Finite Group Databasebeta

← #27181058

Subgroups — #27181058

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#20265689C211yesmaximal, characteristic
2C2×C4⋊C4×D101C211yesmaximal, characteristic, complemented
3#20265514C212yesmaximal, complemented
4#20265509C212yesmaximal, complemented
5#20265497C212yesmaximal, complemented
6#20265492C212yesmaximal, complemented
7#20265220C211yesmaximal, characteristic, complemented
8#20265179C211yesmaximal, characteristic, complemented
9#20265176C211yesmaximal, characteristic, complemented
10#20265140C211yesmaximal, characteristic, complemented
11#428×C101C211yesmaximal, characteristic, complemented
12C2×#13108074C2211yescharacteristic
13C22×C4×D101—21no—
14C22×C4×D101C2211yescharacteristic
15C2×#13108058C2211yescharacteristic
16#15692239—21no—
17#15692235C2211yescharacteristic
18#15692235C2211yescharacteristic
19#15692230C2211yescharacteristic
20#15692230C2211yescharacteristic
21#15692225—21no—
22C4×D404—21no—
23C4×D404—21no—
24C2×C4⋊Dic101C2211yescharacteristic
25C2×#13108053—21no—
26C4⋊C4×D101—21no—
27C4⋊C4×D101—21no—
28C4⋊C4×D101C2212yes—
29C4⋊C4×D101C2212yes—
30#15692188C2212yescomplemented
31#15692186C2212yescomplemented
32#15692180C2212yescomplemented
33#15692178C2212yescomplemented
34#15692177C2212yescomplemented
35#15692175C2212yescomplemented
36#15692167C2212yes—
37#15692167C2212yes—
38C2×M16×C101C2211yescharacteristic
39#117×C101C2211yescharacteristic
40C2×C4⋊C4×C101C2211yescharacteristic
41#107×C101C2212yescomplemented
42#106×C101C2212yescomplemented
43C23×D101D411yescharacteristic
44#13108074—22no—
45#13108074—22no—
46C2×Dic202C2311yescharacteristic
47C2×D404—21no—
48C2×D404—21no—
49C2×D404C2311yescharacteristic
50C2×C22⋊D101—21no—
51C22×Dic101—21no—
52C22×Dic101D411yescharacteristic
53C2×C4×D101—22no—
54C2×C4×D101—22no—
55C2×C4×D101—22no—
56C2×C4×D101—22no—
57C2×C4×D101—21no—
58C2×C4×D101—44no—
59C2×C4×D101D411yescharacteristic
60C2×C4×D101D411yescharacteristic
61C2×C4×D101C2311yescharacteristic
62C2×C4×D101C2311yescharacteristic
63#13108058—22no—
64C2×#11717579C2312yes—
65C4⋊Dic101—21no—
66C4⋊Dic101—21no—
67C4⋊Dic101—21no—
68C4⋊Dic101C2312yes—
69#13108053—22no—
70#13108053—22no—
71#13108051—21no—
72#13108051—21no—
73#13108051—21no—
74#13108051—21no—
75C22×C4×C101—21no—
76C22×C4×C101C2311yescharacteristic
77M16×C101—22no—
78C2×C8×C101C2312yes—
79C4⋊C4×C101—21no—
80C4⋊C4×C101C2311yescharacteristic
81C4⋊C4×C101C2312yes—
82C4⋊C4×C101C2311yescharacteristic
83C22⋊C4×C101—21no—
84C42×C101—21no—
85C23×C101C2×D411yescharacteristic
86C22×D101—22no—
87C22×D101—22no—
88C22×D101—22no—
89C22×D101—21no—
90C22×D101C2×D411yescharacteristic
91C22×D101C2×D411yescharacteristic
92C2×C4×C101—21no—
93C2×C4×C101—21no—
94C2×C4×C101—22no—
95C2×C4×C101—42no—
96C2×C4×C101C2×D411yescharacteristic
97C2×C4×C101#4112yes—
98C2×C4×C101#4112yes—
99C2×C4×C101C2411yescharacteristic
100C22⋊D101—42no—
101C2×Dic101—22no—
102C2×Dic101—21no—
103C2×Dic101—22no—
104C2×Dic101—42no—
105C2×Dic101C2×D411yescharacteristic
106C2×Dic101C2×D411yescharacteristic
107D404—21no—
108D404—21no—
109D404—42no—
110D404—42no—
111C4×D101—22no—
112C4×D101—22no—
113C4×D101—21no—
114C4×D101—21no—
115C4×D101—21no—
116C4×D101—21no—
117C4×D101—44no—
118C4×D101—44no—
119C4×D101—42no—
120Dic202—21no—
121Dic202—21no—
122C8×C101—22no—
123#11717579—22no—
124C22×C101—21no—
125C22×C101—21no—
126C22×C101#12312yes—
127C22×C101C22×D411yescharacteristic
128C2×D101—28no—
129C2×D101—21no—
130C2×D101—21no—
131C2×D101—21no—
132C2×D101—21no—
133C2×D101—41no—
134C2×D101—41no—
135C2×D101—41no—
136C2×D101AGL(1,ℤ/8)12yes—
137C2×D101#13712yes—
138C4×C101—42no—
139C4×C101—41no—
140C4×C101—41no—
141C4×C101#12312yes—
142C4×C101C2×#4111yescharacteristic
143C4×C101C2×#4111yescharacteristic
144Dic101—21no—
145Dic101—21no—
146Dic101—41no—
147Dic101—42no—
148C2×C101—22no—
149C2×C101C2×#12311yescharacteristic
150C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
151C2×C101C2×#13711yescharacteristic
152D101—22no—
153D101—22no—
154D101—81no—
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#428D10111yescharacteristic, complemented
165#417—1011no—
166#417—1011no—
167#407—1011no—
168C2×#41—1011no—
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#117C2×D10111yescharacteristic
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#107C2×D10112yescomplemented
191#106—1012no—
192#106C2×D10112yescomplemented
193#103—1012no—
194#103—1012no—
195#102—1012no—
196#102—1012no—
197C22⋊C8—1012no—
198C22⋊C8—1012no—
199C24—1011no—
200#41—2022no—
201#41—2022no—
202C2×Q8—1011no—
203C2×D4—1011no—
204C2×D4—2021no—
205C2×D4—2021no—
206C2×D4—2021no—
207C22×C4—21no—
208C22×C4—1011no—
209C22×C4—1011no—
210C22×C4—1011no—
211C22×C4—1011no—
212C22×C4—1011no—
213C22×C4—4044no—
214C22×C4—2022no—
215C22×C4—2022no—
216C22×C4—2021no—
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—2022no—
228C4⋊C4—2022no—
229C4⋊C4—2021no—
230C4⋊C4—2021no—
231C4⋊C4—2021no—
232C4⋊C4C22×D10112yes—
233C4⋊C4C22×D10111yescharacteristic
234C4⋊C4C22×D10111yescharacteristic
235C22⋊C4—21no—
236C22⋊C4—2021no—
237C22⋊C4—2021no—
238C22⋊C4—2021no—
239C22⋊C4—2021no—
240C42—21no—
241C23—1011no—
242C23—1011no—
243C23—2022no—
244C23—2022no—
245C23—2021no—
246C23—2022no—
247C23D4×D10111yescharacteristic
248Q8—2021no—
249Q8—2021no—
250D4—4042no—
251D4—4042no—
252D4—4042no—
253D4—2021no—
254D4—2021no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—42no—
259C2×C4—1011no—
260C2×C4—1011no—
261C2×C4—4044no—
262C2×C4—4044no—
263C2×C4—4042no—
264C2×C4—4042no—
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#1310807412yes—
276C2×C4#1310807512yes—
277C2×C4C23×D10111yescharacteristic
278C8—22no—
279C8—2022no—
280C22—21no—
281C22—21no—
282C22—1012no—
283C22—1012no—
284C22—4041no—
285C22—4041no—
286C22—4041no—
287C22—2028no—
288C22—2021no—
289C22—2021no—
290C22—2021no—
291C22—2021no—
292C22#1569224012yes—
293C22C2×D4×D10111yescharacteristic
294C4—42no—
295C4—41no—
296C4—41no—
297C4—4041no—
298C4—4042no—
299C4—2021no—
300C4—2021no—
301C4#1569224012yes—
302C4C2×#1310807411yescharacteristic
303C4C2×#1310807511yescharacteristic
304C2—22no—
305C2—8081no—
306C2—2022no—
307C2—2022no—
308C2C2×#1569224011yescharacteristic
309C2AGL(1,ℤ/8)×D10111yescharacteristic
310C2#137×D10111yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#428×C101C2
socleC22×C101C22×D4
prime radical#20265689C2
hypercenter#428D101
hypocenterC101#3092
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
2. C2×C4⋊C4×D101C2semidirect
3. #20265514C2semidirect
4. #20265509C2semidirect
5. #20265497C2semidirect
6. #20265492C2semidirect
7. #20265220C2semidirect
8. #20265179C2semidirect
9. #20265176C2semidirect
10. #20265140C2semidirect
11. #428×C101C2semidirect
30. #15692188C22semidirect
31. #15692186C22semidirect
32. #15692180C22semidirect
33. #15692178C22semidirect
34. #15692177C22semidirect
35. #15692175C22semidirect
41. #107×C101C22semidirect
42. #106×C101C22semidirect
156. C101#3092semidirect
164. #428D101semidirect
190. #107C2×D101semidirect
192. #106C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202656892
C2×C4⋊C4×D1012
#202655142
#202655092
#202654972
#202654922
#202652202
#202651792
#202651762
#202651402
#428×C1012
#3092101