Finite Group Databasebeta

← #49130520

Subgroups — #49130520

Order 36224.

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

normal subgroups only

This group has 754 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×#17985393C211yesmaximal, characteristic
2C2×#22583642C212yesmaximal, complemented
3C22⋊Q8×D283C211yesmaximal, characteristic, complemented
4#31826630C212yesmaximal, complemented
5#31826626C212yesmaximal, complemented
6#31826621C211yesmaximal, characteristic, complemented
7#31826618C211yesmaximal, characteristic, complemented
8#31826615C211yesmaximal, characteristic, complemented
9#31826465C212yesmaximal, complemented
10#31826427C211yesmaximal, characteristic, complemented
11#451×C283C211yesmaximal, characteristic, complemented
12C2×#17985393—28no—
13C2×#17985393—21no—
14C2×#17985393—21no—
15C2×#17985393C2212yes—
16C2×#17985393C2212yes—
17C2×Q8×D283C2211yescharacteristic, complemented
18C22×D1132—21no—
19C22×D1132C2211yescharacteristic
20C22×C4×D283—21no—
21C22×C4×D283C2211yescharacteristic
22#22583642—28no—
23C2×#17985382C2212yescomplemented
24C2×#17985381C2212yescomplemented
25C2×#17985379C2212yescomplemented
26C2×#17985375C2212yescomplemented
27#22583590C2211yescharacteristic, complemented
28#22583589C2211yescharacteristic
29#22583588—21no—
30#22583587—21no—
31#22583586C2211yescharacteristic, complemented
32C4⋊C4×D283—21no—
33C4⋊C4×D283C2211yescharacteristic, complemented
34C22⋊C4×D283—21no—
35#22583534C2212yescomplemented
36#22583532C2212yescomplemented
37#22583530C2212yescomplemented
38C22⋊(C283⋊C8)C2211yescharacteristic
39(C2×D283)⋊C8C2212yescomplemented
40C22×Q8×C283C2211yescharacteristic
41C2×Q16×C283C2212yescomplemented
42C22⋊Q8×C283C2211yescharacteristic
43#103×C283C2212yescomplemented
44C22⋊C8×C283C2211yescharacteristic
45C23×D283—21no—
46C23×D283D411yescharacteristic
47#17985393—28no—
48#17985393—28no—
49#17985393—24no—
50#17985393—24no—
51#17985393—24no—
52#17985393D418yes—
53C2×Dic566—21no—
54C2×Dic566C2311yescharacteristic
55Q8×D283—24no—
56C2×D1132—28no—
57C2×D1132—22no—
58C2×D1132—22no—
59C2×D1132—24no—
60C2×D1132—21no—
61C2×D1132C2312yes—
62C22×Dic283D411yescharacteristic
63C2×C4×D283—28no—
64C2×C4×D283—22no—
65C2×C4×D283—22no—
66C2×C4×D283—24no—
67C2×C4×D283—21no—
68C2×C4×D283—21no—
69C2×C4×D283D411yescharacteristic
70C2×C4×D283D411yescharacteristic
71C2×C4×D283C2311yescharacteristic
72C2×C4×D283C2311yescharacteristic
73#17985382—24no—
74#17985381—24no—
75#17985379—24no—
76#17985375—24no—
77C2×C283⋊C8C2312yes—
78C4⋊Dic283—21no—
79C4⋊Dic283C2311yescharacteristic
80Dic283⋊C4—21no—
81Dic283⋊C4—21no—
82Dic283⋊C4—21no—
83C22⋊Dic283—21no—
84(C2×D283)⋊C4—21no—
85(C2×D283)⋊C4—21no—
86C2×Q8×C283—24no—
87C2×Q8×C283—21no—
88C2×Q8×C283C2311yescharacteristic
89C2×Q8×C283C2312yes—
90C22×C4×C283—21no—
91C22×C4×C283C2311yescharacteristic
92Q16×C283—24no—
93C2×C8×C283C2312yes—
94C4⋊C4×C283—21no—
95C4⋊C4×C283C2311yescharacteristic
96C22⋊C4×C283—21no—
97C22×D283—24no—
98C22×D283—22no—
99C22×D283—22no—
100C22×D283—22no—
101C22×D283—41no—
102C22×D283—42no—
103C22×D283—42no—
104C22×D283C2×D411yescharacteristic
105C22×D283C2×D411yescharacteristic
106Dic566—22no—
107Dic566—42no—
108D1132—28no—
109D1132—24no—
110D1132—28no—
111D1132—22no—
112D1132C2×D414yes—
113C2×Dic283—22no—
114C2×Dic283—21no—
115C2×Dic283—21no—
116C2×Dic283C2×D411yescharacteristic
117C2×Dic283C2×D411yescharacteristic
118C4×D283—28no—
119C4×D283—22no—
120C4×D283—22no—
121C4×D283—28no—
122C4×D283—22no—
123C4×D283—42no—
124C4×D283—42no—
125C4×D283C2×D414yes—
126C283⋊C8—22no—
127C23×C283C2×D411yescharacteristic
128Q8×C283—24no—
129Q8×C283—22no—
130Q8×C283—22no—
131Q8×C283C2×D414yes—
132C2×C4×C283—24no—
133C2×C4×C283—22no—
134C2×C4×C283—22no—
135C2×C4×C283—21no—
136C2×C4×C283—21no—
137C2×C4×C283C2×D411yescharacteristic
138C2×C4×C283C2411yescharacteristic
139C8×C283—22no—
140C22×C283—21no—
141C22×C283—21no—
142C22×C283#13712yescomplemented
143C22×C283C22×D411yescharacteristic
144C2×D283—21no—
145C2×D283—21no—
146C2×D283—21no—
147C2×D283—24no—
148C2×D283—21no—
149C2×D283—21no—
150C2×D283—44no—
151C2×D283—48no—
152C2×D283—42no—
153C2×D283—42no—
154C2×D283#12012yes—
155C4×C283—24no—
156C4×C283—21no—
157C4×C283—41no—
158C4×C283—41no—
159C4×C283C22×D412yes—
160Dic283—21no—
161Dic283—41no—
162Dic283—41no—
163Dic283#12012yes—
164C2×C283—22no—
165C2×C283C2×#12011yescharacteristic
166C2×C283C2×#13711yescharacteristic
167C2×C283#57511yescharacteristic
168D283—22no—
169D283—41no—
170D283—44no—
171C283#300711yescharacteristic, complemented, Sylow
172#3007—2831nomaximal, Sylow
173C22×#41—2831no—
174C2×#137—2832no—
175C2×C22⋊Q8—2831no—
176#452—2832no—
177#452—2832no—
178#451—2831no—
179#451D28311yescharacteristic, complemented
180#450—2831no—
181#450—2831no—
182#417—2832no—
183#407—2831no—
184C2×#41—2832no—
185C2×#41—2832no—
186C2×#41—5661no—
187C2×#41—5661no—
188C2×#41—5668no—
189C22×Q8—2831no—
190C22×Q8C2×D28311yescharacteristic
191C22×D4—2831no—
192C22×D4—5661no—
193C23×C4—2831no—
194C23×C4—5661no—
195#137—5668no—
196C2×Q16—2832no—
197C2×Q16C2×D28312yescomplemented
198C2×SD16—2832no—
199C2×SD16—2832no—
200C2×M16—2832no—
201C22⋊Q8—2831no—
202C22⋊Q8—2831no—
203C22⋊Q8—2831no—
204C22⋊Q8—5661no—
205C22⋊Q8—5661no—
206C22⋊Q8C2×D28311yescharacteristic
207C2×C4⋊C4—2831no—
208C2×C4⋊C4—5661no—
209C2×C22⋊C4—5661no—
210#103—2832no—
211#103C2×D28312yescomplemented
212#102—2832no—
213#102—2832no—
214C22⋊C8—2831no—
215C22⋊C8—2832no—
216C22⋊C8C2×D28311yescharacteristic
217C24—2831no—
218C24—5661no—
219#41—2838no—
220#41—5664no—
221#41—5664no—
222#41—5668no—
223#41—5668no—
224#41—5664no—
225C2×Q8—24no—
226C2×Q8—21no—
227C2×Q8—2831no—
228C2×Q8—5661no—
229C2×Q8—5664no—
230C2×Q8C22×D28311yescharacteristic
231C2×Q8C22×D28312yes—
232C2×D4—2832no—
233C2×D4—5664no—
234C2×D4—5662no—
235C2×D4—5662no—
236C2×D4—5668no—
237C2×D4—5661no—
238C22×C4—21no—
239C22×C4—2831no—
240C22×C4—2831no—
241C22×C4—2831no—
242C22×C4—2831no—
243C22×C4—2831no—
244C22×C4—5664no—
245C22×C4—5662no—
246C22×C4—5662no—
247C22×C4—5668no—
248C22×C4—5661no—
249C22×C4—5661no—
250C22×C4C22×D28311yescharacteristic
251Q16—24no—
252Q16—5664no—
253SD16—5664no—
254SD16—5664no—
255M16—5664no—
256C2×C8—2832no—
257C2×C8C22×D28312yes—
258C4⋊C4—21no—
259C4⋊C4—2831no—
260C4⋊C4—5661no—
261C4⋊C4—5661no—
262C4⋊C4—5661no—
263C4⋊C4—5661no—
264C4⋊C4C22×D28311yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—5661no—
267C22⋊C4—5661no—
268C22⋊C4—5661no—
269C23—2831no—
270C23—2831no—
271C23—5662no—
272C23—5664no—
273C23—5662no—
274C23—5662no—
275C23—11321no—
276C23—11322no—
277C23—11322no—
278C23D4×D28311yescharacteristic
279Q8—24no—
280Q8—22no—
281Q8—22no—
282Q8—5662no—
283Q8—11322no—
284Q8D4×D28314yes—
285D4—2834no—
286D4—5668no—
287D4—5668no—
288D4—5664no—
289D4—5662no—
290C2×C4—24no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—2831no—
296C2×C4—2834no—
297C2×C4—2831no—
298C2×C4—5662no—
299C2×C4—5668no—
300C2×C4—5662no—
301C2×C4—5668no—
302C2×C4—5662no—
303C2×C4—5662no—
304C2×C4—5661no—
305C2×C4—5661no—
306C2×C4—11322no—
307C2×C4—11322no—
308C2×C4D4×D28311yescharacteristic
309C2×C4C23×D28311yescharacteristic
310C8—22no—
311C8—5662no—
312C22—21no—
313C22—21no—
314C22—2832no—
315C22—5661no—
316C22—5661no—
317C22—5664no—
318C22—5661no—
319C22—5661no—
320C22—5661no—
321C22—11324no—
322C22—11328no—
323C22—11322no—
324C22—11322no—
325C22#2258364212yescomplemented
326C22C2×D4×D28311yescharacteristic
327C4—24no—
328C4—21no—
329C4—41no—
330C4—41no—
331C4—2832no—
332C4—5661no—
333C4—11321no—
334C4—11321no—
335C4C2×D4×D28312yes—
336C2—22no—
337C2—5662no—
338C2—11321no—
339C2—11324no—
340C2#120×D28311yescharacteristic
341C2C2×#2258364211yescharacteristic
342C2#3182726411yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#451×C283C2
socleC22×C283C22×D4
prime radicalC22×#17985393C2
hypercenter#451D283
hypocenterC283#3007
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
2. C2×#22583642C2semidirect
3. C22⋊Q8×D283C2semidirect
4. #31826630C2semidirect
5. #31826626C2semidirect
6. #31826621C2semidirect
7. #31826618C2semidirect
8. #31826615C2semidirect
9. #31826465C2semidirect
10. #31826427C2semidirect
11. #451×C283C2semidirect
17. C2×Q8×D283C22semidirect
23. C2×#17985382C22semidirect
24. C2×#17985381C22semidirect
25. C2×#17985379C22semidirect
26. C2×#17985375C22semidirect
27. #22583590C22semidirect
31. #22583586C22semidirect
33. C4⋊C4×D283C22semidirect
35. #22583534C22semidirect
36. #22583532C22semidirect
37. #22583530C22semidirect
39. (C2×D283)⋊C8C22semidirect
41. C2×Q16×C283C22semidirect
43. #103×C283C22semidirect
142. C22×C283#137semidirect
171. C283#3007semidirect
179. #451D283semidirect
197. C2×Q16C2×D283semidirect
211. #103C2×D283semidirect
325. C22#22583642semidirect

Maximal subgroups

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

subgroupindex
C22×#179853932
C2×#225836422
C22⋊Q8×D2832
#318266302
#318266262
#318266212
#318266182
#318266152
#318264652
#318264272
#451×C2832
#3007283