Finite Group Databasebeta

← #27180763

Subgroups — #27180763

Order 12928.

496 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 143 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×#15692288C211yesmaximal, characteristic, complemented
2C2×M16×D101C211yesmaximal, characteristic, complemented
3#20265689C211yesmaximal, characteristic
4C2×#15692235C211yesmaximal, characteristic, complemented
5#20265436C212yesmaximal, complemented
6#20265424C212yesmaximal, complemented
7#20265419C212yesmaximal, complemented
8#20265407C212yesmaximal, complemented
9#20265185C211yesmaximal, characteristic, complemented
10#20265176C211yesmaximal, characteristic, complemented
11#428×C101C211yesmaximal, characteristic, complemented
12C2×#13108074C2211yescharacteristic
13C22×Dic202C2211yescharacteristic
14C22×C4×D101C2211yescharacteristic
15#15692288—24no—
16C2×Dic404C2212yescomplemented
17C2×#13108063C2212yescomplemented
18C2×#13108058C2211yescharacteristic
19C2×#13108057C2212yes—
20M16×D101—24no—
21C2×C8×D101C2212yes—
22#15692239—21no—
23#15692235—21no—
24#15692235—21no—
25#15692235C2212yes—
26#15692235C2211yescharacteristic
27#15692235C2212yes—
28#15692235C2211yescharacteristic
29#15692230C2211yescharacteristic
30#15692230C2211yescharacteristic
31#15692225—21no—
32C4×D404—21no—
33C4×D404—21no—
34C2×C4⋊Dic101—21no—
35C2×#13108051—21no—
36#15692178C2212yescomplemented
37#15692178C2212yescomplemented
38#15692178C2212yescomplemented
39#15692175C2212yescomplemented
40C2×M16×C101C2211yescharacteristic
41#117×C101C2211yescharacteristic
42C2×C4⋊C4×C101C2211yescharacteristic
43#107×C101C2212yescomplemented
44#106×C101C2212yescomplemented
45C23×D101D411yescharacteristic
46#13108074—22no—
47#13108074—22no—
48C2×Dic202—22no—
49C2×Dic202—22no—
50C2×Dic202—21no—
51C2×Dic202C2312yes—
52C2×Dic202C2311yescharacteristic
53C2×D404—21no—
54C2×D404—21no—
55C2×D404C2311yescharacteristic
56C2×C22⋊D101—21no—
57C22×Dic101—21no—
58C22×Dic101D411yescharacteristic
59C2×C4×D101—22no—
60C2×C4×D101—22no—
61C2×C4×D101—21no—
62C2×C4×D101D411yescharacteristic
63C2×C4×D101D411yescharacteristic
64C2×C4×D101C2311yescharacteristic
65C2×C4×D101C2311yescharacteristic
66Dic404—24no—
67#13108063—24no—
68#13108058—22no—
69#13108057—24no—
70C2×#11717579C2312yes—
71C8×D101—24no—
72C4⋊Dic101—22no—
73C4⋊Dic101—22no—
74C4⋊Dic101—21no—
75C4⋊Dic101—21no—
76#13108051—22no—
77#13108051—22no—
78#13108051—21no—
79#13108051—21no—
80#13108051—21no—
81#13108051—21no—
82C22×C4×C101—21no—
83C22×C4×C101C2311yescharacteristic
84M16×C101D414yescomplemented
85C2×C8×C101C2312yes—
86C4⋊C4×C101—21no—
87C4⋊C4×C101C2311yescharacteristic
88C4⋊C4×C101C2312yes—
89C4⋊C4×C101C2311yescharacteristic
90C22⋊C4×C101—21no—
91C42×C101—21no—
92C23×C101C2×D411yescharacteristic
93C22×D101—22no—
94C22×D101—22no—
95C22×D101—21no—
96C22×D101—41no—
97C22×D101C2×D411yescharacteristic
98C22×D101C2×D411yescharacteristic
99C2×C4×C101—21no—
100C2×C4×C101—21no—
101C2×C4×C101—22no—
102C2×C4×C101—42no—
103C2×C4×C101C2×D411yescharacteristic
104C2×C4×C101C2×D412yes—
105C2×C4×C101#4112yes—
106C2×C4×C101C2411yescharacteristic
107C22⋊D101—42no—
108C2×Dic101—22no—
109C2×Dic101—21no—
110C2×Dic101—22no—
111C2×Dic101—42no—
112C2×Dic101C2×D411yescharacteristic
113C2×Dic101C2×D411yescharacteristic
114D404—21no—
115D404—21no—
116D404—42no—
117D404—42no—
118C4×D101—22no—
119C4×D101—22no—
120C4×D101—21no—
121C4×D101—21no—
122C4×D101—21no—
123C4×D101—21no—
124C4×D101—42no—
125Dic202—22no—
126Dic202—21no—
127Dic202—22no—
128Dic202—21no—
129Dic202—42no—
130C8×C101C2×D414yes—
131#11717579—22no—
132C22×C101—21no—
133C22×C101—21no—
134C22×C101#12112yes—
135C22×C101C22×D411yescharacteristic
136C2×D101—21no—
137C2×D101—21no—
138C2×D101—21no—
139C2×D101—21no—
140C2×D101—21no—
141C2×D101—44no—
142C2×D101—41no—
143C2×D101—41no—
144C2×D101—41no—
145C2×D101#13712yescomplemented
146C4×C101—41no—
147C4×C101—41no—
148C4×C101—42no—
149C4×C101#12112yes—
150C4×C101C22×D411yescharacteristic
151C4×C101C2×#4111yescharacteristic
152Dic101—21no—
153Dic101—21no—
154Dic101—42no—
155Dic101—41no—
156C2×C101—22no—
157C2×C101C2×#12111yescharacteristic
158C2×C101C2×#13711yescharacteristic
159C2×C101#57511yescharacteristic
160D101—22no—
161D101—41no—
162D101—81no—
163#3056—1011nomaximal, Sylow
164C101#305611yescharacteristic, complemented, Sylow
165C2×#137—1011no—
166C22×M16—1011no—
167#525—1011no—
168C2×C22⋊Q8—1011no—
169#470—1012no—
170#468—1012no—
171#467—1012no—
172#465—1012no—
173#428D10111yescharacteristic, complemented
174#419—1011no—
175#417—1011no—
176C2×#41—1011no—
177C22×Q8—1011no—
178C23×C4—1011no—
179#137—2024no—
180C2×Q16—1012no—
181C2×SD16—1012no—
182C2×M16—1011no—
183C2×M16—1012no—
184C2×M16—2024no—
185C2×M16C2×D10111yescharacteristic
186C22×C8—1012no—
187#123—2021no—
188C22⋊Q8—1012no—
189C22⋊Q8—1012no—
190C22⋊Q8—1011no—
191C22⋊Q8—1011no—
192C22⋊Q8—2021no—
193C22⋊Q8—2021no—
194#121—1011no—
195#121—1011no—
196#120—2021no—
197C4×D4—2021no—
198C4×D4—2021no—
199#117C2×D10111yescharacteristic
200C2×C4⋊C4—2021no—
201C2×C4⋊C4C2×D10111yescharacteristic
202C2×C22⋊C4—2021no—
203#107C2×D10112yescomplemented
204#106C2×D10112yescomplemented
205#103—1012no—
206#103—1012no—
207#103—1012no—
208#102—1012no—
209C24—1011no—
210#41—2022no—
211#41—2022no—
212C2×Q8—1012no—
213C2×Q8—1011no—
214C2×Q8—2022no—
215C2×Q8—2022no—
216C2×Q8—2021no—
217C2×D4—1011no—
218C2×D4—2021no—
219C2×D4—2021no—
220C2×D4—2021no—
221C22×C4—21no—
222C22×C4—1011no—
223C22×C4—1011no—
224C22×C4—1011no—
225C22×C4—1011no—
226C22×C4—1011no—
227C22×C4—2021no—
228C22×C4—2021no—
229C22×C4—2022no—
230C22×C4—2022no—
231C22×C4C22×D10111yescharacteristic
232Q16—2024no—
233SD16—2024no—
234M16—2022no—
235M16—2024no—
236M16D40414yescomplemented
237C2×C8—1012no—
238C2×C8—2024no—
239C2×C8C22×D10112yes—
240C4⋊C4—21no—
241C4⋊C4—2022no—
242C4⋊C4—2022no—
243C4⋊C4—2021no—
244C4⋊C4—2021no—
245C4⋊C4C22×D10112yes—
246C4⋊C4C22×D10111yescharacteristic
247C4⋊C4C22×D10111yescharacteristic
248C22⋊C4—21no—
249C22⋊C4—2021no—
250C22⋊C4—2021no—
251C22⋊C4—2022no—
252C22⋊C4—2022no—
253C22⋊C4—2021no—
254C22⋊C4—2021no—
255C42—21no—
256C23—1011no—
257C23—1011no—
258C23—4041no—
259C23—2022no—
260C23—2021no—
261C23—2022no—
262C23D4×D10111yescharacteristic
263Q8—4042no—
264Q8—2022no—
265Q8—2021no—
266Q8—2022no—
267Q8—2021no—
268D4—4042no—
269D4—4042no—
270D4—4042no—
271D4—2021no—
272D4—2021no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—42no—
277C2×C4—1011no—
278C2×C4—1011no—
279C2×C4—4042no—
280C2×C4—4042no—
281C2×C4—2021no—
282C2×C4—2022no—
283C2×C4—2022no—
284C2×C4—2021no—
285C2×C4—2021no—
286C2×C4—2021no—
287C2×C4—2021no—
288C2×C4—2022no—
289C2×C4—2022no—
290C2×C4D4×D10111yescharacteristic
291C2×C4C2×D40412yes—
292C2×C4#1310807512yes—
293C2×C4C23×D10111yescharacteristic
294C8—2022no—
295C8C2×D40414yes—
296C22—21no—
297C22—21no—
298C22—1012no—
299C22—4044no—
300C22—4041no—
301C22—4041no—
302C22—4041no—
303C22—2021no—
304C22—2021no—
305C22—2021no—
306C22—2021no—
307C22—2021no—
308C22#1569223012yes—
309C22C2×D4×D10111yescharacteristic
310C4—41no—
311C4—42no—
312C4—41no—
313C4—4041no—
314C4—4042no—
315C4—2021no—
316C4—2021no—
317C4#1569223012yes—
318C4C22×D40411yescharacteristic
319C4C2×#1310807511yescharacteristic
320C2—22no—
321C2—8081no—
322C2—4041no—
323C2—2022no—
324C2C2×#1569223011yescharacteristic
325C2#137×D10111yescharacteristic
326C2#2026597311yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#428×C101C2
socleC22×C101C22×D4
prime radical#20265689C2
hypercenter#428D101
hypocenterC101#3056
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
1. C2×#15692288C2semidirect
2. C2×M16×D101C2semidirect
4. C2×#15692235C2semidirect
5. #20265436C2semidirect
6. #20265424C2semidirect
7. #20265419C2semidirect
8. #20265407C2semidirect
9. #20265185C2semidirect
10. #20265176C2semidirect
11. #428×C101C2semidirect
16. C2×Dic404C22semidirect
17. C2×#13108063C22semidirect
36. #15692178C22semidirect
37. #15692178C22semidirect
38. #15692178C22semidirect
39. #15692175C22semidirect
43. #107×C101C22semidirect
44. #106×C101C22semidirect
84. M16×C101D4semidirect
145. C2×D101#137semidirect
164. C101#3056semidirect
173. #428D101semidirect
203. #107C2×D101semidirect
204. #106C2×D101semidirect
236. M16D404semidirect

Maximal subgroups

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

subgroupindex
C2×#156922882
C2×M16×D1012
#202656892
C2×#156922352
#202654362
#202654242
#202654192
#202654072
#202651852
#202651762
#428×C1012
#3056101