Finite Group Databasebeta

← #27180528

Subgroups — #27180528

Order 12928.

800 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 145 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×#15692300C211yesmaximal, characteristic, complemented
2#20265736C212yesmaximal, complemented
3#20265366C214yesmaximal, complemented
4#20265363C214yesmaximal, complemented
5C2×#15692170C211yesmaximal, characteristic, complemented
6C2×#15692169C212yesmaximal, complemented
7#534×C101C211yesmaximal, characteristic, complemented
8#15692300—24no—
9#15692300—22no—
10#15692300C2214yes—
11C2×#13108074—21no—
12C2×#13108073—22no—
13C22×C22⋊D101—21no—
14C22×C22⋊D101C2212yes—
15C2×D4×D101—22no—
16#15692238C2214yescomplemented
17#15692236—22no—
18#15692227—24no—
19#15692224C2214yescomplemented
20#15692223—22no—
21C2×C22⋊Dic101C2212yescomplemented
22#15692170C2214yescomplemented
23#15692169C2218yescomplemented
24C22×D4×C101C2211yescharacteristic
25C22×D4×C101C2211yescharacteristic, complemented
26#123×C101—21no—
27#123×C101C2212yescomplemented
28#121×C101—22no—
29#120×C101—21no—
30#120×C101C2212yescomplemented
31C2×C22⋊C4×C101C2211yescharacteristic, complemented
32C23×D101—21no—
33C23×D101D412yescomplemented
34#13108074—24no—
35#13108074—24no—
36#13108073—28no—
37#13108073—28no—
38C2×Dic202—21no—
39C2×D404—21no—
40C2×C22⋊D101—22no—
41C2×C22⋊D101—24no—
42C2×C22⋊D101—22no—
43C2×C22⋊D101—22no—
44C2×C22⋊D101—24no—
45C2×C22⋊D101—22no—
46C2×C22⋊D101—22no—
47C2×C22⋊D101—24no—
48C2×C22⋊D101D414yescomplemented
49C2×C22⋊D101C2314yes—
50D4×D101—28no—
51D4×D101—28no—
52C22×Dic101—21no—
53C22×Dic101—22no—
54C22×Dic101D412yescomplemented
55C2×C4×D101—22no—
56#13108053—24no—
57C22⋊Dic101—22no—
58C22⋊Dic101—24no—
59C22⋊Dic101C2314yescomplemented
60#13108051—24no—
61C24×C101D411yescharacteristic, complemented
62C24×C101C2311yescharacteristic
63C24×C101C2311yescharacteristic, complemented
64C2×D4×C101—22no—
65C2×D4×C101—22no—
66C2×D4×C101—21no—
67C2×D4×C101—21no—
68C2×D4×C101—22no—
69C2×D4×C101—22no—
70C2×D4×C101—22no—
71C2×D4×C101D412yescomplemented
72C2×D4×C101C2312yes—
73C22×C4×C101—21no—
74C22×C4×C101D411yescharacteristic, complemented
75C22×C4×C101C2311yescharacteristic, complemented
76C4⋊C4×C101—22no—
77C22⋊C4×C101—21no—
78C22⋊C4×C101—22no—
79C22⋊C4×C101—22no—
80C22⋊C4×C101C2312yescomplemented
81C23×C101—21no—
82C23×C101—21no—
83C23×C101—21no—
84C23×C101—22no—
85C23×C101—81no—
86C23×C101—81no—
87C23×C101—81no—
88C23×C101C2×D412yes—
89C23×C101C2×D412yes—
90C23×C101C2×D412yes—
91C23×C101C2×D411yescharacteristic
92C23×C101C2×D411yescharacteristic
93C23×C101C2411yescharacteristic
94C22×D101—22no—
95C22×D101—24no—
96C22×D101—22no—
97C22×D101—82no—
98C22×D101—82no—
99C22×D101C2×D414yes—
100D4×C101—24no—
101D4×C101—24no—
102D4×C101—41no—
103D4×C101—42no—
104D4×C101—41no—
105D4×C101—82no—
106C2×C4×C101—21no—
107C2×C4×C101—21no—
108C2×C4×C101—22no—
109C2×C4×C101—21no—
110C2×C4×C101—22no—
111C2×C4×C101—81no—
112C2×C4×C101C2×D412yes—
113C22⋊D101—28no—
114C22⋊D101—24no—
115C22⋊D101—24no—
116C22⋊D101—24no—
117C22⋊D101—24no—
118C22⋊D101—84no—
119C2×Dic101—24no—
120C2×Dic101—22no—
121C2×Dic101—22no—
122C2×Dic101—24no—
123C2×Dic101—82no—
124C2×Dic101C2×D414yes—
125D404—24no—
126C4×D101—28no—
127Dic202—24no—
128C22×C101—22no—
129C22×C101—42no—
130C22×C101—41no—
131C22×C101—42no—
132C22×C101—41no—
133C22×C101—41no—
134C22×C101—42no—
135C22×C101—42no—
136C22×C101—44no—
137C22×C101—42no—
138C22×C101—41no—
139C22×C101—81no—
140C22×C101#12011yescharacteristic
141C22×C101#12012yes—
142C22×C101#12011yescharacteristic
143C22×C101C22×D412yes—
144C22×C101C22×D411yescharacteristic
145C2×D101—24no—
146C2×D101—44no—
147C2×D101—44no—
148C2×D101—82no—
149C2×D101—84no—
150C4×C101—22no—
151C4×C101—42no—
152C4×C101—41no—
153Dic101—24no—
154Dic101—44no—
155C2×C101—22no—
156C2×C101—21no—
157C2×C101—21no—
158C2×C101—22no—
159C2×C101—21no—
160C2×C101—42no—
161C2×C101—81no—
162C2×C101C2×#12011yescharacteristic
163D101—44no—
164#3026—1011nomaximal, Sylow
165C101#302611yescharacteristic, complemented, Sylow
166C2×H(5,𝔽2)—1011no—
167#534—1012no—
168#534D10111yescharacteristic, complemented
169#458—1014no—
170B(4,𝔽2)—1014no—
171C2×#99—1011no—
172C2×#99—1012no—
173H(5,𝔽2)—1014no—
174H(5,𝔽2)—2024no—
175H(5,𝔽2)—2022no—
176C2×#41—2022no—
177C2×#41—2021no—
178C22×D4—1012no—
179C22×D4—2022no—
180C22×D4—2021no—
181C22×D4C2×D10111yescharacteristic
182C22×D4C2×D10111yescharacteristic, complemented
183#123—21no—
184#123—1014no—
185#123—2022no—
186#123C2×D10112yescomplemented
187#121—22no—
188#121—2024no—
189#120—21no—
190#120—1014no—
191#120—2022no—
192#120C2×D10112yescomplemented
193C2×C22⋊C4—1012no—
194C2×C22⋊C4C2×D10111yescharacteristic, complemented
195#99—1014no—
196#99—1018no—
197C24—1012no—
198C24—2021no—
199C24C22⋊D10111yescharacteristic, complemented
200C24C22×D10111yescharacteristic
201C24C22×D10111yescharacteristic, complemented
202#41—2024no—
203#41—2028no—
204#41—2028no—
205#41—2024no—
206C2×Q8—2021no—
207C2×D4—22no—
208C2×D4—22no—
209C2×D4—22no—
210C2×D4—22no—
211C2×D4—21no—
212C2×D4—21no—
213C2×D4—22no—
214C2×D4—1014no—
215C2×D4—1014no—
216C2×D4—2021no—
217C2×D4—2028no—
218C2×D4—2028no—
219C2×D4—2024no—
220C2×D4—2022no—
221C2×D4—2024no—
222C2×D4—2022no—
223C2×D4—2022no—
224C2×D4—2022no—
225C2×D4—2022no—
226C2×D4—2024no—
227C2×D4C22⋊D10112yescomplemented
228C2×D4C22×D10112yes—
229C22×C4—21no—
230C22×C4—1012no—
231C22×C4—2022no—
232C22×C4—2022no—
233C22×C4—2021no—
234C22×C4C22⋊D10111yescharacteristic, complemented
235C22×C4C22×D10111yescharacteristic, complemented
236C4⋊C4—22no—
237C4⋊C4—2024no—
238C22⋊C4—21no—
239C22⋊C4—22no—
240C22⋊C4—22no—
241C22⋊C4—1014no—
242C22⋊C4—2024no—
243C22⋊C4—2024no—
244C22⋊C4—2022no—
245C22⋊C4C22×D10112yescomplemented
246C23—21no—
247C23—21no—
248C23—22no—
249C23—21no—
250C23—81no—
251C23—81no—
252C23—81no—
253C23—8082no—
254C23—8082no—
255C23—1014no—
256C23—2024no—
257C23—2022no—
258C23—2022no—
259C23D4×D10112yes—
260C23D4×D10112yes—
261C23C2×C22⋊D10112yes—
262C23C2×C22⋊D10111yescharacteristic
263C23C2×C22⋊D10111yescharacteristic
264C23C23×D10111yescharacteristic
265Q8—2024no—
266D4—24no—
267D4—24no—
268D4—41no—
269D4—41no—
270D4—42no—
271D4—82no—
272D4—8084no—
273D4—2024no—
274D4—2028no—
275D4—2024no—
276D4—2024no—
277D4—2024no—
278D4—2024no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—81no—
285C2×C4—8082no—
286C2×C4—1014no—
287C2×C4—2028no—
288C2×C4—2024no—
289C2×C4—2024no—
290C2×C4—2022no—
291C2×C4—2022no—
292C2×C4C2×C22⋊D10112yes—
293C22—22no—
294C22—41no—
295C22—42no—
296C22—41no—
297C22—41no—
298C22—42no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22—44no—
303C22—41no—
304C22—81no—
305C22—8082no—
306C22—8084no—
307C22—4044no—
308C22—4044no—
309C22—2024no—
310C22#1569222311yescharacteristic
311C22#1569222311yescharacteristic
312C22#1569222312yes—
313C22C2×D4×D10112yes—
314C22C22×C22⋊D10111yescharacteristic
315C4—22no—
316C4—42no—
317C4—41no—
318C4—4044no—
319C4—2024no—
320C2—21no—
321C2—22no—
322C2—21no—
323C2—21no—
324C2—22no—
325C2—42no—
326C2—81no—
327C2—4044no—
328C2C2×#1569222311yescharacteristic
subgroupG/H
centerC2C2×#15692223
commutator, cosocleC23×C101C24
FrattiniC23C23×D101
Fitting#534×C101C2
socleC2×C101C2×#120
hypercenter#534D101
hypocenterC101#3026
LCSG ▷ C23×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C24 ◁ #534

Complements

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

NH ≅ G/Ntype
1. C2×#15692300C2semidirect
2. #20265736C2semidirect
3. #20265366C2semidirect
4. #20265363C2semidirect
5. C2×#15692170C2semidirect
6. C2×#15692169C2semidirect
7. #534×C101C2semidirect
16. #15692238C22semidirect
19. #15692224C22semidirect
21. C2×C22⋊Dic101C22semidirect
22. #15692170C22semidirect
23. #15692169C22semidirect
25. C22×D4×C101C22semidirect
27. #123×C101C22semidirect
30. #120×C101C22semidirect
31. C2×C22⋊C4×C101C22semidirect
33. C23×D101D4semidirect
48. C2×C22⋊D101D4semidirect
54. C22×Dic101D4semidirect
59. C22⋊Dic101C23semidirect
61. C24×C101D4semidirect
63. C24×C101C23semidirect
71. C2×D4×C101D4semidirect
74. C22×C4×C101D4semidirect
75. C22×C4×C101C23semidirect
80. C22⋊C4×C101C23semidirect
165. C101#3026semidirect
168. #534D101semidirect
182. C22×D4C2×D101semidirect
186. #123C2×D101semidirect
192. #120C2×D101semidirect
194. C2×C22⋊C4C2×D101semidirect
199. C24C22⋊D101semidirect
201. C24C22×D101semidirect
227. C2×D4C22⋊D101semidirect
234. C22×C4C22⋊D101semidirect
235. C22×C4C22×D101semidirect
245. C22⋊C4C22×D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156923002
#202657362
#202653662
#202653632
C2×#156921702
C2×#156921692
#534×C1012
#3026101