Finite Group Databasebeta

← #49131124

Subgroups — #49131124

Order 36224.

376 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31827095C211yesmaximal, characteristic, complemented
2#31826944C211yesmaximal, characteristic
3#31826771C211yesmaximal, characteristic, complemented
4#31826769C211yesmaximal, characteristic, complemented
5#31826767C211yesmaximal, characteristic, complemented
6#31826755C211yesmaximal, characteristic, complemented
7#31826753C211yesmaximal, characteristic, complemented
8#31826751C211yesmaximal, characteristic, complemented
9#31826749C211yesmaximal, characteristic, complemented
10#31826506C211yesmaximal, characteristic, complemented
11#106×D283C211yesmaximal, characteristic, complemented
12#31826490C211yesmaximal, characteristic, complemented
13#31826463C211yesmaximal, characteristic
14#31826458C211yesmaximal, characteristic
15#478×C283C211yesmaximal, characteristic, complemented
16C2×#17985391C2211yescharacteristic
17C2×#17985375C2211yescharacteristic
18C2×C8×D283C2211yescharacteristic
19#22583610C2211yescharacteristic
20#22583609—21no—
21#22583609C2211yescharacteristic
22#22583601C2211yescharacteristic
23#22583600—21no—
24#22583590—21no—
25#22583588—21no—
26C4×Dic566C2211yescharacteristic
27#22583575C2211yescharacteristic
28D4×Dic283C2211yescharacteristic
29C4×C22⋊D283—21no—
30(C22⋊D283)⋊C4—21no—
31#22583568C2211yescharacteristic
32#22583567C2211yescharacteristic
33#22583566—21no—
34C2×Dic283⋊C4—21no—
35C4⋊C4×D283C2211yescharacteristic
36C4⋊C4×D283C2211yescharacteristic
37#22583543C2211yescharacteristic
38#22583542C2211yescharacteristic
39#22583542C2211yescharacteristic
40#22583541C2211yescharacteristic
41#22583541C2211yescharacteristic
42#22583541C2211yescharacteristic
43C4⋊(C283⋊C8)C2211yescharacteristic
44Dic283⋊C8C2211yescharacteristic
45Dic283⋊C8C2211yescharacteristic
46#22583535C2211yescharacteristic
47#22583535C2211yescharacteristic
48#22583533C2211yescharacteristic
49#22583533C2211yescharacteristic
50#22583531C2211yescharacteristic
51#22583531C2211yescharacteristic
52#125×C283C2211yescharacteristic
53C4×D4×C283C2211yescharacteristic
54#107×C283C2211yescharacteristic
55#106×C283C2211yescharacteristic
56C4⋊C8×C283C2211yescharacteristic
57#102×C283C2211yescharacteristic
58#102×C283C2211yescharacteristic
59#17985391—22no—
60#17985391Q814yescomplemented
61C2×Dic566—21no—
62C2×Dic566—21no—
63C2×Dic566C2311yescharacteristic
64C2×C22⋊D283—21no—
65C22×Dic283—21no—
66C22×Dic283—21no—
67C2×C4×D283—21no—
68C2×C4×D283—21no—
69C2×C4×D283C2311yescharacteristic
70#17985375—22no—
71C2×C283⋊C8C2311yescharacteristic
72C2×C283⋊C8C2311yescharacteristic
73C8×D283—22no—
74C4⋊Dic283—21no—
75C4⋊Dic283—21no—
76C4⋊Dic283C2311yescharacteristic
77C4⋊Dic283C2311yescharacteristic
78C4⋊Dic283C2311yescharacteristic
79Dic283⋊C4—21no—
80Dic283⋊C4—21no—
81Dic283⋊C4—21no—
82Dic283⋊C4—21no—
83Dic283⋊C4—21no—
84Dic283⋊C4—21no—
85Dic283⋊C4D411yescharacteristic
86Dic283⋊C4D411yescharacteristic
87C22⋊Dic283—21no—
88(C2×D283)⋊C4—21no—
89(C2×D283)⋊C4D411yescharacteristic
90(C2×D283)⋊C4D411yescharacteristic
91C4×Dic283—21no—
92C4×Dic283—21no—
93C4×Dic283—21no—
94C4×Dic283C2311yescharacteristic
95C2×D4×C283C2311yescharacteristic
96C22×C4×C283—21no—
97C2×C8×C283C2311yescharacteristic
98C2×C8×C283C2311yescharacteristic
99C4⋊C4×C283—21no—
100C4⋊C4×C283—21no—
101C4⋊C4×C283C2311yescharacteristic
102C4⋊C4×C283C2311yescharacteristic
103C4⋊C4×C283C2311yescharacteristic
104C22⋊C4×C283—21no—
105C42×C283C2311yescharacteristic
106C22×D283C2×D411yescharacteristic
107Dic566—21no—
108Dic566—81no—
109Dic566—81no—
110Dic566C2×Q812yes—
111C22⋊D283—24no—
112C2×Dic283—21no—
113C2×Dic283—24no—
114C2×Dic283—21no—
115C2×Dic283—21no—
116C2×Dic283—21no—
117C2×Dic283—81no—
118C2×Dic283C2×D411yescharacteristic
119C2×Dic283C2×D411yescharacteristic
120C2×Dic283C2×D411yescharacteristic
121C4×D283—81no—
122C4×D283—81no—
123C4×D283C2×Q812yes—
124C4×D283#4112yes—
125C283⋊C8—21no—
126C283⋊C8—21no—
127C23×C283—21no—
128D4×C283—21no—
129D4×C283C2×Q812yes—
130C2×C4×C283—21no—
131C2×C4×C283—21no—
132C2×C4×C283—21no—
133C2×C4×C283—81no—
134C2×C4×C283C2×D411yescharacteristic
135C2×C4×C283C2×D411yescharacteristic
136C2×C4×C283C2411yescharacteristic
137C8×C283—21no—
138C8×C283—21no—
139C22×C283—22no—
140C22×C283—41no—
141C22×C283—41no—
142C22×C283C22×D411yescharacteristic
143C2×D283—21no—
144C2×D283—21no—
145C2×D283C22⋊Q812yes—
146C4×C283—21no—
147C4×C283—21no—
148C4×C283—41no—
149C4×C283—41no—
150C4×C283—41no—
151C4×C283C22×Q811yescharacteristic
152C4×C283C2×#4111yescharacteristic
153Dic283—22no—
154Dic283—21no—
155Dic283—21no—
156Dic283—41no—
157Dic283—41no—
158Dic283—41no—
159Dic283C22⋊Q812yes—
160C2×C283—42no—
161C2×C283C2×C22⋊Q811yescharacteristic
162C2×C283#57711yescharacteristic
163C2×C283#57711yescharacteristic
164D283—22no—
165C283#308411yescharacteristic, complemented, Sylow
166#3084—2831nomaximal, Sylow
167#544—2831no—
168#520—2831no—
169#479—2831no—
170#479—2831no—
171#478—2831no—
172#478D28311yescharacteristic, complemented
173#476—2831no—
174#476—2831no—
175#475—2831no—
176#475—2831no—
177#428—2831no—
178C2×#106—2831no—
179#424—2831no—
180#417—2831no—
181#416—2831no—
182C2×#41—2831no—
183C2×M16—2831no—
184C22×C8—2831no—
185C4⋊Q8—2831no—
186C4⋊Q8—2831no—
187C4⋊Q8—5661no—
188#125—2831no—
189#125—5661no—
190#125C2×D28311yescharacteristic
191C22⋊Q8—5661no—
192C22⋊Q8—5661no—
193C4×Q8—2831no—
194C4×Q8—2831no—
195C4×D4—2831no—
196C4×D4—5661no—
197C4×D4—5661no—
198C4×D4C2×D28311yescharacteristic
199#117—2831no—
200#117—2831no—
201#117—5661no—
202C2×C4⋊C4—2831no—
203C2×C4⋊C4—2831no—
204C2×C4⋊C4—5661no—
205#107—2831no—
206#107C2×D28311yescharacteristic
207#106—2831no—
208#106—2831no—
209#106—2831no—
210#106—2831no—
211#106—2831no—
212#106C2×D28311yescharacteristic
213C4⋊C8—2831no—
214C4⋊C8—2831no—
215C4⋊C8—2831no—
216C4⋊C8C2×D28311yescharacteristic
217#103—2831no—
218#103—2831no—
219#103—2831no—
220#103—2831no—
221#102—2831no—
222#102—2831no—
223#102C2×D28311yescharacteristic
224#102C2×D28311yescharacteristic
225#41—2834no—
226#41—5662no—
227C2×Q8—2831no—
228C2×Q8—5661no—
229C2×Q8—5661no—
230C2×D4—5661no—
231C2×D4C22×D28311yescharacteristic
232C22×C4—21no—
233C22×C4—2831no—
234C22×C4—5661no—
235C22×C4—5661no—
236C22×C4—5661no—
237C22×C4—5661no—
238M16—5662no—
239C2×C8—2831no—
240C2×C8—2831no—
241C2×C8—5662no—
242C2×C8C22×D28311yescharacteristic
243C2×C8C22×D28311yescharacteristic
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—2831no—
247C4⋊C4—2831no—
248C4⋊C4—2831no—
249C4⋊C4—2831no—
250C4⋊C4—2831no—
251C4⋊C4—5661no—
252C4⋊C4—5661no—
253C4⋊C4—5661no—
254C4⋊C4—5661no—
255C4⋊C4—5661no—
256C4⋊C4—5661no—
257C4⋊C4—5661no—
258C4⋊C4—5661no—
259C4⋊C4C22×D28311yescharacteristic
260C4⋊C4C22×D28311yescharacteristic
261C4⋊C4C22×D28311yescharacteristic
262C22⋊C4—21no—
263C22⋊C4—2831no—
264C22⋊C4—2831no—
265C22⋊C4—5661no—
266C22⋊C4—5661no—
267C42—2831no—
268C42—5661no—
269C42—5661no—
270C42—5661no—
271C42C22×D28311yescharacteristic
272C23—21no—
273C23—2831no—
274Q8—2832no—
275Q8—5661no—
276Q8—22641no—
277Q8—22641no—
278D4—21no—
279D4—5664no—
280D4Q8×D28312yes—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—81no—
285C2×C4—2831no—
286C2×C4—2832no—
287C2×C4—2832no—
288C2×C4—2831no—
289C2×C4—2831no—
290C2×C4—5664no—
291C2×C4—5661no—
292C2×C4—5661no—
293C2×C4—5661no—
294C2×C4—5661no—
295C2×C4—22641no—
296C2×C4—22641no—
297C2×C4—22641no—
298C2×C4D4×D28311yescharacteristic
299C2×C4D4×D28311yescharacteristic
300C2×C4C23×D28311yescharacteristic
301C8—21no—
302C8—21no—
303C8—5661no—
304C8—5661no—
305C22—22no—
306C22—41no—
307C22—41no—
308C22—2832no—
309C22—5661no—
310C22—5661no—
311C22C2×D4×D28311yescharacteristic
312C4—21no—
313C4—21no—
314C4—41no—
315C4—41no—
316C4—41no—
317C4—2832no—
318C4—5662no—
319C4—5661no—
320C4—5661no—
321C4—11321no—
322C4—11321no—
323C4—11321no—
324C4C2×Q8×D28311yescharacteristic
325C4#41×D28311yescharacteristic
326C2—42no—
327C2—5662no—
328C2C22⋊Q8×D28311yescharacteristic
329C2#3182727211yescharacteristic
330C2#3182727511yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#478×C283C2
socleC22×C283C22×D4
prime radicalC2×#17985391C22
hypercenter#478D283
hypocenterC283#3084
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

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

NH ≅ G/Ntype
1. #31827095C2semidirect
3. #31826771C2semidirect
4. #31826769C2semidirect
5. #31826767C2semidirect
6. #31826755C2semidirect
7. #31826753C2semidirect
8. #31826751C2semidirect
9. #31826749C2semidirect
10. #31826506C2semidirect
11. #106×D283C2semidirect
12. #31826490C2semidirect
15. #478×C283C2semidirect
60. #17985391Q8semidirect
165. C283#3084semidirect
172. #478D283semidirect

Maximal subgroups

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

subgroupindex
#318270952
#318269442
#318267712
#318267692
#318267672
#318267552
#318267532
#318267512
#318267492
#318265062
#106×D2832
#318264902
#318264632
#318264582
#478×C2832
#3084283