Finite Group Databasebeta

← #49131182

Subgroups — #49131182

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#31826972C211yesmaximal, characteristic
2C2×C4⋊C4×D283C211yesmaximal, characteristic, complemented
3#31826799C212yesmaximal, complemented
4#31826794C212yesmaximal, complemented
5#31826782C212yesmaximal, complemented
6#31826777C212yesmaximal, complemented
7#31826507C211yesmaximal, characteristic, complemented
8#31826505C211yesmaximal, characteristic, complemented
9#31826466C211yesmaximal, characteristic, complemented
10#31826428C211yesmaximal, characteristic, complemented
11#417×C283C211yesmaximal, characteristic, complemented
12C22×C4×D283—21no—
13C22×C4×D283C2211yescharacteristic
14C2×#17985376C2211yescharacteristic
15#22583591—21no—
16#22583586C2211yescharacteristic
17#22583586C2211yescharacteristic
18#22583581C2211yescharacteristic
19#22583581C2211yescharacteristic
20#22583578—21no—
21C4×C22⋊D283—21no—
22C4×C22⋊D283—21no—
23#22583569C2211yescharacteristic
24C2×C4⋊Dic283C2211yescharacteristic
25C2×Dic283⋊C4—21no—
26C4⋊C4×D283—21no—
27C4⋊C4×D283—21no—
28C4⋊C4×D283C2212yes—
29C4⋊C4×D283C2212yes—
30#22583544C2212yescomplemented
31#22583543C2212yescomplemented
32#22583542C2212yescomplemented
33#22583541C2212yescomplemented
34#22583534C2212yescomplemented
35#22583531C2212yescomplemented
36(C2×D283)⋊C8C2212yes—
37(C2×D283)⋊C8C2212yes—
38C2×#41×C283C2211yescharacteristic
39C2×M16×C283C2211yescharacteristic
40C2×C4⋊C4×C283C2211yescharacteristic
41#103×C283C2212yescomplemented
42#102×C283C2212yescomplemented
43C23×D283D411yescharacteristic
44C2×C22⋊D283—21no—
45C2×C22⋊D283—21no—
46C22×Dic283—21no—
47C22×Dic283D411yescharacteristic
48C2×C4×D283—22no—
49C2×C4×D283—22no—
50C2×C4×D283—22no—
51C2×C4×D283—22no—
52C2×C4×D283—44no—
53C2×C4×D283D411yescharacteristic
54C2×C4×D283D411yescharacteristic
55C2×C4×D283C2311yescharacteristic
56C2×C4×D283C2311yescharacteristic
57#17985376—22no—
58C2×C283⋊C8C2312yes—
59C4⋊Dic283—21no—
60C4⋊Dic283C2312yes—
61C4⋊Dic283C2311yescharacteristic
62C4⋊Dic283C2311yescharacteristic
63Dic283⋊C4—22no—
64Dic283⋊C4—22no—
65Dic283⋊C4—21no—
66Dic283⋊C4—21no—
67C22⋊Dic283—21no—
68C22⋊Dic283—21no—
69C22⋊Dic283—21no—
70(C2×D283)⋊C4—21no—
71(C2×D283)⋊C4—21no—
72C4×Dic283—21no—
73#41×C283—22no—
74#41×C283—22no—
75C2×Q8×C283C2311yescharacteristic
76C2×D4×C283—21no—
77C2×D4×C283C2311yescharacteristic
78C22×C4×C283—21no—
79C22×C4×C283—21no—
80C22×C4×C283C2311yescharacteristic
81M16×C283—22no—
82C2×C8×C283C2312yes—
83C4⋊C4×C283—21no—
84C4⋊C4×C283C2312yes—
85C22×D283—22no—
86C22×D283—22no—
87C22×D283—22no—
88C22×D283C2×D411yescharacteristic
89C22×D283C2×D411yescharacteristic
90C22⋊D283—42no—
91C22⋊D283—42no—
92C2×Dic283—22no—
93C2×Dic283—21no—
94C2×Dic283—21no—
95C2×Dic283—22no—
96C2×Dic283—42no—
97C2×Dic283C2×D411yescharacteristic
98C2×Dic283C2×D411yescharacteristic
99C4×D283—22no—
100C4×D283—22no—
101C4×D283—21no—
102C4×D283—21no—
103C4×D283—21no—
104C4×D283—21no—
105C4×D283—44no—
106C4×D283—44no—
107C283⋊C8—22no—
108C23×C283—21no—
109C23×C283C2×D411yescharacteristic
110Q8×C283—21no—
111Q8×C283—21no—
112D4×C283—21no—
113D4×C283—21no—
114D4×C283—42no—
115C2×C4×C283—22no—
116C2×C4×C283—21no—
117C2×C4×C283—42no—
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—41no—
127C22×C283—41no—
128C22×C283—41no—
129C22×C283#12312yes—
130C22×C283C22×D411yescharacteristic
131C2×D283—28no—
132C2×D283—21no—
133C2×D283—21no—
134C2×D283—21no—
135C2×D283—21no—
136C2×D283AGL(1,ℤ/8)12yes—
137C2×D283#13712yes—
138C4×C283—41no—
139C4×C283—42no—
140C4×C283#12312yes—
141C4×C283C2×#4111yescharacteristic
142C4×C283C2×#4111yescharacteristic
143Dic283—21no—
144Dic283—21no—
145Dic283—42no—
146Dic283—41no—
147Dic283—41no—
148C2×C283—22no—
149C2×C283—81no—
150C2×C283C2×#12311yescharacteristic
151C2×C283C2×AGL(1,ℤ/8)11yescharacteristic
152C2×C283C2×#13711yescharacteristic
153D283—22no—
154D283—22no—
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#428—2831no—
165#417—2831no—
166#417D28311yescharacteristic, complemented
167#407—2831no—
168C2×#41C2×D28311yescharacteristic
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#117—2831no—
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#107—2832no—
191#106—2832no—
192#106—2832no—
193#103—2832no—
194#103C2×D28312yescomplemented
195#102—2832no—
196#102C2×D28312yescomplemented
197C22⋊C8—2832no—
198C22⋊C8—2832no—
199C24—2831no—
200#41—22no—
201#41—22no—
202C2×Q8C22×D28311yescharacteristic
203C2×D4—21no—
204C2×D4—5661no—
205C2×D4—5661no—
206C2×D4C22×D28311yescharacteristic
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—2831no—
210C22×C4—2831no—
211C22×C4—2831no—
212C22×C4—2831no—
213C22×C4—2831no—
214C22×C4—5662no—
215C22×C4—5662no—
216C22×C4—5662no—
217C22×C4—5662no—
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—2831no—
228C4⋊C4—2831no—
229C4⋊C4—5662no—
230C4⋊C4—5662no—
231C4⋊C4—5661no—
232C4⋊C4—5661no—
233C4⋊C4—5661no—
234C4⋊C4C22×D28312yes—
235C22⋊C4—5661no—
236C22⋊C4—5661no—
237C22⋊C4—5661no—
238C22⋊C4—5661no—
239C22⋊C4—5661no—
240C42—5661no—
241C23—21no—
242C23—2831no—
243C23—2831no—
244C23—5662no—
245C23—5662no—
246C23—5662no—
247C23D4×D28311yescharacteristic
248Q8—21no—
249Q8—21no—
250D4—21no—
251D4—21no—
252D4—42no—
253D4—11322no—
254D4—11322no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—42no—
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—5661no—
270C2×C4—5662no—
271C2×C4—11324no—
272C2×C4—11324no—
273C2×C4—11322no—
274C2×C4D4×D28311yescharacteristic
275C2×C4#1798539112yes—
276C2×C4#1798539212yes—
277C2×C4C23×D28311yescharacteristic
278C8—22no—
279C8—5662no—
280C22—21no—
281C22—21no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—2832no—
286C22—2832no—
287C22—5668no—
288C22—5661no—
289C22—5661no—
290C22—5661no—
291C22—5661no—
292C22#2258359212yes—
293C22C2×D4×D28311yescharacteristic
294C4—41no—
295C4—42no—
296C4—5661no—
297C4—5661no—
298C4—11322no—
299C4—11321no—
300C4—11321no—
301C4#2258359212yes—
302C4C2×#1798539111yescharacteristic
303C4C2×#1798539211yescharacteristic
304C2—22no—
305C2—81no—
306C2—5662no—
307C2—5662no—
308C2C2×#2258359211yescharacteristic
309C2AGL(1,ℤ/8)×D28311yescharacteristic
310C2#137×D28311yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#417×C283C2
socleC22×C283C22×D4
prime radical#31826972C2
hypercenter#417D283
hypocenterC283#3092
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D283C2semidirect
3. #31826799C2semidirect
4. #31826794C2semidirect
5. #31826782C2semidirect
6. #31826777C2semidirect
7. #31826507C2semidirect
8. #31826505C2semidirect
9. #31826466C2semidirect
10. #31826428C2semidirect
11. #417×C283C2semidirect
30. #22583544C22semidirect
31. #22583543C22semidirect
32. #22583542C22semidirect
33. #22583541C22semidirect
34. #22583534C22semidirect
35. #22583531C22semidirect
41. #103×C283C22semidirect
42. #102×C283C22semidirect
155. C283#3092semidirect
166. #417D283semidirect
194. #103C2×D283semidirect
196. #102C2×D283semidirect

Maximal subgroups

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

subgroupindex
#318269722
C2×C4⋊C4×D2832
#318267992
#318267942
#318267822
#318267772
#318265072
#318265052
#318264662
#318264282
#417×C2832
#3092283