Finite Group Databasebeta

← #49130609

Subgroups — #49130609

Order 36224.

724 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 724 conjugacy classes of subgroups — too many to draw. Only its 145 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
1#31827301C211yesmaximal, characteristic, complemented
2#31827263C212yesmaximal, complemented
3#31826647C212yesmaximal, complemented
4#31826642C212yesmaximal, complemented
5#31826637C212yesmaximal, complemented
6#31826632C212yesmaximal, complemented
7C2×#22583538C212yesmaximal, complemented
8#31826449C211yesmaximal, characteristic, complemented
9#532×C283C211yesmaximal, characteristic, complemented
10#22583658C2212yes—
11#22583657—22no—
12#22583656—22no—
13#22583655—22no—
14#22583655C2212yes—
15C2×#17985392—22no—
16C2×#17985392C2212yes—
17#41×D283—24no—
18#22583641C2214yescomplemented
19#22583640—22no—
20#22583636C2214yescomplemented
21#22583635—22no—
22#22583630—24no—
23C2×#17985376C2212yescomplemented
24#22583538C2214yescomplemented
25#22583538C2214yescomplemented
26#22583529C2212yescomplemented
27#22583527C2212yescomplemented
28C2×#41×C283C2211yescharacteristic
29C2×#41×C283C2211yescharacteristic, complemented
30C4⋊Q8×C283C2211yescharacteristic, complemented
31#127×C283C2211yescharacteristic, complemented
32#124×C283C2211yescharacteristic, complemented
33#124×C283C2211yescharacteristic, complemented
34#121×C283—22no—
35C4×D4×C283—22no—
36C2×C42×C283C2211yescharacteristic, complemented
37#17985393—24no—
38#17985392—24no—
39#17985392—24no—
40#17985392—22no—
41#17985392—24no—
42#17985392—22no—
43#17985392D414yescomplemented
44#17985392C2314yes—
45#17985391—28no—
46#17985391—24no—
47C2×Dic566—22no—
48C2×Dic566D412yescomplemented
49Q8×D283—24no—
50C2×D1132—22no—
51C2×D1132D412yescomplemented
52C2×C22⋊D283—24no—
53C2×C22⋊D283—22no—
54D4×D283—28no—
55D4×D283—24no—
56C2×C4×D283—24no—
57C2×C4×D283—22no—
58#17985382—24no—
59#17985380—24no—
60#17985379—24no—
61#17985377—24no—
62#17985376—22no—
63#17985376C2314yes—
64C2×C283⋊C8—22no—
65#41×C283—22no—
66#41×C283—22no—
67#41×C283—22no—
68#41×C283—22no—
69C2×Q8×C283C2311yescharacteristic
70C2×Q8×C283C2311yescharacteristic
71C2×D4×C283—22no—
72C2×D4×C283—21no—
73C2×D4×C283D412yes—
74C2×D4×C283C2311yescharacteristic
75C2×D4×C283C2311yescharacteristic
76C22×C4×C283—21no—
77C22×C4×C283—21no—
78C22×C4×C283D412yes—
79C22×C4×C283C2311yescharacteristic
80C4⋊C4×C283—22no—
81C22⋊C4×C283—22no—
82C22⋊C4×C283—22no—
83C42×C283—21no—
84C42×C283C2311yescharacteristic
85C42×C283C2311yescharacteristic
86C22×D283—24no—
87C22×D283—22no—
88Dic566—24no—
89Dic566—22no—
90Dic566C2×D414yes—
91D1132—24no—
92D1132—22no—
93D1132C2×D414yes—
94C22⋊D283—28no—
95C22⋊D283—24no—
96C22⋊D283—24no—
97C2×Dic283—24no—
98C2×Dic283—22no—
99C4×D283—28no—
100C4×D283—24no—
101C4×D283—24no—
102C283⋊C8—24no—
103C23×C283—21no—
104C23×C283C2×D412yes—
105C23×C283C2×D411yescharacteristic
106Q8×C283—22no—
107Q8×C283—22no—
108D4×C283—24no—
109D4×C283—22no—
110D4×C283—22no—
111D4×C283—44no—
112D4×C283—42no—
113C2×C4×C283—21no—
114C2×C4×C283—22no—
115C2×C4×C283—22no—
116C2×C4×C283—21no—
117C2×C4×C283—21no—
118C2×C4×C283—42no—
119C2×C4×C283—41no—
120C2×C4×C283—41no—
121C2×C4×C283C2×D412yes—
122C2×C4×C283C2×D412yes—
123C2×C4×C283C2×D412yes—
124C2×C4×C283C2×D411yescharacteristic
125C2×C4×C283C2411yescharacteristic
126C22×C283—22no—
127C22×C283—41no—
128C22×C283—42no—
129C22×C283—42no—
130C22×C283—41no—
131C22×C283#12012yes—
132C22×C283C22×D411yescharacteristic
133C2×D283—24no—
134C2×D283—82no—
135C2×D283—84no—
136C4×C283—22no—
137C4×C283—22no—
138C4×C283—41no—
139C4×C283—41no—
140C4×C283#12011yescharacteristic
141C4×C283#12011yescharacteristic
142C4×C283C22×D412yes—
143Dic283—24no—
144C2×C283—21no—
145C2×C283—22no—
146C2×C283—42no—
147C2×C283—81no—
148C2×C283C2×#12011yescharacteristic
149D283—44no—
150C283#301811yescharacteristic, complemented, Sylow
151#3018—2831nomaximal, Sylow
152#585—2831no—
153#575—2832no—
154#532D28311yescharacteristic, complemented
155#456—2832no—
156#455—2832no—
157#454—2832no—
158#453—2832no—
159C2×#104—2832no—
160#413—2831no—
1612−1+4—2832no—
1622−1+4—5662no—
163H(5,𝔽2)—2832no—
164H(5,𝔽2)—5662no—
165H(5,𝔽2)—5662no—
166C2×#41—2832no—
167C2×#41—5664no—
168C2×#41—5662no—
169C2×#41C2×D28311yescharacteristic
170C2×#41C2×D28311yescharacteristic, complemented
171#137—2834no—
172#137—5662no—
173AGL(1,ℤ/8)—2834no—
174AGL(1,ℤ/8)—5662no—
175#135—5664no—
176C2×M16—2832no—
177C4⋊Q8C2×D28311yescharacteristic, complemented
178#127C2×D28311yescharacteristic, complemented
179#124C2×D28311yescharacteristic, complemented
180#124C2×D28311yescharacteristic, complemented
181#121—22no—
182C4×D4—22no—
183C2×C42C2×D28311yescharacteristic, complemented
184#104—2834no—
185#104—2834no—
186#101—2832no—
187#100—2832no—
188#41—22no—
189#41—22no—
190#41—22no—
191#41—22no—
192#41—2834no—
193#41—2834no—
194#41—5664no—
195#41—5664no—
196#41—5664no—
197#41—5668no—
198#41—5664no—
199#41—5662no—
200#41—5662no—
201#41—5664no—
202C2×Q8—2832no—
203C2×Q8—5662no—
204C2×Q8—5664no—
205C2×Q8C22×D28311yescharacteristic
206C2×Q8C22×D28311yescharacteristic
207C2×D4—22no—
208C2×D4—21no—
209C2×D4—2832no—
210C2×D4—5662no—
211C2×D4—5664no—
212C2×D4—5668no—
213C2×D4—5662no—
214C2×D4—5664no—
215C2×D4C22⋊D28312yes—
216C2×D4C22×D28311yescharacteristic
217C2×D4C22×D28311yescharacteristic
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—5664no—
221C22×C4—5662no—
222C22×C4C22⋊D28312yes—
223C22×C4C22×D28311yescharacteristic
224Q16—5664no—
225SD16—5664no—
226SD16—5664no—
227D8—5664no—
228M16—2834no—
229M16—5662no—
230C2×C8—5662no—
231C4⋊C4—22no—
232C22⋊C4—22no—
233C22⋊C4—22no—
234C42—21no—
235C42C22×D28311yescharacteristic
236C42C22×D28311yescharacteristic
237C23—21no—
238C23—5664no—
239C23—5662no—
240C23C2×C22⋊D28312yes—
241C23C2×C22⋊D28311yescharacteristic
242Q8—22no—
243Q8—22no—
244Q8—2834no—
245Q8—5664no—
246Q8—5662no—
247D4—24no—
248D4—22no—
249D4—22no—
250D4—44no—
251D4—42no—
252D4—2834no—
253D4—5664no—
254D4—5668no—
255D4—5664no—
256D4—5664no—
257D4—5662no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—5668no—
267C2×C4—5664no—
268C2×C4—5662no—
269C2×C4—5664no—
270C2×C4—5664no—
271C2×C4D4×D28312yes—
272C2×C4D4×D28312yes—
273C2×C4C2×C22⋊D28312yes—
274C2×C4C2×C22⋊D28311yescharacteristic
275C2×C4C23×D28311yescharacteristic
276C8—5664no—
277C22—22no—
278C22—41no—
279C22—42no—
280C22—42no—
281C22—41no—
282C22—5664no—
283C22—22642no—
284C22—22644no—
285C22#2258357812yes—
286C22C22×C22⋊D28311yescharacteristic
287C4—22no—
288C4—22no—
289C4—41no—
290C4—41no—
291C4—5664no—
292C4#2258357811yescharacteristic
293C4#2258357811yescharacteristic
294C4C2×D4×D28312yes—
295C2—21no—
296C2—22no—
297C2—42no—
298C2—81no—
299C2—11324no—
300C2C2×#2258357811yescharacteristic
subgroupG/H
centerC4#22583578
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#532×C283C2
socleC2×C283C2×#120
hypercenter#532D283
hypocenterC283#3018
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C4 ◁ C22×C4 ◁ #532

Complements

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

NH ≅ G/Ntype
1. #31827301C2semidirect
2. #31827263C2semidirect
3. #31826647C2semidirect
4. #31826642C2semidirect
5. #31826637C2semidirect
6. #31826632C2semidirect
7. C2×#22583538C2semidirect
8. #31826449C2semidirect
9. #532×C283C2semidirect
18. #22583641C22semidirect
20. #22583636C22semidirect
23. C2×#17985376C22semidirect
24. #22583538C22semidirect
25. #22583538C22semidirect
26. #22583529C22semidirect
27. #22583527C22semidirect
29. C2×#41×C283C22semidirect
30. C4⋊Q8×C283C22semidirect
31. #127×C283C22semidirect
32. #124×C283C22semidirect
33. #124×C283C22semidirect
36. C2×C42×C283C22semidirect
43. #17985392D4semidirect
48. C2×Dic566D4semidirect
51. C2×D1132D4semidirect
150. C283#3018semidirect
154. #532D283semidirect
170. C2×#41C2×D283semidirect
177. C4⋊Q8C2×D283semidirect
178. #127C2×D283semidirect
179. #124C2×D283semidirect
180. #124C2×D283semidirect
183. C2×C42C2×D283semidirect

Maximal subgroups

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

subgroupindex
#318273012
#318272632
#318266472
#318266422
#318266372
#318266322
C2×#225835382
#318264492
#532×C2832
#3018283