Finite Group Databasebeta

← #49130660

Subgroups — #49130660

Order 36224.

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×#22583655C211yesmaximal, characteristic, complemented
2#31827027C212yesmaximal, complemented
3#31827024C211yesmaximal, characteristic, complemented
4#31826656C214yesmaximal, complemented
5#31826653C214yesmaximal, complemented
6C2×#22583524C212yesmaximal, complemented
7C2×#99×C283C211yesmaximal, characteristic, complemented
8#22583655—24no—
9#22583655—22no—
10#22583655C2214yes—
11C2×#17985392—21no—
12C2×#17985391—22no—
13C22×C22⋊D283—21no—
14C22×C22⋊D283C2212yes—
15C22×C22⋊D283C2211yescharacteristic, complemented
16C2×D4×D283—22no—
17#22583594—22no—
18#22583594C2214yescomplemented
19#22583593—21no—
20#22583591C2212yescomplemented
21#22583584—24no—
22#22583582—22no—
23#22583580—22no—
24#22583580C2214yescomplemented
25#22583579—21no—
26#22583578C2212yescomplemented
27C2×C22⋊Dic283C2211yescharacteristic, complemented
28#22583524C2218yescomplemented
29C22×D4×C283C2211yescharacteristic
30C2×C22⋊C4×C283C2212yescomplemented
31#99×C283C2214yescomplemented
32C23×D283—21no—
33C23×D283D412yescomplemented
34C23×D283C2311yescharacteristic, complemented
35#17985392—24no—
36#17985392—24no—
37#17985391—28no—
38#17985391—28no—
39C2×Dic566—21no—
40C2×D1132—21no—
41C2×D1132—24no—
42C2×C22⋊D283—22no—
43C2×C22⋊D283—24no—
44C2×C22⋊D283—22no—
45C2×C22⋊D283—22no—
46C2×C22⋊D283—24no—
47C2×C22⋊D283—22no—
48C2×C22⋊D283—22no—
49C2×C22⋊D283—21no—
50C2×C22⋊D283—22no—
51C2×C22⋊D283—21no—
52C2×C22⋊D283—22no—
53C2×C22⋊D283—22no—
54C2×C22⋊D283D414yescomplemented
55C2×C22⋊D283C2314yes—
56D4×D283—28no—
57D4×D283—28no—
58C22×Dic283—21no—
59C22×Dic283—21no—
60C22×Dic283D412yescomplemented
61C22×Dic283C2311yescharacteristic, complemented
62C2×C4×D283—22no—
63C4⋊Dic283—24no—
64Dic283⋊C4—22no—
65C22⋊Dic283—21no—
66C22⋊Dic283—22no—
67C22⋊Dic283C2312yescomplemented
68(C2×D283)⋊C4—24no—
69(C2×D283)⋊C4—24no—
70(C2×D283)⋊C4—22no—
71C24×C283D411yescharacteristic, complemented
72C24×C283C2311yescharacteristic
73C2×D4×C283—22no—
74C2×D4×C283—22no—
75C2×D4×C283D412yescomplemented
76C2×D4×C283C2312yes—
77C22×C4×C283—22no—
78C22×C4×C283D411yescharacteristic, complemented
79C22⋊C4×C283—22no—
80C22⋊C4×C283C2314yescomplemented
81C22×D283—22no—
82C22×D283—24no—
83C22×D283—22no—
84C22×D283—21no—
85C22×D283—22no—
86C22×D283—81no—
87C22×D283—82no—
88C22×D283—82no—
89C22×D283C2×D414yes—
90Dic566—24no—
91D1132—24no—
92D1132—84no—
93C22⋊D283—28no—
94C22⋊D283—24no—
95C22⋊D283—24no—
96C22⋊D283—24no—
97C22⋊D283—24no—
98C22⋊D283—41no—
99C22⋊D283—41no—
100C22⋊D283—42no—
101C22⋊D283—82no—
102C2×Dic283—24no—
103C2×Dic283—22no—
104C2×Dic283—22no—
105C2×Dic283—22no—
106C2×Dic283—22no—
107C2×Dic283—21no—
108C2×Dic283—81no—
109C2×Dic283C2×D414yes—
110C4×D283—28no—
111C23×C283—21no—
112C23×C283—21no—
113C23×C283—81no—
114C23×C283—81no—
115C23×C283C2×D412yes—
116C23×C283C2×D412yes—
117C23×C283C2×D412yes—
118C23×C283C2×D411yescharacteristic
119C23×C283C2×D411yescharacteristic
120C23×C283C2411yescharacteristic
121D4×C283—24no—
122D4×C283—24no—
123C2×C4×C283—21no—
124C2×C4×C283—21no—
125C2×C4×C283—24no—
126C2×C4×C283—82no—
127C2×C4×C283C2×D412yes—
128C22×C283—22no—
129C22×C283—42no—
130C22×C283—41no—
131C22×C283—41no—
132C22×C283—42no—
133C22×C283—41no—
134C22×C283—42no—
135C22×C283—42no—
136C22×C283—81no—
137C22×C283#12011yescharacteristic
138C22×C283#12011yescharacteristic
139C22×C283#12012yes—
140C22×C283C22×D412yes—
141C22×C283C22×D411yescharacteristic
142C2×D283—24no—
143C2×D283—44no—
144C2×D283—44no—
145C2×D283—42no—
146C2×D283—44no—
147C2×D283—41no—
148C2×D283—82no—
149C2×D283—84no—
150C4×C283—22no—
151C4×C283—44no—
152Dic283—24no—
153Dic283—42no—
154Dic283—41no—
155C2×C283—21no—
156C2×C283—22no—
157C2×C283—21no—
158C2×C283—22no—
159C2×C283—21no—
160C2×C283—42no—
161C2×C283C2×#12011yescharacteristic
162D283—44no—
163D283—81no—
164C283#302611yescharacteristic, complemented, Sylow
165#3026—2831nomaximal, Sylow
166C2×H(5,𝔽2)—2831no—
167#534—2831no—
168#534—2832no—
169#458—2834no—
170B(4,𝔽2)—2834no—
171C2×#99—2832no—
172C2×#99D28311yescharacteristic, complemented
173H(5,𝔽2)—2834no—
174H(5,𝔽2)—5664no—
175H(5,𝔽2)—5662no—
176C2×#41—5662no—
177C2×#41—5661no—
178C22×D4—2832no—
179C22×D4—2831no—
180C22×D4—5662no—
181C22×D4—5661no—
182C22×D4C2×D28311yescharacteristic
183#123—2832no—
184#123—2834no—
185#123—5662no—
186#123—5661no—
187#121—5664no—
188#121—5662no—
189#120—2832no—
190#120—2834no—
191#120—5662no—
192#120—5661no—
193C2×C22⋊C4—2831no—
194C2×C22⋊C4C2×D28312yescomplemented
195#99—2838no—
196#99C2×D28314yescomplemented
197C24—2832no—
198C24—2831no—
199C24—5661no—
200C24D113211yescharacteristic, complemented
201C24C22×D28311yescharacteristic
202#41—5664no—
203#41—5668no—
204#41—5668no—
205#41—5664no—
206C2×Q8—5661no—
207C2×D4—22no—
208C2×D4—22no—
209C2×D4—2834no—
210C2×D4—2834no—
211C2×D4—5661no—
212C2×D4—5664no—
213C2×D4—5668no—
214C2×D4—5668no—
215C2×D4—5662no—
216C2×D4—5664no—
217C2×D4—5662no—
218C2×D4—5662no—
219C2×D4—5662no—
220C2×D4—5662no—
221C2×D4—5664no—
222C2×D4—5662no—
223C2×D4—5662no—
224C2×D4—5661no—
225C2×D4—5661no—
226C2×D4—5662no—
227C2×D4D113212yescomplemented
228C2×D4C22×D28312yes—
229C22×C4—22no—
230C22×C4—2832no—
231C22×C4—2831no—
232C22×C4—5662no—
233C22×C4—5661no—
234C22×C4—5661no—
235C22×C4D113211yescharacteristic, complemented
236C4⋊C4—5664no—
237C4⋊C4—5662no—
238C22⋊C4—22no—
239C22⋊C4—2832no—
240C22⋊C4—5664no—
241C22⋊C4—5664no—
242C22⋊C4—5661no—
243C22⋊C4—5662no—
244C22⋊C4—5662no—
245C22⋊C4C22×D28314yescomplemented
246C23—21no—
247C23—21no—
248C23—81no—
249C23—81no—
250C23—2834no—
251C23—5664no—
252C23—5662no—
253C23—5662no—
254C23—5662no—
255C23—5661no—
256C23—22641no—
257C23—22642no—
258C23—22642no—
259C23D4×D28312yes—
260C23D4×D28312yes—
261C23C2×D113212yes—
262C23C2×D113211yescharacteristic
263C23C2×D113211yescharacteristic
264C23C23×D28311yescharacteristic
265Q8—5664no—
266D4—24no—
267D4—24no—
268D4—5664no—
269D4—5668no—
270D4—5664no—
271D4—5664no—
272D4—5664no—
273D4—5664no—
274D4—11321no—
275D4—11321no—
276D4—11322no—
277D4—22644no—
278D4—22642no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—24no—
282C2×C4—82no—
283C2×C4—2834no—
284C2×C4—5668no—
285C2×C4—5664no—
286C2×C4—5662no—
287C2×C4—5662no—
288C2×C4—5662no—
289C2×C4—5662no—
290C2×C4—5661no—
291C2×C4—22641no—
292C2×C4C2×D113212yes—
293C22—22no—
294C22—42no—
295C22—41no—
296C22—41no—
297C22—42no—
298C22—41no—
299C22—42no—
300C22—42no—
301C22—81no—
302C22—5664no—
303C22—11324no—
304C22—11324no—
305C22—11324no—
306C22—11322no—
307C22—11321no—
308C22—22642no—
309C22—22644no—
310C22#2258358011yescharacteristic
311C22#2258358011yescharacteristic
312C22#2258358012yes—
313C22C2×D4×D28312yes—
314C22C22×D113211yescharacteristic
315C4—22no—
316C4—44no—
317C4—5664no—
318C4—11322no—
319C4—11321no—
320C2—22no—
321C2—21no—
322C2—21no—
323C2—21no—
324C2—22no—
325C2—42no—
326C2—11324no—
327C2—22641no—
328C2C2×#2258358011yescharacteristic
subgroupG/H
centerC2C2×#22583580
commutator, cosocleC23×C283C24
FrattiniC23C23×D283
FittingC2×#99×C283C2
socleC2×C283C2×#120
hypercenterC2×#99D283
hypocenterC283#3026
LCSG ▷ C23×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C2 ◁ C24 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
1. C2×#22583655C2semidirect
2. #31827027C2semidirect
3. #31827024C2semidirect
4. #31826656C2semidirect
5. #31826653C2semidirect
6. C2×#22583524C2semidirect
7. C2×#99×C283C2semidirect
15. C22×C22⋊D283C22semidirect
18. #22583594C22semidirect
20. #22583591C22semidirect
24. #22583580C22semidirect
26. #22583578C22semidirect
27. C2×C22⋊Dic283C22semidirect
28. #22583524C22semidirect
30. C2×C22⋊C4×C283C22semidirect
31. #99×C283C22semidirect
33. C23×D283D4semidirect
34. C23×D283C23semidirect
54. C2×C22⋊D283D4semidirect
60. C22×Dic283D4semidirect
61. C22×Dic283C23semidirect
67. C22⋊Dic283C23semidirect
71. C24×C283D4semidirect
75. C2×D4×C283D4semidirect
78. C22×C4×C283D4semidirect
80. C22⋊C4×C283C23semidirect
164. C283#3026semidirect
172. C2×#99D283semidirect
194. C2×C22⋊C4C2×D283semidirect
196. #99C2×D283semidirect
200. C24D1132semidirect
227. C2×D4D1132semidirect
235. C22×C4D1132semidirect
245. C22⋊C4C22×D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225836552
#318270272
#318270242
#318266562
#318266532
C2×#225835242
C2×#99×C2832
#3026283