Finite Group Databasebeta

← #49131203

Subgroups — #49131203

Order 36224.

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

normal subgroups only

This group has 462 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×#22583585C211yesmaximal, characteristic
2#117×D283C211yesmaximal, characteristic, complemented
3#31826792C212yesmaximal, complemented
4#31826791C212yesmaximal, complemented
5#31826785C212yesmaximal, complemented
6#31826780C212yesmaximal, complemented
7#31826505C211yesmaximal, characteristic, complemented
8#31826471C211yesmaximal, characteristic, complemented
9#31826469C211yesmaximal, characteristic, complemented
10#31826428C211yesmaximal, characteristic, complemented
11#428×C283C211yesmaximal, characteristic, complemented
12C22×D1132—21no—
13C22×D1132C2211yescharacteristic
14C22×C4×D283C2211yescharacteristic
15C2×#17985376C2211yescharacteristic
16#22583585—21no—
17#22583585—21no—
18#22583585C2212yes—
19#22583585C2212yes—
20#22583569C2211yescharacteristic
21#22583568C2211yescharacteristic, complemented
22#22583568C2211yescharacteristic, complemented
23#22583567—21no—
24#22583566—21no—
25C4⋊C4×D283C2211yescharacteristic, complemented
26C4⋊C4×D283C2211yescharacteristic, complemented
27C2×(C2×D283)⋊C4—21no—
28C22⋊C4×D283—21no—
29C42×D283—21no—
30#22583543C2212yescomplemented
31#22583541C2212yescomplemented
32#22583532C2212yescomplemented
33#22583532C2212yescomplemented
34#22583530C2212yescomplemented
35#22583530C2212yescomplemented
36(C2×D283)⋊C8C2212yescomplemented
37(C2×D283)⋊C8C2212yescomplemented
38C2×M16×C283C2211yescharacteristic
39#117×C283C2211yescharacteristic
40C2×C4⋊C4×C283C2211yescharacteristic
41#107×C283C2212yescomplemented
42#106×C283C2212yescomplemented
43C23×D283—21no—
44C23×D283D411yescharacteristic
45C2×D1132—22no—
46C2×D1132—22no—
47C2×D1132—21no—
48C2×D1132—22no—
49C2×D1132—22no—
50C2×D1132—44no—
51C2×D1132C2312yes—
52C22×Dic283D411yescharacteristic
53C2×C4×D283—21no—
54C2×C4×D283—21no—
55C2×C4×D283—22no—
56C2×C4×D283—22no—
57C2×C4×D283D411yescharacteristic
58C2×C4×D283D411yescharacteristic
59C2×C4×D283C2311yescharacteristic
60C2×C4×D283C2311yescharacteristic
61#17985376—22no—
62C2×C283⋊C8C2312yes—
63C4⋊Dic283C2311yescharacteristic
64C4⋊Dic283C2311yescharacteristic
65Dic283⋊C4—21no—
66Dic283⋊C4—21no—
67C22⋊Dic283—21no—
68(C2×D283)⋊C4—21no—
69(C2×D283)⋊C4—21no—
70(C2×D283)⋊C4—22no—
71(C2×D283)⋊C4—22no—
72C4×Dic283—21no—
73C4×Dic283—21no—
74C4×Dic283—21no—
75C22×C4×C283—21no—
76C22×C4×C283C2311yescharacteristic
77M16×C283—22no—
78C2×C8×C283C2312yes—
79C4⋊C4×C283—21no—
80C4⋊C4×C283C2311yescharacteristic
81C4⋊C4×C283C2311yescharacteristic
82C4⋊C4×C283C2312yes—
83C22⋊C4×C283—21no—
84C42×C283—21no—
85C22×D283—22no—
86C22×D283—22no—
87C22×D283—22no—
88C22×D283—41no—
89C22×D283—42no—
90C22×D283—42no—
91C22×D283—42no—
92C22×D283C2×D411yescharacteristic
93C22×D283C2×D411yescharacteristic
94D1132—22no—
95D1132—22no—
96D1132—44no—
97D1132—44no—
98D1132—42no—
99C2×Dic283—22no—
100C2×Dic283—21no—
101C2×Dic283—21no—
102C2×Dic283C2×D411yescharacteristic
103C2×Dic283C2×D411yescharacteristic
104C4×D283—22no—
105C4×D283—22no—
106C4×D283—21no—
107C4×D283—21no—
108C4×D283—21no—
109C4×D283—21no—
110C4×D283—42no—
111C4×D283—42no—
112C283⋊C8—22no—
113C23×C283C2×D411yescharacteristic
114C2×C4×C283—21no—
115C2×C4×C283—21no—
116C2×C4×C283—22no—
117C2×C4×C283—42no—
118C2×C4×C283C2×D411yescharacteristic
119C2×C4×C283#4112yes—
120C2×C4×C283#4112yes—
121C2×C4×C283C2411yescharacteristic
122C8×C283—22no—
123C22×C283—21no—
124C22×C283—21no—
125C22×C283#12312yes—
126C22×C283C22×D411yescharacteristic
127C2×D283—21no—
128C2×D283—21no—
129C2×D283—21no—
130C2×D283—21no—
131C2×D283—21no—
132C2×D283—44no—
133C2×D283—42no—
134C2×D283—42no—
135C2×D283—42no—
136C2×D283—84no—
137C2×D283AGL(1,ℤ/8)12yes—
138C4×C283—42no—
139C4×C283—41no—
140C4×C283—41no—
141C4×C283#12312yes—
142C4×C283C2×#4111yescharacteristic
143C4×C283C2×#4111yescharacteristic
144Dic283—21no—
145Dic283—21no—
146Dic283—41no—
147Dic283—41no—
148C2×C283—22no—
149C2×C283C2×#12311yescharacteristic
150C2×C283C2×AGL(1,ℤ/8)11yescharacteristic
151C2×C283#57511yescharacteristic
152D283—22no—
153D283—41no—
154D283—82no—
155C283#309411yescharacteristic, complemented, Sylow
156#3094—2831nomaximal, Sylow
157C2×#121—2831no—
158C2×#117—2831no—
159#482—2832no—
160#482—2832no—
161#481—2832no—
162#480—2832no—
163#428—2831no—
164#428D28311yescharacteristic, complemented
165#418—2831no—
166#418—2831no—
167#407—2831no—
168C22×D4—2831no—
169C22×D4—5661no—
170C23×C4—2831no—
171C2×M16—2831no—
172C2×M16C2×D28311yescharacteristic
173#121—2832no—
174#121—2832no—
175#121—5661no—
176#121—5661no—
177#117—2831no—
178#117—2831no—
179#117—2831no—
180#117—5661no—
181#117—5661no—
182#117C2×D28311yescharacteristic
183C2×C4⋊C4—2831no—
184C2×C4⋊C4—2831no—
185C2×C4⋊C4C2×D28311yescharacteristic
186C2×C22⋊C4—5661no—
187C2×C22⋊C4—5661no—
188C2×C42—5661no—
189#107—2832no—
190#107C2×D28312yescomplemented
191#106—2832no—
192#106C2×D28312yescomplemented
193#102—2832no—
194#102—2832no—
195#102—2832no—
196#102—2832no—
197C22⋊C8—2832no—
198C22⋊C8—2832no—
199C24—2831no—
200C24—5661no—
201C2×D4—2832no—
202C2×D4—5662no—
203C2×D4—5662no—
204C2×D4—5661no—
205C2×D4—5662no—
206C2×D4—5662no—
207C2×D4—11324no—
208C22×C4—21no—
209C22×C4—2831no—
210C22×C4—2831no—
211C22×C4—2831no—
212C22×C4—2831no—
213C22×C4—2831no—
214C22×C4—5661no—
215C22×C4—5661no—
216C22×C4—5662no—
217C22×C4—5662no—
218C22×C4C22×D28311yescharacteristic
219M16—22no—
220M16—5662no—
221C2×C8—2832no—
222C2×C8C22×D28312yes—
223C4⋊C4—21no—
224C4⋊C4—2831no—
225C4⋊C4—2831no—
226C4⋊C4—5661no—
227C4⋊C4—5661no—
228C4⋊C4C22×D28311yescharacteristic
229C4⋊C4C22×D28311yescharacteristic
230C4⋊C4C22×D28312yes—
231C22⋊C4—21no—
232C22⋊C4—5662no—
233C22⋊C4—5662no—
234C22⋊C4—5661no—
235C22⋊C4—5661no—
236C22⋊C4—5661no—
237C42—21no—
238C42—5661no—
239C42—5661no—
240C42—5661no—
241C23—2831no—
242C23—2831no—
243C23—5662no—
244C23—5662no—
245C23—5662no—
246C23—11321no—
247C23—11322no—
248C23—11322no—
249C23—11322no—
250C23D4×D28311yescharacteristic
251D4—5662no—
252D4—5662no—
253D4—11324no—
254D4—11324no—
255D4—11322no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—42no—
260C2×C4—2831no—
261C2×C4—2831no—
262C2×C4—5662no—
263C2×C4—5662no—
264C2×C4—5662no—
265C2×C4—5661no—
266C2×C4—5661no—
267C2×C4—5661no—
268C2×C4—5661no—
269C2×C4—5661no—
270C2×C4—5661no—
271C2×C4—11322no—
272C2×C4—11322no—
273C2×C4D4×D28311yescharacteristic
274C2×C4#1798539212yes—
275C2×C4#1798539312yes—
276C2×C4C23×D28311yescharacteristic
277C8—22no—
278C8—5662no—
279C22—21no—
280C22—21no—
281C22—2832no—
282C22—5661no—
283C22—5661no—
284C22—5661no—
285C22—5661no—
286C22—5661no—
287C22—11324no—
288C22—11322no—
289C22—11322no—
290C22—11322no—
291C22—22644no—
292C22#2258359512yes—
293C22C2×D4×D28311yescharacteristic
294C4—42no—
295C4—41no—
296C4—41no—
297C4—5661no—
298C4—5661no—
299C4—11321no—
300C4—11321no—
301C4#2258359512yes—
302C4C2×#1798539211yescharacteristic
303C4C2×#1798539311yescharacteristic
304C2—22no—
305C2—5662no—
306C2—11321no—
307C2—22642no—
308C2C2×#2258359511yescharacteristic
309C2AGL(1,ℤ/8)×D28311yescharacteristic
310C2#3182726411yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#428×C283C2
socleC22×C283C22×D4
prime radicalC2×#22583585C2
hypercenter#428D283
hypocenterC283#3094
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. #117×D283C2semidirect
3. #31826792C2semidirect
4. #31826791C2semidirect
5. #31826785C2semidirect
6. #31826780C2semidirect
7. #31826505C2semidirect
8. #31826471C2semidirect
9. #31826469C2semidirect
10. #31826428C2semidirect
11. #428×C283C2semidirect
21. #22583568C22semidirect
22. #22583568C22semidirect
25. C4⋊C4×D283C22semidirect
26. C4⋊C4×D283C22semidirect
30. #22583543C22semidirect
31. #22583541C22semidirect
32. #22583532C22semidirect
33. #22583532C22semidirect
34. #22583530C22semidirect
35. #22583530C22semidirect
36. (C2×D283)⋊C8C22semidirect
37. (C2×D283)⋊C8C22semidirect
41. #107×C283C22semidirect
42. #106×C283C22semidirect
155. C283#3094semidirect
164. #428D283semidirect
190. #107C2×D283semidirect
192. #106C2×D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225835852
#117×D2832
#318267922
#318267912
#318267852
#318267802
#318265052
#318264712
#318264692
#318264282
#428×C2832
#3094283