Finite Group Databasebeta

← #27180712

Subgroups — #27180712

Order 12928.

520 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 137 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×#15692275C211yesmaximal, characteristic, complemented
2C22×C8×D101C211yesmaximal, characteristic, complemented
3#20265689C211yesmaximal, characteristic, complemented
4#20265684C211yesmaximal, characteristic, complemented
5#20265417C212yesmaximal, complemented
6#20265412C212yesmaximal, complemented
7#20265409C212yesmaximal, complemented
8#20265405C212yesmaximal, complemented
9#20265213C211yesmaximal, characteristic, complemented
10#20265174C211yesmaximal, characteristic, complemented
11#416×C101C211yesmaximal, characteristic, complemented
12C2×#13108074C2211yescharacteristic
13C22×C4×D101C2211yescharacteristic
14#15692275—24no—
15C2×#13108064C2211yescharacteristic, complemented
16C2×#13108062C2211yescharacteristic, complemented
17C2×#13108061C2211yescharacteristic, complemented
18C2×#13108059C2211yescharacteristic, complemented
19C22×#11717579C2211yescharacteristic
20C2×C8×D101—24no—
21C2×C8×D101C2212yes—
22C2×C8×D101C2212yes—
23#15692239—21no—
24#15692236—21no—
25#15692235C2212yes—
26#15692231C2212yes—
27#15692230C2212yes—
28#15692226C2212yes—
29#15692225—21no—
30#15692223—21no—
31C4×D404—22no—
32C4×C22⋊D101—22no—
33#15692214C2211yescharacteristic
34#15692188C2212yescomplemented
35#15692186C2212yescomplemented
36#15692180C2212yescomplemented
37#15692177C2212yescomplemented
38C2×#41×C101C2211yescharacteristic
39C22×C8×C101C2211yescharacteristic
40#117×C101C2211yescharacteristic
41#103×C101C2212yescomplemented
42#102×C101C2212yescomplemented
43C23×D101D411yescharacteristic
44#13108074—22no—
45#13108074—22no—
46C2×Dic202C2311yescharacteristic
47C2×D404—22no—
48C2×D404C2311yescharacteristic
49C2×C22⋊D101—21no—
50C2×C22⋊D101—22no—
51C22×Dic101D411yescharacteristic
52C2×C4×D101—22no—
53C2×C4×D101—22no—
54C2×C4×D101—21no—
55C2×C4×D101D412yes—
56C2×C4×D101C2312yes—
57#13108064—22no—
58#13108062—22no—
59#13108061—22no—
60#13108059—22no—
61C2×#11717579D414yescomplemented
62C2×#11717579C2311yescharacteristic
63C2×#11717579C2311yescharacteristic
64C8×D101—24no—
65C8×D101—24no—
66C4⋊Dic101—22no—
67C4⋊Dic101C2312yes—
68#13108053—22no—
69C22⋊Dic101—21no—
70C22⋊Dic101—22no—
71#13108051—22no—
72#13108051—22no—
73#13108051—22no—
74C4×Dic101—21no—
75#41×C101—22no—
76#41×C101—22no—
77C2×Q8×C101C2311yescharacteristic
78C2×D4×C101—21no—
79C2×D4×C101C2311yescharacteristic
80C22×C4×C101—21no—
81C22×C4×C101C2311yescharacteristic
82C2×C8×C101—22no—
83C2×C8×C101C2312yes—
84C4⋊C4×C101C2312yes—
85C22⋊C4×C101—21no—
86C42×C101—21no—
87C23×C101—21no—
88C23×C101C2×D411yescharacteristic
89C22×D101—22no—
90C22×D101—24no—
91C22×D101—21no—
92C22×D101C2×D412yes—
93Q8×C101—21no—
94Q8×C101—21no—
95D4×C101—21no—
96D4×C101—21no—
97D4×C101—42no—
98C2×C4×C101—22no—
99C2×C4×C101—21no—
100C2×C4×C101—42no—
101C2×C4×C101C2×D412yes—
102C2×C4×C101C2×D411yescharacteristic
103C2×C4×C101#4112yes—
104C2×C4×C101C2411yescharacteristic
105C22⋊D101—42no—
106C22⋊D101—44no—
107C2×Dic101—22no—
108C2×Dic101—22no—
109C2×Dic101—21no—
110C2×Dic101C2×D412yes—
111D404—21no—
112D404—21no—
113D404—44no—
114C4×D101—24no—
115C4×D101—22no—
116C4×D101—22no—
117C4×D101—42no—
118Dic202—21no—
119Dic202—21no—
120C8×C101—22no—
121#11717579C2×D412yes—
122#11717579C2×D412yes—
123C22×C101—22no—
124C22×C101—41no—
125C22×C101—42no—
126C22×C101#12112yes—
127C22×C101C22×D411yescharacteristic
128C2×D101—28no—
129C2×D101—22no—
130C2×D101—22no—
131C2×D101—41no—
132C2×D101—42no—
133C2×D101#13514yescomplemented
134C4×C101—41no—
135C4×C101—42no—
136C4×C101#12112yes—
137C4×C101C22×D411yescharacteristic
138C4×C101C2×#4111yescharacteristic
139Dic101—22no—
140Dic101—41no—
141Dic101—42no—
142C2×C101—22no—
143C2×C101—81no—
144C2×C101C2×#12111yescharacteristic
145C2×C101C2×#13512yes—
146D101—24no—
147D101—81no—
148#3050—1011nomaximal, Sylow
149C101#305011yescharacteristic, complemented, Sylow
150C2×#135—1011no—
151C23×C8—1011no—
152#525—1011no—
153#525—1011no—
154#467—1012no—
155#466—1012no—
156#465—1012no—
157#465—1012no—
158#427—1011no—
159#416—1011no—
160#416D10111yescharacteristic, complemented
161C2×#41—1011no—
162C2×#41C2×D10111yescharacteristic
163C23×C4—1011no—
164#135—2024no—
165C2×Q16—1011no—
166C2×SD16—1011no—
167C2×SD16—1011no—
168C2×D8—1011no—
169C22×C8—1011no—
170C22×C8—1012no—
171C22×C8—1012no—
172C22×C8—2024no—
173C22×C8C2×D10111yescharacteristic
174#123—2021no—
175#123—2021no—
176C22⋊Q8—1012no—
177C22⋊Q8—1012no—
178#121—1012no—
179#121—1012no—
180#120—2021no—
181#120—2021no—
182C4×D4—2022no—
183C4×D4—2022no—
184#117—1011no—
185#117C2×D10111yescharacteristic
186#107—1012no—
187#106—1012no—
188#103—1012no—
189#103C2×D10112yescomplemented
190#102—1012no—
191#102C2×D10112yescomplemented
192C24—1011no—
193#41—22no—
194#41—22no—
195#41—2022no—
196#41—2022no—
197C2×Q8—1011no—
198C2×Q8C22×D10111yescharacteristic
199C2×D4—21no—
200C2×D4—1011no—
201C2×D4—2022no—
202C2×D4—2022no—
203C2×D4—2021no—
204C2×D4C22×D10111yescharacteristic
205C22×C4—21no—
206C22×C4—1011no—
207C22×C4—1012no—
208C22×C4—1012no—
209C22×C4—2021no—
210C22×C4—2022no—
211C22×C4—2022no—
212C22×C4C22×D10111yescharacteristic
213Q16—2022no—
214SD16—2022no—
215SD16—2022no—
216D8—2022no—
217C2×C8—22no—
218C2×C8—1014no—
219C2×C8—1011no—
220C2×C8—1011no—
221C2×C8—2024no—
222C2×C8—2024no—
223C2×C8C22×D10112yes—
224C4⋊C4—1012no—
225C4⋊C4—2022no—
226C4⋊C4—2022no—
227C4⋊C4C22×D10112yes—
228C22⋊C4—21no—
229C22⋊C4—2022no—
230C22⋊C4—2022no—
231C22⋊C4—2022no—
232C22⋊C4—2022no—
233C22⋊C4—2021no—
234C42—21no—
235C42—2021no—
236C23—21no—
237C23—1012no—
238C23—2024no—
239C23—2021no—
240C23—2022no—
241C23D4×D10111yescharacteristic
242Q8—21no—
243Q8—21no—
244Q8—2021no—
245Q8—2021no—
246D4—21no—
247D4—21no—
248D4—42no—
249D4—4044no—
250D4—4044no—
251D4—4042no—
252D4—2021no—
253D4—2021no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—42no—
257C2×C4—1012no—
258C2×C4—4042no—
259C2×C4—2022no—
260C2×C4—2021no—
261C2×C4—2022no—
262C2×C4—2022no—
263C2×C4—2022no—
264C2×C4—2024no—
265C2×C4D4×D10112yes—
266C2×C4D4×D10111yescharacteristic
267C2×C4#1310807412yes—
268C2×C4C23×D10111yescharacteristic
269C8—22no—
270C8—1012no—
271C8—1012no—
272C22—22no—
273C22—41no—
274C22—42no—
275C22—1014no—
276C22—4041no—
277C22—4042no—
278C22—2028no—
279C22—2022no—
280C22—2022no—
281C22#1569222812yes—
282C22C2×D4×D10111yescharacteristic
283C4—41no—
284C4—42no—
285C4—4041no—
286C4—4042no—
287C4—2022no—
288C4#1569222812yes—
289C4C2×D4×D10111yescharacteristic
290C4C2×#1310807411yescharacteristic
291C2—22no—
292C2—81no—
293C2—8081no—
294C2—2024no—
295C2C2×#1569222811yescharacteristic
296C2#135×D10112yes—
subgroupG/H
centerC2×C4D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#416×C101C2
socleC22×C101C22×D4
hypercenter#416D101
hypocenterC101#3050
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. C2×#15692275C2semidirect
2. C22×C8×D101C2semidirect
3. #20265689C2semidirect
4. #20265684C2semidirect
5. #20265417C2semidirect
6. #20265412C2semidirect
7. #20265409C2semidirect
8. #20265405C2semidirect
9. #20265213C2semidirect
10. #20265174C2semidirect
11. #416×C101C2semidirect
15. C2×#13108064C22semidirect
16. C2×#13108062C22semidirect
17. C2×#13108061C22semidirect
18. C2×#13108059C22semidirect
34. #15692188C22semidirect
35. #15692186C22semidirect
36. #15692180C22semidirect
37. #15692177C22semidirect
41. #103×C101C22semidirect
42. #102×C101C22semidirect
61. C2×#11717579D4semidirect
133. C2×D101#135semidirect
149. C101#3050semidirect
160. #416D101semidirect
189. #103C2×D101semidirect
191. #102C2×D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922752
C22×C8×D1012
#202656892
#202656842
#202654172
#202654122
#202654092
#202654052
#202652132
#202651742
#416×C1012
#3050101