Finite Group Databasebeta

← #49130523

Subgroups — #49130523

Order 36224.

754 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 153 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×#17985392C211yesmaximal, characteristic
2C2×#22583643C212yesmaximal, complemented
3C2×#22583589C211yesmaximal, characteristic, complemented
4#31826629C214yesmaximal, complemented
5#31826623C212yesmaximal, complemented
6#31826618C212yesmaximal, complemented
7#31826467C212yesmaximal, complemented
8#407×C283C211yesmaximal, characteristic, complemented
9C2×#17985392—28no—
10C2×#17985392—21no—
11C2×#17985392—21no—
12C2×#17985392C2212yes—
13C2×#17985392C2212yes—
14C22×Dic566C2211yescharacteristic
15C22×Dic566C2211yescharacteristic, complemented
16C22×D1132C2211yescharacteristic
17C22×C22⋊D283—21no—
18C22×C4×D283—21no—
19#22583643—28no—
20C2×Dic1132C2214yescomplemented
21C2×#17985381C2214yescomplemented
22#22583589—21no—
23#22583589—21no—
24#22583589C2212yes—
25#22583589C2212yescomplemented
26C2×C4⋊Dic283C2211yescharacteristic, complemented
27C2×Dic283⋊C4—21no—
28C2×C22⋊Dic283—21no—
29#22583535C2214yescomplemented
30#22583532C2214yescomplemented
31C23×C4×C283C2211yescharacteristic
32C2×M16×C283C2212yescomplemented
33C22⋊C8×C283C2212yes—
34C22⋊C8×C283C2212yescomplemented
35C23×D283—21no—
36#17985392—28no—
37#17985392—28no—
38#17985392—24no—
39#17985392—24no—
40#17985392—24no—
41#17985392D418yes—
42C2×Dic566—24no—
43C2×Dic566—21no—
44C2×Dic566—21no—
45C2×Dic566—24no—
46C2×Dic566C2312yes—
47C2×Dic566C2312yes—
48C2×D1132—24no—
49C2×D1132—21no—
50C2×D1132C2312yes—
51C2×C22⋊D283—28no—
52C2×C22⋊D283—22no—
53C2×C22⋊D283—22no—
54C22×Dic283—21no—
55C22×Dic283—21no—
56C22×Dic283—21no—
57C2×C4×D283—28no—
58C2×C4×D283—22no—
59C2×C4×D283—22no—
60Dic1132—28no—
61#17985381—28no—
62C4⋊Dic283—21no—
63C4⋊Dic283C2312yes—
64Dic283⋊C4—22no—
65Dic283⋊C4—22no—
66C22⋊Dic283—22no—
67C22⋊Dic283—22no—
68C24×C283D411yescharacteristic
69C22×C4×C283—24no—
70C22×C4×C283D411yescharacteristic
71C22×C4×C283D411yescharacteristic
72C22×C4×C283D411yescharacteristic
73C22×C4×C283C2311yescharacteristic
74C22×C4×C283C2311yescharacteristic
75C22×C4×C283C2311yescharacteristic
76M16×C283—24no—
77C2×C8×C283C2314yes—
78C22×D283—24no—
79C22×D283—22no—
80C22×D283—42no—
81C22×D283—42no—
82Dic566—24no—
83Dic566—22no—
84Dic566—24no—
85Dic566—42no—
86Dic566C2×D414yes—
87D1132—24no—
88D1132—22no—
89D1132C2×D414yes—
90C22⋊D283—28no—
91C22⋊D283—28no—
92C2×Dic283—24no—
93C2×Dic283—22no—
94C2×Dic283—22no—
95C2×Dic283—22no—
96C2×Dic283—42no—
97C2×Dic283—42no—
98C4×D283—28no—
99C4×D283—28no—
100C23×C283—22no—
101C23×C283—22no—
102C23×C283—41no—
103C23×C283C2×D411yescharacteristic
104C23×C283C2×D411yescharacteristic
105C23×C283C2×D411yescharacteristic
106C2×C4×C283—22no—
107C2×C4×C283—22no—
108C2×C4×C283—22no—
109C2×C4×C283—22no—
110C2×C4×C283—22no—
111C2×C4×C283C2×D414yes—
112C2×C4×C283C2×D411yescharacteristic
113C2×C4×C283C2×D411yescharacteristic
114C2×C4×C283C2×D411yescharacteristic
115C2×C4×C283C2411yescharacteristic
116C8×C283—24no—
117C22×C283—21no—
118C22×C283—21no—
119C22×C283—21no—
120C22×C283—21no—
121C22×C283—21no—
122C22×C283—21no—
123C22×C283—21no—
124C22×C283—44no—
125C22×C283#12012yes—
126C22×C283#13712yescomplemented
127C22×C283C22×D411yescharacteristic
128C2×D283—24no—
129C2×D283—48no—
130C2×D283—42no—
131C2×D283—42no—
132C4×C283—21no—
133C4×C283—21no—
134C4×C283#12012yes—
135C4×C283C22×D412yes—
136Dic283—24no—
137Dic283—42no—
138Dic283—42no—
139C2×C283—22no—
140C2×C283—22no—
141C2×C283—41no—
142C2×C283C2×#12011yescharacteristic
143C2×C283C2×#13711yescharacteristic
144C2×C283#57511yescharacteristic
145D283—44no—
146C283#300711yescharacteristic, complemented, Sylow
147#3007—2831nomaximal, Sylow
148C22×#41—2831no—
149C2×#137—2832no—
150C2×C22⋊Q8—2831no—
151#452—2834no—
152#451—2832no—
153#450—2832no—
154#417—2832no—
155#407D28311yescharacteristic, complemented
156C2×#41—2832no—
157C2×#41—2832no—
158C2×#41—5668no—
159C2×#41—5661no—
160C2×#41—5661no—
161C22×Q8—2831no—
162C22×Q8—2831no—
163C22×D4—2831no—
164C22×D4—5661no—
165C23×C4—5661no—
166C23×C4C2×D28311yescharacteristic
167#137—5668no—
168C2×Q16—2834no—
169C2×SD16—2834no—
170C2×M16C2×D28312yescomplemented
171C22⋊Q8—2832no—
172C22⋊Q8—2832no—
173C22⋊Q8—5661no—
174C22⋊Q8—5661no—
175C2×C4⋊C4—2831no—
176C2×C4⋊C4—5661no—
177C2×C22⋊C4—5661no—
178#103—2834no—
179#102—2834no—
180C22⋊C8C2×D28312yes—
181C22⋊C8C2×D28312yescomplemented
182C24—5661no—
183C24D113211yescharacteristic
184#41—2838no—
185#41—5664no—
186#41—5664no—
187#41—5668no—
188#41—5668no—
189#41—5664no—
190C2×Q8—2832no—
191C2×Q8—2832no—
192C2×Q8—5664no—
193C2×Q8—5661no—
194C2×Q8—5661no—
195C2×Q8—5664no—
196C2×D4—2832no—
197C2×D4—5664no—
198C2×D4—5668no—
199C2×D4—5662no—
200C2×D4—5662no—
201C2×D4—5661no—
202C22×C4—24no—
203C22×C4—5668no—
204C22×C4—5661no—
205C22×C4—5662no—
206C22×C4—5662no—
207C22×C4—5661no—
208C22×C4—5661no—
209C22×C4D113211yescharacteristic
210C22×C4D113211yescharacteristic
211C22×C4D113211yescharacteristic
212C22×C4C22×D28311yescharacteristic
213C22×C4C22×D28311yescharacteristic
214C22×C4C22×D28311yescharacteristic
215Q16—5668no—
216SD16—5668no—
217M16—24no—
218C2×C8C22×D28314yes—
219C4⋊C4—2832no—
220C4⋊C4—5662no—
221C4⋊C4—5662no—
222C4⋊C4—5661no—
223C22⋊C4—5662no—
224C22⋊C4—5662no—
225C23—22no—
226C23—22no—
227C23—41no—
228C23—5664no—
229C23—5662no—
230C23—11322no—
231C23—11322no—
232C23C2×D113211yescharacteristic
233C23C2×D113211yescharacteristic
234C23C2×D113211yescharacteristic
235Q8—2834no—
236Q8—5664no—
237Q8—5662no—
238Q8—5664no—
239Q8—11322no—
240D4—2834no—
241D4—5664no—
242D4—5668no—
243D4—5668no—
244D4—5662no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—5668no—
251C2×C4—5664no—
252C2×C4—5662no—
253C2×C4—5668no—
254C2×C4—5662no—
255C2×C4—5662no—
256C2×C4—11322no—
257C2×C4—11322no—
258C2×C4D4×D28314yes—
259C2×C4C2×D113211yescharacteristic
260C2×C4C2×D113211yescharacteristic
261C2×C4C2×D113211yescharacteristic
262C2×C4C23×D28311yescharacteristic
263C8—24no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—44no—
272C22—5664no—
273C22—11328no—
274C22—11322no—
275C22—11322no—
276C22#2258358012yes—
277C22#2258364312yescomplemented
278C22C22×D113211yescharacteristic
279C4—21no—
280C4—21no—
281C4—5664no—
282C4—11322no—
283C4—11322no—
284C4#2258358012yes—
285C4C2×D4×D28312yes—
286C2—22no—
287C2—22no—
288C2—41no—
289C2—11324no—
290C2C2×#2258358011yescharacteristic
291C2C2×#2258364311yescharacteristic
292C2#3182726511yescharacteristic
subgroupG/H
centerC22C22×D1132
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#407×C283C2
socleC22×C283C22×D4
prime radicalC22×#17985392C2
hypercenter#407D283
hypocenterC283#3007
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C2×#22583643C2semidirect
3. C2×#22583589C2semidirect
4. #31826629C2semidirect
5. #31826623C2semidirect
6. #31826618C2semidirect
7. #31826467C2semidirect
8. #407×C283C2semidirect
15. C22×Dic566C22semidirect
20. C2×Dic1132C22semidirect
21. C2×#17985381C22semidirect
25. #22583589C22semidirect
26. C2×C4⋊Dic283C22semidirect
29. #22583535C22semidirect
30. #22583532C22semidirect
32. C2×M16×C283C22semidirect
34. C22⋊C8×C283C22semidirect
126. C22×C283#137semidirect
146. C283#3007semidirect
155. #407D283semidirect
170. C2×M16C2×D283semidirect
181. C22⋊C8C2×D283semidirect
277. C22#22583643semidirect

Maximal subgroups

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

subgroupindex
C22×#179853922
C2×#225836432
C2×#225835892
#318266292
#318266232
#318266182
#318264672
#407×C2832
#3007283