Finite Group Databasebeta

← #49130564

Subgroups — #49130564

Order 36224.

764 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C2×#22583655C211yesmaximal, characteristic, complemented
2C2×#22583630C211yesmaximal, characteristic, complemented
3#31827043C211yesmaximal, characteristic, complemented
4#31826624C212yesmaximal, complemented
5#31826614C212yesmaximal, complemented
6#31826607C212yesmaximal, complemented
7#31826597C212yesmaximal, complemented
8#31826467C211yesmaximal, characteristic, complemented
9C2×#22583531C211yesmaximal, characteristic, complemented
10#31826433C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C283C211yesmaximal, characteristic, complemented
12#22583655—24no—
13#22583655—24no—
14C2×#17985392C2211yescharacteristic
15C2×#17985392C2211yescharacteristic
16C2×#17985391—21no—
17C2×#17985391C2212yes—
18C22×C22⋊D283—22no—
19C2×D4×D283—21no—
20C2×D4×D283C2212yes—
21#22583630—24no—
22C2×#17985382C2211yescharacteristic, complemented
23C2×#17985380C2211yescharacteristic, complemented
24C2×#17985379C2211yescharacteristic, complemented
25C2×#17985377C2211yescharacteristic, complemented
26C22×C283⋊C8C2211yescharacteristic, complemented
27#22583589—21no—
28#22583586C2212yescomplemented
29#22583584—21no—
30#22583582—22no—
31#22583581C2212yescomplemented
32C2×C4⋊Dic283C2211yescharacteristic, complemented
33#22583535C2212yescomplemented
34#22583532C2212yescomplemented
35#22583531C2212yescomplemented
36#22583531C2212yescomplemented
37(C2×D283)⋊C8C2212yescomplemented
38(C2×D283)⋊C8C2212yescomplemented
39C2×#41×C283C2211yescharacteristic, complemented
40C22×D4×C283C2211yescharacteristic
41AGL(1,ℤ/8)×C283—24no—
42C2×SD16×C283C2212yescomplemented
43C2×D8×C283C2212yescomplemented
44C2×M16×C283C2211yescharacteristic, complemented
45C23×D283—22no—
46#17985392—22no—
47#17985392—22no—
48#17985392—22no—
49#17985392D414yescomplemented
50#17985391—28no—
51#17985391—24no—
52#17985391—24no—
53C2×Dic566D411yescharacteristic
54C2×Dic566C2311yescharacteristic
55C2×D1132D411yescharacteristic
56C2×D1132C2311yescharacteristic
57C2×C22⋊D283—216no—
58C2×C22⋊D283—22no—
59C2×C22⋊D283—22no—
60C2×C22⋊D283—22no—
61C2×C22⋊D283—21no—
62C2×C22⋊D283D412yes—
63D4×D283—28no—
64D4×D283—24no—
65D4×D283—24no—
66C22×Dic283—22no—
67C22×Dic283—21no—
68C2×C4×D283—21no—
69C2×C4×D283C2312yes—
70#17985382—22no—
71#17985380—22no—
72#17985379—22no—
73#17985377—22no—
74C2×C283⋊C8—22no—
75C2×C283⋊C8C2311yescharacteristic
76C2×C283⋊C8C2311yescharacteristic
77C4⋊Dic283—21no—
78C4⋊Dic283C2312yes—
79Dic283⋊C4—22no—
80C22⋊Dic283—22no—
81(C2×D283)⋊C4—22no—
82C24×C283—21no—
83#41×C283—22no—
84#41×C283—22no—
85C2×Q8×C283C2311yescharacteristic
86C2×D4×C283—21no—
87C2×D4×C283—21no—
88C2×D4×C283—22no—
89C2×D4×C283D414yescomplemented
90C2×D4×C283C2311yescharacteristic
91C2×D4×C283C2312yes—
92C22×C4×C283—21no—
93C22×C4×C283C2311yescharacteristic
94SD16×C283—24no—
95D8×C283—24no—
96M16×C283—22no—
97C2×C8×C283C2312yes—
98C22×D283—21no—
99C22×D283—28no—
100C22×D283—82no—
101C22×D283—82no—
102C22×D283C2×D412yes—
103Dic566—22no—
104Dic566—21no—
105Dic566C2×D412yes—
106D1132—22no—
107D1132—21no—
108D1132C2×D412yes—
109C22⋊D283—28no—
110C22⋊D283—28no—
111C22⋊D283—24no—
112C22⋊D283—24no—
113C22⋊D283—82no—
114C2×Dic283—28no—
115C2×Dic283—21no—
116C2×Dic283—22no—
117C2×Dic283—81no—
118C2×Dic283C2×D412yes—
119C4×D283—24no—
120C4×D283—22no—
121C4×D283—22no—
122C283⋊C8—21no—
123C283⋊C8—21no—
124C23×C283—22no—
125C23×C283—24no—
126C23×C283—21no—
127C23×C283—81no—
128C23×C283—81no—
129C23×C283C2×D411yescharacteristic
130Q8×C283—21no—
131Q8×C283—21no—
132D4×C283—24no—
133D4×C283—22no—
134D4×C283—21no—
135D4×C283—21no—
136D4×C283—42no—
137D4×C283C2×D414yes—
138C2×C4×C283—21no—
139C2×C4×C283—42no—
140C2×C4×C283C2×D411yescharacteristic
141C2×C4×C283C2×D412yes—
142C2×C4×C283C2×D412yes—
143C2×C4×C283C2411yescharacteristic
144C8×C283—22no—
145C22×C283—21no—
146C22×C283—21no—
147C22×C283—24no—
148C22×C283—42no—
149C22×C283—42no—
150C22×C283—41no—
151C22×C283—41no—
152C22×C283—41no—
153C22×C283—84no—
154C22×C283#12012yes—
155C22×C283C22×D411yescharacteristic
156C2×D283—22no—
157C2×D283—22no—
158C2×D283—22no—
159C2×D283—22no—
160C2×D283—41no—
161C2×D283—41no—
162C2×D283—88no—
163C4×C283—41no—
164C4×C283#12012yes—
165C4×C283C22×D411yescharacteristic
166C4×C283C22×D411yescharacteristic
167Dic283—22no—
168Dic283—22no—
169Dic283—42no—
170C2×C283—22no—
171C2×C283—44no—
172C2×C283—81no—
173C2×C283C2×#12011yescharacteristic
174C2×C283#57611yescharacteristic
175C2×C283#57711yescharacteristic
176D283—42no—
177D283—42no—
178C283#301211yescharacteristic, complemented, Sylow
179#3012—2831nomaximal, Sylow
180C2×H(5,𝔽2)—2831no—
181C2×AGL(1,ℤ/8)D28311yescharacteristic, complemented
182C2×#135—2831no—
183#537—2831no—
184#452—2832no—
185#450—2832no—
186#449—2832no—
187#447—2832no—
188#417—2831no—
189C2×#102—2831no—
190#408—2831no—
191H(5,𝔽2)—5664no—
192H(5,𝔽2)—5664no—
193C2×#41—2831no—
194C2×#41—2832no—
195C2×#41—2831no—
196C2×#41—5661no—
197C2×#41C2×D28311yescharacteristic, complemented
198C22×D4—2832no—
199C22×D4—5661no—
200C22×D4—5662no—
201C22×D4C2×D28311yescharacteristic
202AGL(1,ℤ/8)—24no—
203#135—5664no—
204C2×Q16—2831no—
205C2×SD16—2831no—
206C2×SD16—2831no—
207C2×SD16C2×D28312yescomplemented
208C2×D8—2831no—
209C2×D8C2×D28312yescomplemented
210C2×M16C2×D28311yescharacteristic, complemented
211C22×C8—2831no—
212C22⋊Q8—2832no—
213C22⋊Q8—5661no—
214#121—2832no—
215#121—5661no—
216#121—5662no—
217C2×C4⋊C4—2831no—
218#103—2832no—
219#102—2832no—
220#102—2832no—
221#102—2832no—
222C22⋊C8—2832no—
223C22⋊C8—2832no—
224C24—21no—
225C24—5662no—
226#41—22no—
227#41—22no—
228#41—2834no—
229#41—5664no—
230#41—5668no—
231#41—5664no—
232#41—5662no—
233#41—5662no—
234#41—5662no—
235C2×Q8—2831no—
236C2×Q8—2831no—
237C2×Q8C22×D28311yescharacteristic
238C2×D4—21no—
239C2×D4—21no—
240C2×D4—22no—
241C2×D4—2832no—
242C2×D4—2831no—
243C2×D4—2831no—
244C2×D4—5662no—
245C2×D4—5664no—
246C2×D4—5668no—
247C2×D4—5662no—
248C2×D4—56616no—
249C2×D4—5661no—
250C2×D4—5664no—
251C2×D4—5662no—
252C2×D4C22⋊D28314yescomplemented
253C2×D4C22×D28311yescharacteristic
254C2×D4C22×D28312yes—
255C22×C4—21no—
256C22×C4—2832no—
257C22×C4—5662no—
258C22×C4—5661no—
259C22×C4—5661no—
260C22×C4C22×D28311yescharacteristic
261Q16—5662no—
262SD16—24no—
263SD16—5662no—
264SD16—5662no—
265D8—24no—
266D8—5662no—
267M16—22no—
268C2×C8—2831no—
269C2×C8—2831no—
270C2×C8—5662no—
271C2×C8C22×D28312yes—
272C4⋊C4—2832no—
273C4⋊C4—5662no—
274C4⋊C4—5661no—
275C22⋊C4—5662no—
276C22⋊C4—5662no—
277C23—22no—
278C23—24no—
279C23—21no—
280C23—81no—
281C23—81no—
282C23—2832no—
283C23—5668no—
284C23—5661no—
285C23—22642no—
286C23—22642no—
287C23D4×D28311yescharacteristic
288Q8—21no—
289Q8—21no—
290Q8—2832no—
291Q8—5662no—
292Q8—5661no—
293D4—24no—
294D4—22no—
295D4—21no—
296D4—21no—
297D4—42no—
298D4—2832no—
299D4—5668no—
300D4—5668no—
301D4—5664no—
302D4—5664no—
303D4—5662no—
304D4—5661no—
305D4—22642no—
306D4C2×C22⋊D28314yes—
307C2×C4—21no—
308C2×C4—42no—
309C2×C4—2832no—
310C2×C4—5668no—
311C2×C4—5661no—
312C2×C4—5664no—
313C2×C4—5662no—
314C2×C4—5662no—
315C2×C4—5662no—
316C2×C4—22641no—
317C2×C4D4×D28311yescharacteristic
318C2×C4D4×D28312yes—
319C2×C4C2×C22⋊D28312yes—
320C2×C4C23×D28311yescharacteristic
321C8—22no—
322C8—5661no—
323C8—5661no—
324C22—21no—
325C22—21no—
326C22—24no—
327C22—42no—
328C22—42no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—84no—
333C22—5662no—
334C22—5662no—
335C22—5662no—
336C22—5662no—
337C22—11321no—
338C22—11321no—
339C22—22648no—
340C22#2258357812yes—
341C22C2×D4×D28311yescharacteristic
342C4—41no—
343C4—5662no—
344C4—5662no—
345C4—11322no—
346C4#2258357812yes—
347C4C2×D4×D28311yescharacteristic
348C4C22×C22⋊D28311yescharacteristic
349C2—22no—
350C2—44no—
351C2—81no—
352C2—11322no—
353C2—11322no—
354C2C2×#2258357811yescharacteristic
355C2#3182726911yescharacteristic
356C2#3182727511yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
FittingC2×AGL(1,ℤ/8)×C283C2
socleC22×C283C22×D4
hypercenterC2×AGL(1,ℤ/8)D283
hypocenterC283#3012
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#22583655C2semidirect
2. C2×#22583630C2semidirect
3. #31827043C2semidirect
4. #31826624C2semidirect
5. #31826614C2semidirect
6. #31826607C2semidirect
7. #31826597C2semidirect
8. #31826467C2semidirect
9. C2×#22583531C2semidirect
10. #31826433C2semidirect
11. C2×AGL(1,ℤ/8)×C283C2semidirect
22. C2×#17985382C22semidirect
23. C2×#17985380C22semidirect
24. C2×#17985379C22semidirect
25. C2×#17985377C22semidirect
26. C22×C283⋊C8C22semidirect
28. #22583586C22semidirect
31. #22583581C22semidirect
32. C2×C4⋊Dic283C22semidirect
33. #22583535C22semidirect
34. #22583532C22semidirect
35. #22583531C22semidirect
36. #22583531C22semidirect
37. (C2×D283)⋊C8C22semidirect
38. (C2×D283)⋊C8C22semidirect
39. C2×#41×C283C22semidirect
42. C2×SD16×C283C22semidirect
43. C2×D8×C283C22semidirect
44. C2×M16×C283C22semidirect
49. #17985392D4semidirect
89. C2×D4×C283D4semidirect
178. C283#3012semidirect
181. C2×AGL(1,ℤ/8)D283semidirect
197. C2×#41C2×D283semidirect
207. C2×SD16C2×D283semidirect
209. C2×D8C2×D283semidirect
210. C2×M16C2×D283semidirect
252. C2×D4C22⋊D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225836552
C2×#225836302
#318270432
#318266242
#318266142
#318266072
#318265972
#318264672
C2×#225835312
#318264332
C2×AGL(1,ℤ/8)×C2832
#3012283