Finite Group Databasebeta

← #49130549

Subgroups — #49130549

Order 36224.

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

normal subgroups only

This group has 732 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×#22583657C211yesmaximal, characteristic, complemented
2C2×#22583638C211yesmaximal, characteristic, complemented
3C22×#17985381C211yesmaximal, characteristic, complemented
4#31827029C211yesmaximal, characteristic, complemented
5#31826612C214yesmaximal, complemented
6#31826606C214yesmaximal, complemented
7#31826475C211yesmaximal, characteristic, complemented
8#31826462C211yesmaximal, characteristic, complemented
9#408×C283C211yesmaximal, characteristic, complemented
10#22583657—24no—
11#22583657—24no—
12C2×#17985392C2211yescharacteristic
13C2×#17985392C2212yescomplemented
14C2×#17985391—22no—
15C2×#17985391—21no—
16C22×Dic566C2211yescharacteristic
17C22×Dic566C2212yescomplemented
18C2×Q8×D283—21no—
19C22×D1132C2211yescharacteristic
20#22583638—24no—
21C2×#17985381—24no—
22C2×#17985381C2214yes—
23C2×#17985381C2212yes—
24C2×D2264C2212yes—
25#22583606—21no—
26#22583599—21no—
27#22583584C2214yes—
28#22583578—22no—
29#22583569C2211yescharacteristic
30#22583535C2212yescomplemented
31#22583535C2214yescomplemented
32#22583532C2212yescomplemented
33C2×#41×C283C2211yescharacteristic
34C2×M16×C283C2211yescharacteristic
35C22×C8×C283C2211yescharacteristic
36C22⋊C8×C283C2214yescomplemented
37C23×D283—21no—
38#17985392—24no—
39#17985392—24no—
40#17985392—22no—
41#17985392D414yescomplemented
42#17985391—216no—
43#17985391—28no—
44C2×Dic566—24no—
45C2×Dic566—24no—
46C2×Dic566—21no—
47C2×Dic566—22no—
48C2×Dic566—22no—
49C2×Dic566D414yescomplemented
50C2×Dic566C2312yes—
51C2×Dic566C2311yescharacteristic
52Q8×D283—28no—
53C2×D1132—22no—
54C2×D1132—22no—
55C2×D1132—21no—
56C2×D1132C2312yes—
57C2×D1132C2311yescharacteristic
58C2×C22⋊D283—21no—
59C2×C22⋊D283—22no—
60C2×C22⋊D283—24no—
61C22×Dic283—21no—
62C22×Dic283—22no—
63C2×C4×D283—21no—
64C2×C4×D283—22no—
65#17985381—28no—
66#17985381—24no—
67D2264—24no—
68C4⋊Dic283C2312yes—
69C22⋊Dic283—21no—
70(C2×D283)⋊C4—24no—
71C4×Dic283—21no—
72#41×C283—22no—
73#41×C283—22no—
74C2×Q8×C283D411yescharacteristic
75C2×D4×C283D412yes—
76C2×D4×C283D411yescharacteristic
77C22×C4×C283C2312yes—
78C22×C4×C283C2311yescharacteristic
79M16×C283—22no—
80C2×C8×C283—22no—
81C2×C8×C283C2312yes—
82C2×C8×C283C2312yes—
83C22×D283—21no—
84C22×D283—22no—
85C22×D283—42no—
86C22×D283—82no—
87Dic566—28no—
88Dic566—24no—
89Dic566—21no—
90Dic566—22no—
91Dic566C2×D412yes—
92Dic566C2×D414yes—
93D1132—21no—
94D1132—22no—
95D1132—22no—
96D1132—42no—
97D1132C2×D412yes—
98C22⋊D283—28no—
99C22⋊D283—24no—
100C22⋊D283—84no—
101C2×Dic283—28no—
102C2×Dic283—24no—
103C2×Dic283—21no—
104C2×Dic283—22no—
105C2×Dic283—22no—
106C4×D283—28no—
107C4×D283—24no—
108C23×C283C2×D412yes—
109C23×C283C2×D411yescharacteristic
110Q8×C283—22no—
111D4×C283—24no—
112D4×C283—22no—
113C2×C4×C283—22no—
114C2×C4×C283—22no—
115C2×C4×C283C2×D412yes—
116C2×C4×C283C2×D412yes—
117C2×C4×C283C2×D412yes—
118C2×C4×C283C2×D411yescharacteristic
119C2×C4×C283C2411yescharacteristic
120C8×C283—22no—
121C8×C283—22no—
122C22×C283—22no—
123C22×C283—22no—
124C22×C283—24no—
125C22×C283#12012yes—
126C22×C283C22×D411yescharacteristic
127C2×D283—22no—
128C2×D283—42no—
129C2×D283—42no—
130C2×D283—44no—
131C2×D283—84no—
132C4×C283—22no—
133C4×C283#12012yes—
134C4×C283C22×D411yescharacteristic
135C4×C283C22×D411yescharacteristic
136Dic283—24no—
137Dic283—22no—
138Dic283—42no—
139C2×C283—22no—
140C2×C283—42no—
141C2×C283C2×#12011yescharacteristic
142C2×C283#57712yes—
143D283—42no—
144D283—82no—
145C283#301011yescharacteristic, complemented, Sylow
146#3010—2831nomaximal, Sylow
147C2×2−1+4—2831no—
148C2×AGL(1,ℤ/8)—2831no—
149C22×SD16—2831no—
150#535—2831no—
151#449—2834no—
152#448—2834no—
153#419—2831no—
154#416—2831no—
155#408D28311yescharacteristic, complemented
1562−1+4—5664no—
1572−1+4—5664no—
158C2×#41—2831no—
159C2×#41—2832no—
160C2×#41—5662no—
161C2×#41—5661no—
162C2×#41C2×D28311yescharacteristic
163C22×Q8—2831no—
164C22×Q8—2832no—
165C22×Q8—5661no—
166C22×D4—2831no—
167AGL(1,ℤ/8)—5664no—
168C2×SD16—2834no—
169C2×SD16—2832no—
170C2×SD16—5664no—
171C2×D8—2832no—
172C2×M16C2×D28311yescharacteristic
173C22×C8C2×D28311yescharacteristic
174#127—5661no—
175#124—5661no—
176#121—2834no—
177#120—5662no—
178#117—2831no—
179#103—2832no—
180#103—2834no—
181#102—2832no—
182C22⋊C8C2×D28314yescomplemented
183C24—5661no—
184#41—22no—
185#41—22no—
186#41—2834no—
187#41—5664no—
188#41—5664no—
189#41—56616no—
190#41—5668no—
191#41—5662no—
192C2×Q8—2834no—
193C2×Q8—2832no—
194C2×Q8—2831no—
195C2×Q8—5664no—
196C2×Q8—5664no—
197C2×Q8—5668no—
198C2×Q8—5662no—
199C2×Q8—5662no—
200C2×Q8—5661no—
201C2×Q8D113211yescharacteristic
202C2×D4—2831no—
203C2×D4—2832no—
204C2×D4—5662no—
205C2×D4—5661no—
206C2×D4—5664no—
207C2×D4—5662no—
208C2×D4—5662no—
209C2×D4—5661no—
210C2×D4D113212yes—
211C2×D4D113211yescharacteristic
212C22×C4—5662no—
213C22×C4—5661no—
214C22×C4—5662no—
215C22×C4—5661no—
216C22×C4C22×D28312yes—
217C22×C4C22×D28311yescharacteristic
218SD16—5668no—
219SD16—5664no—
220D8—5664no—
221M16—22no—
222C2×C8—22no—
223C2×C8C22×D28312yes—
224C2×C8C22×D28312yes—
225C4⋊C4—2832no—
226C22⋊C4—5661no—
227C22⋊C4—5664no—
228C42—5661no—
229C23—5661no—
230C23—5662no—
231C23—11322no—
232C23—22642no—
233C23C2×D113212yes—
234C23C2×D113211yescharacteristic
235Q8—22no—
236Q8—2834no—
237Q8—2832no—
238Q8—5668no—
239Q8—5664no—
240Q8—5662no—
241Q8—5661no—
242D4—24no—
243D4—22no—
244D4—2832no—
245D4—5668no—
246D4—5664no—
247D4—5661no—
248D4—5662no—
249D4—5662no—
250D4—11322no—
251D4—22644no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—5668no—
255C2×C4—5668no—
256C2×C4—5664no—
257C2×C4—5662no—
258C2×C4—5661no—
259C2×C4—5664no—
260C2×C4—5662no—
261C2×C4D4×D28312yes—
262C2×C4D4×D28312yes—
263C2×C4C2×D113212yes—
264C2×C4C2×D113211yescharacteristic
265C2×C4C23×D28311yescharacteristic
266C8—22no—
267C8—22no—
268C22—22no—
269C22—22no—
270C22—24no—
271C22—5662no—
272C22—11322no—
273C22—11322no—
274C22—11324no—
275C22—22644no—
276C22#2258358012yes—
277C22C22×D113211yescharacteristic
278C4—22no—
279C4—5664no—
280C4—5662no—
281C4—11322no—
282C4#2258358012yes—
283C4C2×D4×D28311yescharacteristic
284C4C2×D4×D28311yescharacteristic
285C2—22no—
286C2—42no—
287C2—11322no—
288C2—22642no—
289C2C2×#2258358011yescharacteristic
290C2#3182727712yes—
subgroupG/H
centerC22C22×D1132
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#408×C283C2
socleC22×C283C22×D4
hypercenter#408D283
hypocenterC283#3010
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
1. C2×#22583657C2semidirect
2. C2×#22583638C2semidirect
3. C22×#17985381C2semidirect
4. #31827029C2semidirect
5. #31826612C2semidirect
6. #31826606C2semidirect
7. #31826475C2semidirect
8. #31826462C2semidirect
9. #408×C283C2semidirect
13. C2×#17985392C22semidirect
17. C22×Dic566C22semidirect
30. #22583535C22semidirect
31. #22583535C22semidirect
32. #22583532C22semidirect
36. C22⋊C8×C283C22semidirect
41. #17985392D4semidirect
49. C2×Dic566D4semidirect
145. C283#3010semidirect
155. #408D283semidirect
182. C22⋊C8C2×D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225836572
C2×#225836382
C22×#179853812
#318270292
#318266122
#318266062
#318264752
#318264622
#408×C2832
#3010283