Finite Group Databasebeta

← #49130512

Subgroups — #49130512

Order 36224.

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

normal subgroups only

This group has 806 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, complemented
2C2×#22583637C212yesmaximal, complemented
3#121×D283C211yesmaximal, characteristic, complemented
4#31826613C212yesmaximal, complemented
5#31826609C212yesmaximal, complemented
6#31826604C211yesmaximal, characteristic, complemented
7#31826601C211yesmaximal, characteristic, complemented
8#31826598C211yesmaximal, characteristic, complemented
9#31826465C212yesmaximal, complemented
10#31826427C211yesmaximal, characteristic, complemented
11#448×C283C211yesmaximal, characteristic, complemented
12C2×#17985393—28no—
13C2×#17985393—21no—
14C2×#17985393—21no—
15C2×#17985393C2212yes—
16C2×#17985393C2212yescomplemented
17C22×D1132—21no—
18C22×D1132C2211yescharacteristic, complemented
19C2×D4×D283—21no—
20C2×D4×D283C2211yescharacteristic, complemented
21C22×C4×D283—21no—
22C22×C4×D283C2211yescharacteristic
23#22583637—28no—
24C2×#17985379C2212yescomplemented
25C2×D2264C2212yescomplemented
26C2×#17985377C2212yescomplemented
27C2×#17985375C2212yescomplemented
28#22583585C2211yescharacteristic, complemented
29#22583584C2211yescharacteristic
30#22583583—21no—
31#22583582—21no—
32#22583581C2211yescharacteristic, complemented
33C4⋊C4×D283C2211yescharacteristic, complemented
34C22⋊C4×D283—21no—
35#22583534C2212yescomplemented
36#22583532C2212yescomplemented
37#22583530C2212yescomplemented
38C22⋊(C283⋊C8)C2211yescharacteristic, complemented
39(C2×D283)⋊C8C2212yescomplemented
40C22×Q8×C283C2211yescharacteristic, complemented
41C2×SD16×C283C2212yescomplemented
42#121×C283C2211yescharacteristic
43#103×C283C2212yescomplemented
44C22⋊C8×C283C2211yescharacteristic, complemented
45C23×D283—21no—
46C23×D283—21no—
47C23×D283D411yescharacteristic
48#17985393—28no—
49#17985393—28no—
50#17985393—24no—
51#17985393—24no—
52#17985393—24no—
53#17985393D418yescomplemented
54C2×D1132—28no—
55C2×D1132—22no—
56C2×D1132—22no—
57C2×D1132—24no—
58C2×D1132—21no—
59C2×D1132—21no—
60C2×D1132C2311yescharacteristic
61C2×D1132C2312yes—
62C2×C22⋊D283—21no—
63C2×C22⋊D283—21no—
64C2×C22⋊D283—21no—
65D4×D283—24no—
66D4×D283—44no—
67C22×Dic283D411yescharacteristic
68C2×C4×D283—28no—
69C2×C4×D283—22no—
70C2×C4×D283—22no—
71C2×C4×D283—24no—
72C2×C4×D283—21no—
73C2×C4×D283D411yescharacteristic
74C2×C4×D283D411yescharacteristic
75C2×C4×D283C2311yescharacteristic
76C2×C4×D283C2311yescharacteristic
77#17985379—24no—
78D2264—24no—
79#17985377—24no—
80#17985375—24no—
81C2×C283⋊C8C2312yes—
82C4⋊Dic283C2311yescharacteristic
83Dic283⋊C4—21no—
84C22⋊Dic283—21no—
85(C2×D283)⋊C4—21no—
86(C2×D283)⋊C4—21no—
87C2×Q8×C283—24no—
88C2×Q8×C283—21no—
89C2×Q8×C283C2312yes—
90C2×D4×C283—21no—
91C2×D4×C283C2311yescharacteristic
92C22×C4×C283—21no—
93C22×C4×C283C2311yescharacteristic
94SD16×C283—24no—
95C2×C8×C283C2312yes—
96C4⋊C4×C283C2311yescharacteristic
97C22⋊C4×C283—21no—
98C22×D283—24no—
99C22×D283—22no—
100C22×D283—22no—
101C22×D283—22no—
102C22×D283—21no—
103C22×D283—41no—
104C22×D283—42no—
105C22×D283—42no—
106C22×D283—42no—
107C22×D283—42no—
108C22×D283—42no—
109C22×D283C2×D411yescharacteristic
110C22×D283C2×D411yescharacteristic
111D1132—28no—
112D1132—24no—
113D1132—28no—
114D1132—22no—
115D1132—22no—
116D1132—42no—
117D1132C2×D414yescomplemented
118C22⋊D283—42no—
119C22⋊D283—42no—
120C22⋊D283—42no—
121C2×Dic283—22no—
122C2×Dic283—21no—
123C2×Dic283C2×D411yescharacteristic
124C2×Dic283C2×D411yescharacteristic
125C4×D283—28no—
126C4×D283—22no—
127C4×D283—22no—
128C4×D283—28no—
129C4×D283—22no—
130C4×D283—42no—
131C4×D283C2×D414yes—
132C283⋊C8—22no—
133C23×C283—21no—
134C23×C283C2×D411yescharacteristic
135Q8×C283—24no—
136Q8×C283—22no—
137Q8×C283C2×D414yescomplemented
138D4×C283—22no—
139D4×C283—42no—
140C2×C4×C283—24no—
141C2×C4×C283—22no—
142C2×C4×C283—22no—
143C2×C4×C283—21no—
144C2×C4×C283C2×D411yescharacteristic
145C2×C4×C283C2411yescharacteristic
146C8×C283—22no—
147C22×C283—21no—
148C22×C283—21no—
149C22×C283—41no—
150C22×C283—41no—
151C22×C283—41no—
152C22×C283AGL(1,ℤ/8)12yescomplemented
153C22×C283C22×D411yescharacteristic
154C2×D283—21no—
155C2×D283—21no—
156C2×D283—21no—
157C2×D283—24no—
158C2×D283—21no—
159C2×D283—21no—
160C2×D283—44no—
161C2×D283—48no—
162C2×D283—42no—
163C2×D283—42no—
164C2×D283—41no—
165C2×D283—41no—
166C2×D283—41no—
167C2×D283—84no—
168C2×D283#12012yes—
169C4×C283—24no—
170C4×C283—21no—
171C4×C283—41no—
172C4×C283C22×D412yes—
173Dic283—21no—
174Dic283—41no—
175Dic283#12012yes—
176C2×C283—22no—
177C2×C283—81no—
178C2×C283C2×#12011yescharacteristic
179C2×C283C2×AGL(1,ℤ/8)11yescharacteristic
180C2×C283#57511yescharacteristic
181D283—22no—
182D283—41no—
183D283—44no—
184D283—81no—
185C283#300611yescharacteristic, complemented, Sylow
186#3006—2831nomaximal, Sylow
187C22×#41—2831no—
188C2×AGL(1,ℤ/8)—2832no—
189C2×#121—2831no—
190#449—2832no—
191#449—2832no—
192#448—2831no—
193#448D28311yescharacteristic, complemented
194#447—2831no—
195#447—2831no—
196#417—2832no—
197#407—2831no—
198C2×#41—2832no—
199C2×#41—2832no—
200C2×#41—5661no—
201C2×#41—5661no—
202C2×#41—5668no—
203C22×Q8C2×D28311yescharacteristic, complemented
204C22×D4—2831no—
205C22×D4—2831no—
206C22×D4—5661no—
207C22×D4—5661no—
208C23×C4—2831no—
209C23×C4—5661no—
210AGL(1,ℤ/8)—5668no—
211C2×SD16—2832no—
212C2×SD16C2×D28312yescomplemented
213C2×D8—2832no—
214C2×D8—2832no—
215C2×M16—2832no—
216#121—2831no—
217#121—2831no—
218#121—2831no—
219#121—5661no—
220#121—5661no—
221#121C2×D28311yescharacteristic
222C2×C4⋊C4—2831no—
223C2×C22⋊C4—5661no—
224#103—2832no—
225#103C2×D28312yescomplemented
226#102—2832no—
227#102—2832no—
228C22⋊C8—2831no—
229C22⋊C8—2832no—
230C22⋊C8C2×D28311yescharacteristic, complemented
231C24—2831no—
232C24—5661no—
233C24—5661no—
234#41—2838no—
235#41—5664no—
236#41—5664no—
237#41—5668no—
238#41—5668no—
239#41—5664no—
240C2×Q8—24no—
241C2×Q8—21no—
242C2×Q8C22×D28312yes—
243C2×D4—21no—
244C2×D4—2832no—
245C2×D4—2831no—
246C2×D4—5664no—
247C2×D4—5662no—
248C2×D4—5662no—
249C2×D4—5668no—
250C2×D4—5661no—
251C2×D4—5664no—
252C2×D4—5661no—
253C2×D4—5661no—
254C2×D4—5661no—
255C2×D4—5661no—
256C2×D4—11324no—
257C2×D4C22×D28311yescharacteristic
258C22×C4—21no—
259C22×C4—2831no—
260C22×C4—2831no—
261C22×C4—2831no—
262C22×C4—2831no—
263C22×C4—2831no—
264C22×C4—5664no—
265C22×C4—5662no—
266C22×C4—5662no—
267C22×C4—5668no—
268C22×C4—5661no—
269C22×C4C22×D28311yescharacteristic
270SD16—24no—
271SD16—5664no—
272D8—5664no—
273D8—5664no—
274M16—5664no—
275C2×C8—2832no—
276C2×C8C22×D28312yes—
277C4⋊C4—2831no—
278C4⋊C4—5661no—
279C4⋊C4C22×D28311yescharacteristic
280C22⋊C4—21no—
281C22⋊C4—5661no—
282C22⋊C4—5661no—
283C22⋊C4—5661no—
284C23—21no—
285C23—2831no—
286C23—2831no—
287C23—5662no—
288C23—5664no—
289C23—5662no—
290C23—5662no—
291C23—5661no—
292C23—11321no—
293C23—11322no—
294C23—11322no—
295C23—11322no—
296C23—11322no—
297C23—11322no—
298C23D4×D28311yescharacteristic
299Q8—24no—
300Q8—22no—
301Q8D4×D28314yescomplemented
302D4—22no—
303D4—42no—
304D4—2834no—
305D4—5668no—
306D4—5668no—
307D4—5664no—
308D4—5662no—
309D4—5662no—
310D4—11322no—
311D4—11322no—
312D4—11322no—
313D4—11322no—
314C2×C4—24no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—2831no—
319C2×C4—2834no—
320C2×C4—2831no—
321C2×C4—5662no—
322C2×C4—5668no—
323C2×C4—5662no—
324C2×C4—5668no—
325C2×C4—5662no—
326C2×C4—5662no—
327C2×C4—5661no—
328C2×C4—11322no—
329C2×C4D4×D28311yescharacteristic
330C2×C4C23×D28311yescharacteristic
331C8—22no—
332C8—5662no—
333C22—21no—
334C22—21no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—2832no—
339C22—5661no—
340C22—5661no—
341C22—5664no—
342C22—5661no—
343C22—5661no—
344C22—5661no—
345C22—11324no—
346C22—11328no—
347C22—11322no—
348C22—11322no—
349C22—11321no—
350C22—11321no—
351C22—11321no—
352C22—22644no—
353C22#2258363712yescomplemented
354C22C2×D4×D28311yescharacteristic
355C4—24no—
356C4—21no—
357C4—41no—
358C4—2832no—
359C4—5661no—
360C4—11321no—
361C4C2×D4×D28312yes—
362C2—22no—
363C2—81no—
364C2—5662no—
365C2—11321no—
366C2—11324no—
367C2—22641no—
368C2#120×D28311yescharacteristic
369C2C2×#2258363711yescharacteristic
370C2#3182726411yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#448×C283C2
socleC22×C283C22×D4
hypercenter#448D283
hypocenterC283#3006
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C22×#17985393C2semidirect
2. C2×#22583637C2semidirect
3. #121×D283C2semidirect
4. #31826613C2semidirect
5. #31826609C2semidirect
6. #31826604C2semidirect
7. #31826601C2semidirect
8. #31826598C2semidirect
9. #31826465C2semidirect
10. #31826427C2semidirect
11. #448×C283C2semidirect
16. C2×#17985393C22semidirect
18. C22×D1132C22semidirect
20. C2×D4×D283C22semidirect
24. C2×#17985379C22semidirect
25. C2×D2264C22semidirect
26. C2×#17985377C22semidirect
27. C2×#17985375C22semidirect
28. #22583585C22semidirect
32. #22583581C22semidirect
33. C4⋊C4×D283C22semidirect
35. #22583534C22semidirect
36. #22583532C22semidirect
37. #22583530C22semidirect
38. C22⋊(C283⋊C8)C22semidirect
39. (C2×D283)⋊C8C22semidirect
40. C22×Q8×C283C22semidirect
41. C2×SD16×C283C22semidirect
43. #103×C283C22semidirect
44. C22⋊C8×C283C22semidirect
53. #17985393D4semidirect
117. D1132C2×D4semidirect
137. Q8×C283C2×D4semidirect
152. C22×C283AGL(1,ℤ/8)semidirect
185. C283#3006semidirect
193. #448D283semidirect
203. C22×Q8C2×D283semidirect
212. C2×SD16C2×D283semidirect
225. #103C2×D283semidirect
230. C22⋊C8C2×D283semidirect
301. Q8D4×D283semidirect
353. C22#22583637semidirect

Maximal subgroups

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

subgroupindex
C22×#179853932
C2×#225836372
#121×D2832
#318266132
#318266092
#318266042
#318266012
#318265982
#318264652
#318264272
#448×C2832
#3006283