Finite Group Databasebeta

← #49130474

Subgroups — #49130474

Order 36224.

764 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C22×#17985391C211yesmaximal, characteristic, complemented
2C2×#22583629C212yesmaximal, complemented
3#31826975C211yesmaximal, characteristic, complemented
4#31826628C212yesmaximal, complemented
5#31826623C211yesmaximal, characteristic, complemented
6#31826616C211yesmaximal, characteristic, complemented
7#31826607C212yesmaximal, complemented
8#31826603C211yesmaximal, characteristic, complemented
9#31826458C212yesmaximal, complemented
10C22⋊C8×D283C211yesmaximal, characteristic, complemented
11#447×C283C211yesmaximal, characteristic, complemented
12C2×#17985391—28no—
13C2×#17985391—21no—
14C2×#17985391—21no—
15C2×#17985391C2212yes—
16C2×#17985391C2212yes—
17C2×#17985391C2211yescharacteristic, complemented
18C22×Dic566C2211yescharacteristic
19C22×C22⋊D283—21no—
20C23×Dic283—21no—
21C22×C4×D283C2211yescharacteristic
22#22583629—28no—
23C2×Dic1132C2212yescomplemented
24C2×#17985380C2212yescomplemented
25C2×#17985380C2212yescomplemented
26C2×C8×D283C2212yescomplemented
27#22583592—21no—
28#22583590C2211yescharacteristic, complemented
29#22583589C2211yescharacteristic
30#22583581C2211yescharacteristic, complemented
31#22583578—21no—
32D4×Dic283—21no—
33(C22⋊D283)⋊C4—21no—
34#22583568C2211yescharacteristic, complemented
35#22583535C2212yescomplemented
36#22583533C2212yescomplemented
37#22583531C2212yescomplemented
38C22⋊(C283⋊C8)C2211yescharacteristic
39(C2×D283)⋊C8C2212yescomplemented
40C22×D4×C283C2211yescharacteristic
41C2×D8×C283C2212yescomplemented
42#121×C283C2211yescharacteristic
43#102×C283C2212yescomplemented
44C22⋊C8×C283C2211yescharacteristic
45C23×D283D411yescharacteristic
46#17985391—28no—
47#17985391—28no—
48#17985391—24no—
49#17985391—24no—
50#17985391—24no—
51#17985391—24no—
52#17985391D418yescomplemented
53C2×Dic566—24no—
54C2×Dic566—21no—
55C2×Dic566C2311yescharacteristic
56C2×Dic566C2312yes—
57C2×C22⋊D283—28no—
58C2×C22⋊D283—22no—
59C2×C22⋊D283—22no—
60C2×C22⋊D283—21no—
61C2×C22⋊D283—21no—
62C22×Dic283—28no—
63C22×Dic283—22no—
64C22×Dic283—22no—
65C22×Dic283—21no—
66C22×Dic283—21no—
67C22×Dic283D411yescharacteristic
68C2×C4×D283—24no—
69C2×C4×D283D411yescharacteristic
70C2×C4×D283D411yescharacteristic
71C2×C4×D283C2311yescharacteristic
72C2×C4×D283C2311yescharacteristic
73Dic1132—24no—
74#17985380—24no—
75#17985380—24no—
76C2×C283⋊C8C2312yes—
77C8×D283—24no—
78C4⋊Dic283—21no—
79C4⋊Dic283C2311yescharacteristic
80Dic283⋊C4—21no—
81C22⋊Dic283—21no—
82C22⋊Dic283—21no—
83(C2×D283)⋊C4—21no—
84(C2×D283)⋊C4—21no—
85C4×Dic283—21no—
86C24×C283—21no—
87C2×D4×C283—24no—
88C2×D4×C283—21no—
89C2×D4×C283—21no—
90C2×D4×C283C2311yescharacteristic
91C2×D4×C283C2312yes—
92C22×C4×C283C2311yescharacteristic
93D8×C283—24no—
94C2×C8×C283C2312yes—
95C4⋊C4×C283C2311yescharacteristic
96C22⋊C4×C283—21no—
97C22×D283—22no—
98C22×D283—22no—
99C22×D283—41no—
100C22×D283C2×D411yescharacteristic
101C22×D283C2×D411yescharacteristic
102Dic566—24no—
103Dic566—22no—
104Dic566—22no—
105Dic566C2×D414yes—
106C22⋊D283—28no—
107C22⋊D283—28no—
108C22⋊D283—42no—
109C22⋊D283—42no—
110C2×Dic283—24no—
111C2×Dic283—28no—
112C2×Dic283—22no—
113C2×Dic283—22no—
114C2×Dic283—28no—
115C2×Dic283—21no—
116C2×Dic283—21no—
117C2×Dic283—42no—
118C2×Dic283C2×D411yescharacteristic
119C2×Dic283C2×D411yescharacteristic
120C4×D283—22no—
121C4×D283—22no—
122C4×D283—22no—
123C4×D283C2×D414yes—
124C283⋊C8—22no—
125C23×C283—24no—
126C23×C283—22no—
127C23×C283—21no—
128C23×C283—42no—
129C23×C283—42no—
130C23×C283C2×D411yescharacteristic
131D4×C283—24no—
132D4×C283—22no—
133D4×C283—22no—
134D4×C283—42no—
135D4×C283C2×D414yes—
136C2×C4×C283—22no—
137C2×C4×C283—21no—
138C2×C4×C283C2×D411yescharacteristic
139C2×C4×C283C2411yescharacteristic
140C8×C283—22no—
141C22×C283—21no—
142C22×C283—21no—
143C22×C283—24no—
144C22×C283—48no—
145C22×C283—42no—
146C22×C283—42no—
147C22×C283—41no—
148C22×C283—41no—
149C22×C283—41no—
150C22×C283#13512yescomplemented
151C22×C283C22×D411yescharacteristic
152C2×D283—21no—
153C2×D283—21no—
154C2×D283—21no—
155C2×D283—21no—
156C2×D283—21no—
157C2×D283—44no—
158C2×D283#12012yes—
159C4×C283—21no—
160C4×C283—41no—
161C4×C283C22×D412yes—
162Dic283—21no—
163Dic283—24no—
164Dic283—41no—
165Dic283—41no—
166Dic283#12012yes—
167C2×C283—22no—
168C2×C283—44no—
169C2×C283—81no—
170C2×C283C2×#12011yescharacteristic
171C2×C283C2×#13511yescharacteristic
172C2×C283#57511yescharacteristic
173D283—22no—
174D283—41no—
175C283#300211yescharacteristic, complemented, Sylow
176#3002—2831nomaximal, Sylow
177C22×#41—2831no—
178C2×#135—2832no—
179#525—2831no—
180#452—2832no—
181#451—2831no—
182#450—2831no—
183#449—2832no—
184#448—2831no—
185#447D28311yescharacteristic, complemented
186#416—2832no—
187C2×C22⋊C8—2831no—
188C2×#41—2832no—
189C2×#41—2831no—
190C2×#41—2832no—
191C2×#41—5661no—
192C2×#41—5661no—
193C2×#41—5668no—
194C22×Q8—2831no—
195C22×D4—5661no—
196C22×D4C2×D28311yescharacteristic
197C23×C4—2831no—
198C23×C4—5661no—
199#135—5668no—
200C2×Q16—2832no—
201C2×SD16—2832no—
202C2×SD16—2832no—
203C2×D8C2×D28312yescomplemented
204C22×C8—2832no—
205#123—5661no—
206C22⋊Q8—2831no—
207C22⋊Q8—2831no—
208#121—2831no—
209#121C2×D28311yescharacteristic
210#120—5661no—
211C4×D4—5661no—
212C4×D4—5661no—
213#117—2831no—
214#103—2832no—
215#103—2832no—
216#102—2832no—
217#102C2×D28312yescomplemented
218C22⋊C8—2831no—
219C22⋊C8—2832no—
220C22⋊C8C2×D28311yescharacteristic
221C24—21no—
222C24—2831no—
223#41—2838no—
224#41—5664no—
225#41—5664no—
226#41—5664no—
227#41—5668no—
228#41—5668no—
229#41—5664no—
230C2×Q8—2832no—
231C2×Q8—2831no—
232C2×Q8—5664no—
233C2×Q8—5661no—
234C2×D4—24no—
235C2×D4—21no—
236C2×D4—21no—
237C2×D4—5662no—
238C2×D4—5662no—
239C2×D4—5668no—
240C2×D4—5661no—
241C2×D4—5661no—
242C2×D4C22×D28311yescharacteristic
243C2×D4C22×D28312yes—
244C22×C4—2831no—
245C22×C4—2831no—
246C22×C4—2831no—
247C22×C4—2831no—
248C22×C4—2831no—
249C22×C4—5664no—
250C22×C4—5662no—
251C22×C4—5662no—
252C22×C4—5661no—
253C22×C4—5668no—
254C22×C4—5661no—
255C22×C4C22×D28311yescharacteristic
256Q16—5664no—
257SD16—5664no—
258SD16—5664no—
259D8—24no—
260C2×C8—2832no—
261C2×C8—5664no—
262C2×C8C22×D28312yes—
263C4⋊C4—2831no—
264C4⋊C4—5661no—
265C4⋊C4—5661no—
266C4⋊C4C22×D28311yescharacteristic
267C22⋊C4—21no—
268C22⋊C4—5661no—
269C22⋊C4—5661no—
270C22⋊C4—5661no—
271C22⋊C4—5661no—
272C42—5661no—
273C23—24no—
274C23—22no—
275C23—21no—
276C23—42no—
277C23—42no—
278C23—2831no—
279C23—2831no—
280C23—5662no—
281C23—5662no—
282C23—11321no—
283C23D4×D28311yescharacteristic
284Q8—2834no—
285Q8—5664no—
286Q8—5662no—
287Q8—5662no—
288D4—24no—
289D4—22no—
290D4—22no—
291D4—42no—
292D4—5668no—
293D4—5668no—
294D4—11322no—
295D4—11322no—
296D4D4×D28314yes—
297C2×C4—22no—
298C2×C4—21no—
299C2×C4—2831no—
300C2×C4—2834no—
301C2×C4—2831no—
302C2×C4—5662no—
303C2×C4—5662no—
304C2×C4—5662no—
305C2×C4—5668no—
306C2×C4—5664no—
307C2×C4—5668no—
308C2×C4—5662no—
309C2×C4—5662no—
310C2×C4—5661no—
311C2×C4—5661no—
312C2×C4—11322no—
313C2×C4D4×D28311yescharacteristic
314C2×C4C23×D28311yescharacteristic
315C8—22no—
316C8—5662no—
317C22—21no—
318C22—21no—
319C22—24no—
320C22—48no—
321C22—42no—
322C22—42no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—2832no—
327C22—5661no—
328C22—5661no—
329C22—5661no—
330C22—5661no—
331C22—5661no—
332C22—11324no—
333C22#2258362912yescomplemented
334C22C2×D4×D28311yescharacteristic
335C4—21no—
336C4—41no—
337C4—2832no—
338C4—5661no—
339C4—5664no—
340C4—11321no—
341C4—11321no—
342C4C2×D4×D28312yes—
343C2—22no—
344C2—44no—
345C2—81no—
346C2—5662no—
347C2—11321no—
348C2#120×D28311yescharacteristic
349C2C2×#2258362911yescharacteristic
350C2#3182726111yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#447×C283C2
socleC22×C283C22×D4
hypercenter#447D283
hypocenterC283#3002
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. C22×#17985391C2semidirect
2. C2×#22583629C2semidirect
3. #31826975C2semidirect
4. #31826628C2semidirect
5. #31826623C2semidirect
6. #31826616C2semidirect
7. #31826607C2semidirect
8. #31826603C2semidirect
9. #31826458C2semidirect
10. C22⋊C8×D283C2semidirect
11. #447×C283C2semidirect
17. C2×#17985391C22semidirect
23. C2×Dic1132C22semidirect
24. C2×#17985380C22semidirect
25. C2×#17985380C22semidirect
26. C2×C8×D283C22semidirect
28. #22583590C22semidirect
30. #22583581C22semidirect
34. #22583568C22semidirect
35. #22583535C22semidirect
36. #22583533C22semidirect
37. #22583531C22semidirect
39. (C2×D283)⋊C8C22semidirect
41. C2×D8×C283C22semidirect
43. #102×C283C22semidirect
52. #17985391D4semidirect
150. C22×C283#135semidirect
175. C283#3002semidirect
185. #447D283semidirect
203. C2×D8C2×D283semidirect
217. #102C2×D283semidirect
333. C22#22583629semidirect

Maximal subgroups

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

subgroupindex
C22×#179853912
C2×#225836292
#318269752
#318266282
#318266232
#318266162
#318266072
#318266032
#318264582
C22⋊C8×D2832
#447×C2832
#3002283