1012no—
160#483—1012no—
161#481—1012no—
162#480—1012no—
163#428—1011no—
164#428—1011no—
165#417—1011no—
166#417D10111yescharacteristic, complemented
167#407—1011no—
168C2×#41C2×D10111yescharacteristic
169C23×C4—1011no—
170C23×C4—2021no—
171C2×M16—1011no—
172C2×M16C2×D10111yescharacteristic
173#123—2021no—
174C22⋊Q8—1011no—
175C22⋊Q8—1011no—
176#121—1011no—
177#121—1011no—
178#120—2021no—
179C4×D4—2021no—
180C4×D4—2021no—
181#117—1011no—
182C2×C4⋊C4—1011no—
183C2×C4⋊C4—1012no—
184C2×C4⋊C4—1012no—
185C2×C4⋊C4—2021no—
186C2×C4⋊C4—2021no—
187C2×C4⋊C4—2021no—
188C2×C4⋊C4C2×D10111yescharacteristic
189#107—1012no—
190#107—1012no—
191#106—1012no—
192#106—1012no—
193#103—1012no—
194#103C2×D10112yescomplemented
195#102—1012no—
196#102C2×D10112yescomplemented
197C22⋊C8—1012no—
198C22⋊C8—1012no—
199C24—1011no—
200#41—22no—
201#41—22no—
202C2×Q8C22×D10111yescharacteristic
203C2×D4—21no—
204C2×D4—2021no—
205C2×D4—2021no—
206C2×D4C22×D10111yescharacteristic
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—1011no—
210C22×C4—1011no—
211C22×C4—1011no—
212C22×C4—1011no—
213C22×C4—1011no—
214C22×C4—4044no—
215C22×C4—2022no—
216C22×C4—2022no—
217C22×C4—2021no—
218C22×C4—2022no—
219C22×C4—2022no—
220C22×C4C22×D10111yescharacteristic
221M16—22no—
222M16—2022no—
223C2×C8—1012no—
224C2×C8C22×D10112yes—
225C4⋊C4—21no—
226C4⋊C4—1012no—
227C4⋊C4—1011no—
228C4⋊C4—1011no—
229C4⋊C4—2022no—
230C4⋊C4—2022no—
231C4⋊C4—2021no—
232C4⋊C4—2021no—
233C4⋊C4—2021no—
234C4⋊C4C22×D10112yes—
235C22⋊C4—2021no—
236C22⋊C4—2021no—
237C22⋊C4—2021no—
238C22⋊C4—2021no—
239C22⋊C4—2021no—
240C42—2021no—
241C23—21no—
242C23—1011no—
243C23—1011no—
244C23—2022no—
245C23—2022no—
246C23—2022no—
247C23D4×D10111yescharacteristic
248Q8—21no—
249Q8—21no—
250D4—21no—
251D4—21no—
252D4—42no—
253D4—4042no—
254D4—4042no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—1011no—
260C2×C4—1011no—
261C2×C4—4044no—
262C2×C4—4044no—
263C2×C4—4042no—
264C2×C4—2021no—
265C2×C4—2021no—
266C2×C4—2022no—
267C2×C4—2022no—
268C2×C4—2021no—
269C2×C4—2021no—
270C2×C4—2021no—
271C2×C4—2021no—
272C2×C4—2022no—
273C2×C4—2022no—
274C2×C4D4×D10111yescharacteristic
275C2×C4#1310807312yes—
276C2×C4#1310807412yes—
277C2×C4C23×D10111yescharacteristic
278C8—22no—
279C8—2022no—
280C22—21no—
281C22—21no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—1012no—
286C22—1012no—
287C22—2028no—
288C22—2021no—
289C22—2021no—
290C22—2021no—
291C22—2021no—
292C22#1569223712yes—
293C22C2×D4×D10111yescharacteristic
294C4—41no—
295C4—42no—
296C4—4042no—
297C4—4041no—
298C4—4041no—
299C4—2021no—
300C4—2021no—
301C4#1569223712yes—
302C4C2×#1310807311yescharacteristic
303C4C2×#1310807411yescharacteristic
304C2—22no—
305C2—81no—
306C2—2022no—
307C2—2022no—
308C2C2×#1569223711yescharacteristic
309C2AGL(1,ℤ/8)×D10111yescharacteristic
310C2#137×D10111yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#417×C101C2
socleC22×C101C22×D4
prime radical#20265684C2
hypercenter#417D101
hypocenterC101#3092
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D101C2semidirect
3. #20265511C2semidirect
4. #20265506C2semidirect
5. #20265494C2semidirect
6. #20265489C2semidirect
7. #20265219C2semidirect
8. #20265217C2semidirect
9. #20265178C2semidirect
10. #20265140C2semidirect
11. #417×C101C2semidirect
30. #15692189C22semidirect
31. #15692188C22semidirect
32. #15692187C22semidirect
33. #15692186C22semidirect
34. #15692179C22semidirect
35. #15692176C22semidirect
41. #103×C101C22semidirect
42. #102×C101C22semidirect
156. C101#3092semidirect
166. #417D101semidirect
194. #103C2×D101semidirect
196. #102C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202656842
C2×C4⋊C4×D1012
#202655112
#202655062
#202654942
#202654892
#202652192
#202652172
#202651782
#202651402
#417×C1012
#3092101