Finite Group Databasebeta

← #49131187

Subgroups — #49131187

Order 36224.

458 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 129 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×C4⋊C4×D283C211yesmaximal, characteristic, complemented
3#31826802C212yesmaximal, complemented
4#31826797C212yesmaximal, complemented
5#31826785C212yesmaximal, complemented
6#31826780C212yesmaximal, 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×D283—21no—
14C22×C4×D283C2211yescharacteristic
15C2×#17985376C2211yescharacteristic
16#22583594—21no—
17#22583590C2211yescharacteristic
18#22583590C2211yescharacteristic
19#22583585C2211yescharacteristic
20#22583585C2211yescharacteristic
21#22583580—21no—
22C4×D1132—21no—
23C4×D1132—21no—
24C2×C4⋊Dic283C2211yescharacteristic
25C2×Dic283⋊C4—21no—
26C4⋊C4×D283—21no—
27C4⋊C4×D283—21no—
28C4⋊C4×D283C2212yes—
29C4⋊C4×D283C2212yes—
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—22no—
54C2×C4×D283—22no—
55C2×C4×D283—21no—
56C2×C4×D283—22no—
57C2×C4×D283—22no—
58C2×C4×D283—44no—
59C2×C4×D283D411yescharacteristic
60C2×C4×D283D411yescharacteristic
61C2×C4×D283C2311yescharacteristic
62C2×C4×D283C2311yescharacteristic
63#17985376—22no—
64C2×C283⋊C8C2312yes—
65C4⋊Dic283—21no—
66C4⋊Dic283—21no—
67C4⋊Dic283—21no—
68C4⋊Dic283C2312yes—
69Dic283⋊C4—22no—
70Dic283⋊C4—22no—
71(C2×D283)⋊C4—21no—
72(C2×D283)⋊C4—21no—
73(C2×D283)⋊C4—21no—
74(C2×D283)⋊C4—21no—
75C22×C4×C283—21no—
76C22×C4×C283C2311yescharacteristic
77M16×C283—22no—
78C2×C8×C283C2312yes—
79C4⋊C4×C283—21no—
80C4⋊C4×C283C2312yes—
81C4⋊C4×C283C2311yescharacteristic
82C4⋊C4×C283C2311yescharacteristic
83C22⋊C4×C283—21no—
84C42×C283—21no—
85C22×D283—22no—
86C22×D283—22no—
87C22×D283—22no—
88C22×D283—21no—
89C22×D283C2×D411yescharacteristic
90C22×D283C2×D411yescharacteristic
91Dic566—21no—
92Dic566—21no—
93D1132—21no—
94D1132—21no—
95D1132—42no—
96D1132—42no—
97C22⋊D283—42no—
98C2×Dic283—22no—
99C2×Dic283—21no—
100C2×Dic283—22no—
101C2×Dic283—42no—
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—44no—
111C4×D283—44no—
112C4×D283—42no—
113C283⋊C8—22no—
114C23×C283C2×D411yescharacteristic
115C2×C4×C283—21no—
116C2×C4×C283—21no—
117C2×C4×C283—22no—
118C2×C4×C283—42no—
119C2×C4×C283C2×D411yescharacteristic
120C2×C4×C283#4112yes—
121C2×C4×C283#4112yes—
122C2×C4×C283C2411yescharacteristic
123C8×C283—22no—
124C22×C283—21no—
125C22×C283—21no—
126C22×C283#12312yes—
127C22×C283C22×D411yescharacteristic
128C2×D283—28no—
129C2×D283—21no—
130C2×D283—21no—
131C2×D283—21no—
132C2×D283—21no—
133C2×D283—41no—
134C2×D283—41no—
135C2×D283—41no—
136C2×D283AGL(1,ℤ/8)12yes—
137C2×D283#13712yes—
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—42no—
148C2×C283—22no—
149C2×C283C2×#12311yescharacteristic
150C2×C283C2×AGL(1,ℤ/8)11yescharacteristic
151C2×C283C2×#13711yescharacteristic
152D283—22no—
153D283—22no—
154D283—81no—
155C283#309211yescharacteristic, complemented, Sylow
156#3092—2831nomaximal, Sylow
157#525—2831no—
158C22×C4⋊C4—2831no—
159#484—2832no—
160#483—2832no—
161#481—2832no—
162#480—2832no—
163#428—2831no—
164#428D28311yescharacteristic, complemented
165#417—2831no—
166#417—2831no—
167#407—2831no—
168C2×#41—2831no—
169C23×C4—2831no—
170C23×C4—5661no—
171C2×M16—2831no—
172C2×M16C2×D28311yescharacteristic
173#123—5661no—
174C22⋊Q8—2831no—
175C22⋊Q8—2831no—
176#121—2831no—
177#121—2831no—
178#120—5661no—
179C4×D4—5661no—
180C4×D4—5661no—
181#117C2×D28311yescharacteristic
182C2×C4⋊C4—2831no—
183C2×C4⋊C4—2832no—
184C2×C4⋊C4—2832no—
185C2×C4⋊C4—5661no—
186C2×C4⋊C4—5661no—
187C2×C4⋊C4—5661no—
188C2×C4⋊C4C2×D28311yescharacteristic
189#107—2832no—
190#107C2×D28312yescomplemented
191#106—2832no—
192#106C2×D28312yescomplemented
193#103—2832no—
194#103—2832no—
195#102—2832no—
196#102—2832no—
197C22⋊C8—2832no—
198C22⋊C8—2832no—
199C24—2831no—
200#41—5662no—
201#41—5662no—
202C2×Q8—2831no—
203C2×D4—2831no—
204C2×D4—5661no—
205C2×D4—5661no—
206C2×D4—5661no—
207C22×C4—21no—
208C22×C4—2831no—
209C22×C4—2831no—
210C22×C4—2831no—
211C22×C4—2831no—
212C22×C4—2831no—
213C22×C4—5662no—
214C22×C4—5662no—
215C22×C4—5662no—
216C22×C4—5662no—
217C22×C4—5661no—
218C22×C4—5661no—
219C22×C4—11324no—
220C22×C4C22×D28311yescharacteristic
221M16—22no—
222M16—5662no—
223C2×C8—2832no—
224C2×C8C22×D28312yes—
225C4⋊C4—21no—
226C4⋊C4—2832no—
227C4⋊C4—5662no—
228C4⋊C4—5662no—
229C4⋊C4—5661no—
230C4⋊C4—5661no—
231C4⋊C4—5661no—
232C4⋊C4C22×D28312yes—
233C4⋊C4C22×D28311yescharacteristic
234C4⋊C4C22×D28311yescharacteristic
235C22⋊C4—21no—
236C22⋊C4—5661no—
237C22⋊C4—5661no—
238C22⋊C4—5661no—
239C22⋊C4—5661no—
240C42—21no—
241C23—2831no—
242C23—2831no—
243C23—5662no—
244C23—5662no—
245C23—5662no—
246C23—5661no—
247C23D4×D28311yescharacteristic
248Q8—5661no—
249Q8—5661no—
250D4—5661no—
251D4—5661no—
252D4—11322no—
253D4—11322no—
254D4—11322no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—42no—
259C2×C4—2831no—
260C2×C4—2831no—
261C2×C4—5662no—
262C2×C4—5662no—
263C2×C4—5662no—
264C2×C4—5661no—
265C2×C4—5661no—
266C2×C4—5661no—
267C2×C4—5661no—
268C2×C4—5661no—
269C2×C4—5662no—
270C2×C4—11324no—
271C2×C4—11324no—
272C2×C4—11322no—
273C2×C4—11322no—
274C2×C4D4×D28311yescharacteristic
275C2×C4#1798539212yes—
276C2×C4#1798539312yes—
277C2×C4C23×D28311yescharacteristic
278C8—22no—
279C8—5662no—
280C22—21no—
281C22—21no—
282C22—2832no—
283C22—2832no—
284C22—5668no—
285C22—5661no—
286C22—5661no—
287C22—5661no—
288C22—5661no—
289C22—11321no—
290C22—11321no—
291C22—11321no—
292C22#2258359512yes—
293C22C2×D4×D28311yescharacteristic
294C4—42no—
295C4—41no—
296C4—41no—
297C4—5661no—
298C4—5661no—
299C4—11321no—
300C4—11322no—
301C4#2258359512yes—
302C4C2×#1798539211yescharacteristic
303C4C2×#1798539311yescharacteristic
304C2—22no—
305C2—5662no—
306C2—5662no—
307C2—22641no—
308C2C2×#2258359511yescharacteristic
309C2AGL(1,ℤ/8)×D28311yescharacteristic
310C2#137×D28311yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#428×C283C2
socleC22×C283C22×D4
prime radical#31826977C2
hypercenter#428D283
hypocenterC283#3092
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×C4⋊C4×D283C2semidirect
3. #31826802C2semidirect
4. #31826797C2semidirect
5. #31826785C2semidirect
6. #31826780C2semidirect
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
155. C283#3092semidirect
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
#318269772
C2×C4⋊C4×D2832
#318268022
#318267972
#318267852
#318267802
#318265082
#318264672
#318264642
#318264282
#428×C2832
#3092283