Finite Group Databasebeta

← #49131195

Subgroups — #49131195

Order 36224.

414 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 125 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
1#31826977C211yesmaximal, characteristic
2C2×#22583568C211yesmaximal, characteristic, complemented
3#31826809C212yesmaximal, complemented
4#31826808C212yesmaximal, complemented
5#31826792C212yesmaximal, complemented
6#31826791C212yesmaximal, complemented
7#31826508C211yesmaximal, characteristic, complemented
8#31826467C211yesmaximal, characteristic, complemented
9#31826464C211yesmaximal, characteristic, complemented
10#31826428C211yesmaximal, characteristic, complemented
11#428×C283C211yesmaximal, characteristic, complemented
12C2×#17985392C2211yescharacteristic
13C22×C4×D283C2211yescharacteristic
14C2×#17985376C2211yescharacteristic
15#22583594—21no—
16#22583590C2211yescharacteristic
17#22583590C2211yescharacteristic
18#22583585C2211yescharacteristic
19#22583585C2211yescharacteristic
20#22583580—21no—
21C4×D1132—21no—
22C4×D1132—21no—
23#22583568—21no—
24#22583568—21no—
25#22583568C2212yes—
26#22583568C2212yes—
27C2×C4⋊Dic283C2211yescharacteristic
28C2×(C2×D283)⋊C4—21no—
29C2×C4×Dic283—21no—
30#22583543C2212yescomplemented
31#22583541C2212yescomplemented
32#22583535C2212yescomplemented
33#22583533C2212yescomplemented
34#22583532C2212yescomplemented
35#22583530C2212yescomplemented
36(C2×D283)⋊C8C2212yes—
37(C2×D283)⋊C8C2212yes—
38C2×M16×C283C2211yescharacteristic
39#117×C283C2211yescharacteristic
40C2×C4⋊C4×C283C2211yescharacteristic
41#107×C283C2212yescomplemented
42#106×C283C2212yescomplemented
43C23×D283D411yescharacteristic
44#17985392—22no—
45#17985392—22no—
46C2×Dic566C2311yescharacteristic
47C2×D1132—21no—
48C2×D1132—21no—
49C2×D1132C2311yescharacteristic
50C2×C22⋊D283—21no—
51C22×Dic283—21no—
52C22×Dic283D411yescharacteristic
53C2×C4×D283—21no—
54C2×C4×D283—22no—
55C2×C4×D283—22no—
56C2×C4×D283D411yescharacteristic
57C2×C4×D283D411yescharacteristic
58C2×C4×D283C2311yescharacteristic
59C2×C4×D283C2311yescharacteristic
60#17985376—22no—
61C2×C283⋊C8C2312yes—
62C4⋊Dic283—21no—
63C4⋊Dic283—21no—
64C4⋊Dic283—21no—
65C4⋊Dic283C2312yes—
66(C2×D283)⋊C4—22no—
67(C2×D283)⋊C4—22no—
68(C2×D283)⋊C4—21no—
69(C2×D283)⋊C4—21no—
70(C2×D283)⋊C4—21no—
71(C2×D283)⋊C4—21no—
72C4×Dic283—22no—
73C4×Dic283—22no—
74C22×C4×C283—21no—
75C22×C4×C283C2311yescharacteristic
76M16×C283—22no—
77C2×C8×C283C2312yes—
78C4⋊C4×C283—21no—
79C4⋊C4×C283C2312yes—
80C4⋊C4×C283C2311yescharacteristic
81C4⋊C4×C283C2311yescharacteristic
82C22⋊C4×C283—21no—
83C42×C283—21no—
84C22×D283—22no—
85C22×D283—21no—
86C22×D283—41no—
87C22×D283—41no—
88C22×D283C2×D411yescharacteristic
89C22×D283C2×D411yescharacteristic
90Dic566—21no—
91Dic566—21no—
92D1132—21no—
93D1132—21no—
94D1132—42no—
95D1132—42no—
96C22⋊D283—42no—
97C2×Dic283—22no—
98C2×Dic283—22no—
99C2×Dic283—21no—
100C2×Dic283—42no—
101C2×Dic283C2×D411yescharacteristic
102C2×Dic283C2×D411yescharacteristic
103C4×D283—22no—
104C4×D283—22no—
105C4×D283—21no—
106C4×D283—21no—
107C4×D283—21no—
108C4×D283—21no—
109C4×D283—42no—
110C283⋊C8—22no—
111C23×C283C2×D411yescharacteristic
112C2×C4×C283—21no—
113C2×C4×C283—21no—
114C2×C4×C283—22no—
115C2×C4×C283—42no—
116C2×C4×C283C2×D411yescharacteristic
117C2×C4×C283#4112yes—
118C2×C4×C283#4112yes—
119C2×C4×C283C2411yescharacteristic
120C8×C283—22no—
121C22×C283—21no—
122C22×C283—21no—
123C22×C283#12312yes—
124C22×C283C22×D411yescharacteristic
125C2×D283—21no—
126C2×D283—21no—
127C2×D283—21no—
128C2×D283—21no—
129C2×D283—21no—
130C2×D283—21no—
131C2×D283—44no—
132C2×D283—41no—
133C2×D283—41no—
134C2×D283—41no—
135C4×C283—42no—
136C4×C283—41no—
137C4×C283—41no—
138C4×C283#12312yes—
139C4×C283C2×#4111yescharacteristic
140C4×C283C2×#4111yescharacteristic
141Dic283—21no—
142Dic283—21no—
143Dic283—41no—
144Dic283—42no—
145C2×C283—22no—
146C2×C283C2×#12311yescharacteristic
147C2×C283#57511yescharacteristic
148C2×C283#57511yescharacteristic
149D283—41no—
150D283—41no—
151D283—81no—
152C283#309311yescharacteristic, complemented, Sylow
153#3093—2831nomaximal, Sylow
154#525—2831no—
155C2×#117—2831no—
156#485—2832no—
157#485—2832no—
158#482—2832no—
159#482—2832no—
160#428—2831no—
161#428D28311yescharacteristic, complemented
162#417—2831no—
163#417—2831no—
164#407—2831no—
165C2×#41—2831no—
166C23×C4—2831no—
167C2×M16—2831no—
168C2×M16C2×D28311yescharacteristic
169#123—5661no—
170C22⋊Q8—2831no—
171C22⋊Q8—2831no—
172#121—2831no—
173#121—2831no—
174#120—5661no—
175C4×D4—5661no—
176C4×D4—5661no—
177#117—2832no—
178#117—2832no—
179#117—5661no—
180#117—5661no—
181#117C2×D28311yescharacteristic
182C2×C4⋊C4—2831no—
183C2×C4⋊C4C2×D28311yescharacteristic
184C2×C22⋊C4—5661no—
185C2×C42—5661no—
186#107—2832no—
187#107C2×D28312yescomplemented
188#106—2832no—
189#106C2×D28312yescomplemented
190#103—2832no—
191#103—2832no—
192#102—2832no—
193#102—2832no—
194C22⋊C8—2832no—
195C22⋊C8—2832no—
196C24—2831no—
197#41—5662no—
198#41—5662no—
199C2×Q8—2831no—
200C2×D4—2831no—
201C2×D4—5661no—
202C2×D4—5661no—
203C2×D4—5661no—
204C22×C4—21no—
205C22×C4—2831no—
206C22×C4—2831no—
207C22×C4—2831no—
208C22×C4—2831no—
209C22×C4—2831no—
210C22×C4—5662no—
211C22×C4—5662no—
212C22×C4—5661no—
213C22×C4—5661no—
214C22×C4C22×D28311yescharacteristic
215M16—22no—
216M16—5662no—
217C2×C8—2832no—
218C2×C8C22×D28312yes—
219C4⋊C4—21no—
220C4⋊C4—2832no—
221C4⋊C4—5661no—
222C4⋊C4—5661no—
223C4⋊C4—5661no—
224C4⋊C4C22×D28312yes—
225C4⋊C4C22×D28311yescharacteristic
226C4⋊C4C22×D28311yescharacteristic
227C22⋊C4—21no—
228C22⋊C4—5661no—
229C22⋊C4—5661no—
230C22⋊C4—5662no—
231C22⋊C4—5662no—
232C22⋊C4—5661no—
233C22⋊C4—5661no—
234C42—21no—
235C42—5662no—
236C42—5662no—
237C23—2831no—
238C23—2831no—
239C23—5662no—
240C23—5661no—
241C23—11321no—
242C23—11321no—
243C23D4×D28311yescharacteristic
244Q8—5661no—
245Q8—5661no—
246D4—5661no—
247D4—5661no—
248D4—11322no—
249D4—11322no—
250D4—11322no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—42no—
255C2×C4—2831no—
256C2×C4—2831no—
257C2×C4—5662no—
258C2×C4—5662no—
259C2×C4—5662no—
260C2×C4—5661no—
261C2×C4—5661no—
262C2×C4—5661no—
263C2×C4—5661no—
264C2×C4—5662no—
265C2×C4—5661no—
266C2×C4—11322no—
267C2×C4—11322no—
268C2×C4D4×D28311yescharacteristic
269C2×C4#1798539212yes—
270C2×C4#1798539312yes—
271C2×C4C23×D28311yescharacteristic
272C8—22no—
273C8—5662no—
274C22—21no—
275C22—21no—
276C22—5661no—
277C22—5661no—
278C22—5661no—
279C22—5661no—
280C22—5661no—
281C22—5661no—
282C22—11324no—
283C22—11321no—
284C22—11321no—
285C22—11321no—
286C22#2258359512yes—
287C22C2×D4×D28311yescharacteristic
288C4—42no—
289C4—41no—
290C4—41no—
291C4—5661no—
292C4—5661no—
293C4—11321no—
294C4—11322no—
295C4#2258359512yes—
296C4C2×#1798539211yescharacteristic
297C4C2×#1798539311yescharacteristic
298C2—22no—
299C2—11321no—
300C2—11321no—
301C2—22641no—
302C2C2×#2258359511yescharacteristic
303C2#3182726111yescharacteristic
304C2#3182726411yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#428×C283C2
socleC22×C283C22×D4
prime radical#31826977C2
hypercenter#428D283
hypocenterC283#3093
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
2. C2×#22583568C2semidirect
3. #31826809C2semidirect
4. #31826808C2semidirect
5. #31826792C2semidirect
6. #31826791C2semidirect
7. #31826508C2semidirect
8. #31826467C2semidirect
9. #31826464C2semidirect
10. #31826428C2semidirect
11. #428×C283C2semidirect
30. #22583543C22semidirect
31. #22583541C22semidirect
32. #22583535C22semidirect
33. #22583533C22semidirect
34. #22583532C22semidirect
35. #22583530C22semidirect
41. #107×C283C22semidirect
42. #106×C283C22semidirect
152. C283#3093semidirect
161. #428D283semidirect
187. #107C2×D283semidirect
189. #106C2×D283semidirect

Maximal subgroups

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

subgroupindex
#318269772
C2×#225835682
#318268092
#318268082
#318267922
#318267912
#318265082
#318264672
#318264642
#318264282
#428×C2832
#3093283