Finite Group Databasebeta

← #27180778

Subgroups — #27180778

Order 12928.

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

normal subgroups only

This group has 524 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×D8×D101C211yesmaximal, characteristic, complemented
2C2×#15692263C211yesmaximal, characteristic, complemented
3#20265743C212yesmaximal, complemented
4#20265431C211yesmaximal, characteristic, complemented
5#20265430C211yesmaximal, characteristic, complemented
6#20265427C211yesmaximal, characteristic, complemented
7#20265412C212yesmaximal, complemented
8#20265411C212yesmaximal, complemented
9#20265212C211yesmaximal, characteristic, complemented
10#20265180C212yesmaximal, complemented
11#469×C101C211yesmaximal, characteristic, complemented
12C2×#13108074C2211yescharacteristic
13C2×D4×D101C2212yes—
14C2×D808C2211yescharacteristic
15C2×#13108059C2212yes—
16D8×D101—24no—
17#15692263—24no—
18C2×#13108057C2211yescharacteristic
19C2×#13108057C2211yescharacteristic, complemented
20C22×#11717579C2211yescharacteristic, complemented
21C2×C8×D101C2211yescharacteristic
22C2×C8×D101C2211yescharacteristic, complemented
23#15692251—22no—
24#15692241—22no—
25#15692230C2212yescomplemented
26#15692229C2212yescomplemented
27#15692226C2212yescomplemented
28#15692225—22no—
29#15692223—22no—
30#15692213C2212yes—
31#15692188C2212yescomplemented
32#15692187C2211yescharacteristic, complemented
33#15692177C2212yescomplemented
34#15692176C2212yescomplemented
35#15692175C2212yescomplemented
36C2×D8×C101C2211yescharacteristic
37C2×M16×C101C2211yescharacteristic, complemented
38#121×C101C2212yescomplemented
39#106×C101C2211yescharacteristic, complemented
40#102×C101C2212yescomplemented
41C23×D101—22no—
42#13108074—22no—
43#13108074—22no—
44C2×Dic202D411yescharacteristic
45C2×D404—22no—
46C2×D404D411yescharacteristic
47C2×D404C2312yes—
48C2×C22⋊D101—22no—
49C2×C22⋊D101—22no—
50C2×C22⋊D101—22no—
51C2×C22⋊D101D411yescharacteristic
52C2×C22⋊D101D411yescharacteristic
53D4×D101—24no—
54D4×D101—24no—
55C2×C4×D101C2311yescharacteristic
56C2×C4×D101C2311yescharacteristic
57D808—22no—
58#13108059—24no—
59#13108057—22no—
60#13108057—22no—
61C2×#11717579—22no—
62C2×#11717579C2311yescharacteristic
63C2×#11717579C2311yescharacteristic
64C8×D101—22no—
65C8×D101D414yescomplemented
66C4⋊Dic101C2312yes—
67C22⋊Dic101—22no—
68#13108051—22no—
69#13108051—22no—
70#13108051—22no—
71C4×Dic101—22no—
72C2×D4×C101—22no—
73C2×D4×C101C2312yes—
74C22×C4×C101C2311yescharacteristic
75D8×C101—22no—
76M16×C101—22no—
77C2×C8×C101C2311yescharacteristic
78C2×C8×C101C2311yescharacteristic
79C4⋊C4×C101C2312yes—
80C22⋊C4×C101—22no—
81C23×C101—22no—
82C23×C101C2×D411yescharacteristic
83C22×D101—22no—
84C22×D101—44no—
85C22×D101—82no—
86C22×D101—82no—
87C22×D101C2×D411yescharacteristic
88C22×D101C2×D411yescharacteristic
89D4×C101—22no—
90D4×C101—22no—
91D4×C101—82no—
92C2×C4×C101—21no—
93C2×C4×C101—21no—
94C2×C4×C101—22no—
95C2×C4×C101C2×D411yescharacteristic
96C2×C4×C101C2411yescharacteristic
97C22⋊D101—22no—
98C22⋊D101—22no—
99C22⋊D101—44no—
100C22⋊D101—82no—
101C22⋊D101—82no—
102C2×Dic101—22no—
103C2×Dic101C2×D411yescharacteristic
104C2×Dic101C2×D411yescharacteristic
105D404—22no—
106D404—22no—
107D404—22no—
108D404—82no—
109C4×D101—21no—
110C4×D101—21no—
111C4×D101C2×D412yes—
112C4×D101#4112yes—
113Dic202—22no—
114C8×C101—21no—
115C8×C101C2×D412yes—
116#11717579—21no—
117#11717579C2×D412yes—
118C22×C101—21no—
119C22×C101—22no—
120C22×C101—42no—
121C22×C101—42no—
122C22×C101—42no—
123C22×C101C22×D411yescharacteristic
124C2×D101—22no—
125C2×D101—21no—
126C2×D101—22no—
127C2×D101—42no—
128C2×D101—42no—
129C2×D101—42no—
130C2×D101—88no—
131C2×D101#12112yes—
132C4×C101—21no—
133C4×C101—42no—
134C4×C101C22×D411yescharacteristic
135C4×C101C2×#4111yescharacteristic
136Dic101—21no—
137Dic101—42no—
138Dic101#12112yes—
139C2×C101—41no—
140C2×C101—82no—
141C2×C101C2×#12111yescharacteristic
142C2×C101#57612yes—
143D101—22no—
144D101—41no—
145D101—82no—
146#3058—1011nomaximal, Sylow
147C101#305811yescharacteristic, complemented, Sylow
148C22×D8—1011no—
149C2×#131—1011no—
150#535—1012no—
151#469—1011no—
152#469—1011no—
153#469—1011no—
154#469D10111yescharacteristic, complemented
155#466—1012no—
156#466—1012no—
157#427—1011no—
158#418—1012no—
159C2×#41—1011no—
160C22×D4—1012no—
161C2×D8—1012no—
162C2×D8—1011no—
163C2×D8—2024no—
164C2×D8C2×D10111yescharacteristic
165#131—2024no—
166C2×M16—1011no—
167C2×M16—1011no—
168C2×M16C2×D10111yescharacteristic, complemented
169C22×C8—1011no—
170C22×C8—1011no—
171C22×C8—1011no—
172#127—2022no—
173#124—2022no—
174#121—1012no—
175#121—1012no—
176#121—1012no—
177#121C2×D10112yescomplemented
178#120—2022no—
179#120—2022no—
180#117—1012no—
181#107—1012no—
182#106—1011no—
183#106C2×D10111yescharacteristic, complemented
184#102—1012no—
185#102—1012no—
186#102—1012no—
187#102C2×D10112yescomplemented
188C24—2022no—
189#41—2022no—
190#41—2022no—
191C2×Q8—1011no—
192C2×D4—22no—
193C2×D4—1011no—
194C2×D4—1011no—
195C2×D4—1011no—
196C2×D4—1012no—
197C2×D4—2022no—
198C2×D4—2024no—
199C2×D4—2024no—
200C2×D4—2022no—
201C2×D4—2022no—
202C2×D4—2022no—
203C2×D4C22×D10112yes—
204C22×C4—1011no—
205C22×C4—1011no—
206C22×C4C22×D10111yescharacteristic
207D8—22no—
208D8—2024no—
209D8—2022no—
210M16—22no—
211M16—2022no—
212M16—2022no—
213C2×C8—1011no—
214C2×C8—1014no—
215C2×C8—1011no—
216C2×C8—2022no—
217C2×C8—2022no—
218C2×C8C22×D10111yescharacteristic
219C2×C8C22×D10111yescharacteristic
220C4⋊C4—1012no—
221C4⋊C4C22×D10112yes—
222C22⋊C4—22no—
223C22⋊C4—2022no—
224C22⋊C4—2022no—
225C22⋊C4—2022no—
226C22⋊C4—2022no—
227C42—2022no—
228C23—22no—
229C23—8082no—
230C23—8082no—
231C23—1011no—
232C23—1011no—
233C23—4044no—
234C23—2022no—
235C23D4×D10111yescharacteristic
236Q8—2022no—
237D4—22no—
238D4—22no—
239D4—82no—
240D4—8082no—
241D4—8082no—
242D4—8082no—
243D4—4044no—
244D4—2022no—
245D4—2022no—
246D4—2022no—
247D4—2022no—
248D4—2022no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—1011no—
253C2×C4—1012no—
254C2×C4—1012no—
255C2×C4—1011no—
256C2×C4—2022no—
257C2×C4—2021no—
258C2×C4—2021no—
259C2×C4D4×D10111yescharacteristic
260C2×C4C23×D10111yescharacteristic
261C8—21no—
262C8—1012no—
263C8—2021no—
264C8D4×D10112yes—
265C22—21no—
266C22—22no—
267C22—42no—
268C22—42no—
269C22—42no—
270C22—8088no—
271C22—1012no—
272C22—4042no—
273C22—4042no—
274C22—4042no—
275C22—2021no—
276C22—2022no—
277C22—2022no—
278C22C2×D4×D10111yescharacteristic
279C4—21no—
280C4—42no—
281C4—1012no—
282C4—4042no—
283C4—2021no—
284C4C2×D4×D10111yescharacteristic
285C4#41×D10111yescharacteristic
286C2—41no—
287C2—82no—
288C2—8082no—
289C2—4041no—
290C2—2022no—
291C2#121×D10111yescharacteristic
292C2#2026598112yes—
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#469×C101C2
socleC22×C101C22×D4
hypercenter#469D101
hypocenterC101#3058
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. C2×D8×D101C2semidirect
2. C2×#15692263C2semidirect
3. #20265743C2semidirect
4. #20265431C2semidirect
5. #20265430C2semidirect
6. #20265427C2semidirect
7. #20265412C2semidirect
8. #20265411C2semidirect
9. #20265212C2semidirect
10. #20265180C2semidirect
11. #469×C101C2semidirect
19. C2×#13108057C22semidirect
20. C22×#11717579C22semidirect
22. C2×C8×D101C22semidirect
25. #15692230C22semidirect
26. #15692229C22semidirect
27. #15692226C22semidirect
31. #15692188C22semidirect
32. #15692187C22semidirect
33. #15692177C22semidirect
34. #15692176C22semidirect
35. #15692175C22semidirect
37. C2×M16×C101C22semidirect
38. #121×C101C22semidirect
39. #106×C101C22semidirect
40. #102×C101C22semidirect
65. C8×D101D4semidirect
147. C101#3058semidirect
154. #469D101semidirect
168. C2×M16C2×D101semidirect
177. #121C2×D101semidirect
183. #106C2×D101semidirect
187. #102C2×D101semidirect

Maximal subgroups

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

subgroupindex
C2×D8×D1012
C2×#156922632
#202657432
#202654312
#202654302
#202654272
#202654122
#202654112
#202652122
#202651802
#469×C1012
#3058101