Finite Group Databasebeta

← #49124404

Subgroups — #49124404

Order 36224.

512 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 173 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×#22583591C211yesmaximal, characteristic
2#31826368C211yesmaximal, characteristic
3#31826362C212yesmaximal, complemented
4#31826359C212yesmaximal, complemented
5#31826353C212yesmaximal, complemented
6#31826347C212yesmaximal, complemented
7#31826338C212yesmaximal, complemented
8#31826323C212yesmaximal, complemented
9#386×C283C211yesmaximal, characteristic
10C23×Dic283C2211yescharacteristic
11#22583591C418yescomplemented
12C2×C4⋊Dic283C2211yescharacteristic
13C2×Dic283⋊C4C2212yes—
14C2×Dic283⋊C4C2212yes—
15C2×Dic283⋊C4C2212yes—
16C2×Dic283⋊C4C2212yes—
17C2×C22⋊Dic283C2212yes—
18C2×C22⋊Dic283C2212yes—
19C2×C22⋊Dic283C2212yes—
20C2×C22⋊Dic283C2211yescharacteristic
21C2×C4×Dic283C2211yescharacteristic
22C2×C4×Dic283C2212yes—
23C283⋊#95C2212yes—
24C283⋊#95C2212yes—
25C283⋊#95C2212yes—
26C283⋊#95C2212yes—
27C22×D4×C283C2211yescharacteristic
28C23×C4×C283C2211yescharacteristic
29C2×C22⋊C4×C283C2212yes—
30C2×C22⋊C4×C283C2211yescharacteristic
31#95×C283C2212yes—
32C22×Dic283—41no—
33C22×Dic283—41no—
34C22×Dic283C2×C414yescomplemented
35C22×Dic283C2312yes—
36C22×Dic283C2312yes—
37C22×Dic283C2312yes—
38C22×Dic283C2312yes—
39C4⋊Dic283—41no—
40Dic283⋊C4—42no—
41Dic283⋊C4—42no—
42Dic283⋊C4—42no—
43Dic283⋊C4C2×C418yescomplemented
44C22⋊Dic283—42no—
45C22⋊Dic283—42no—
46C22⋊Dic283C2×C418yes—
47C22⋊Dic283C2×C414yescomplemented
48C4×Dic283—41no—
49C4×Dic283—42no—
50C24×C283C2311yescharacteristic
51C24×C283C2311yescharacteristic
52C2×D4×C283—41no—
53C2×D4×C283—41no—
54C2×D4×C283C2×C414yes—
55C22×C4×C283—42no—
56C22×C4×C283—41no—
57C22×C4×C283C2312yes—
58C22×C4×C283C2312yes—
59C22×C4×C283C2311yescharacteristic
60C22⋊C4×C283—42no—
61C22⋊C4×C283—41no—
62C2×Dic283—22no—
63C2×Dic283—22no—
64C2×Dic283—22no—
65C2×Dic283—22no—
66C2×Dic283—22no—
67C2×Dic283—22no—
68C2×Dic283—22no—
69C2×Dic283—22no—
70C2×Dic283—22no—
71C2×Dic283—22no—
72C2×Dic283—44no—
73C2×Dic283C22×C414yes—
74C2×Dic283C22×C414yes—
75C23×C283—21no—
76C23×C283—22no—
77C23×C283—21no—
78C23×C283—22no—
79C23×C283—21no—
80C23×C283—21no—
81C23×C283—22no—
82C23×C283—22no—
83C23×C283C22×C412yes—
84C23×C283C22×C412yes—
85C23×C283C2411yescharacteristic
86D4×C283—44no—
87C2×C4×C283—21no—
88C2×C4×C283—21no—
89C2×C4×C283—22no—
90C2×C4×C283—22no—
91C2×C4×C283—22no—
92C2×C4×C283—22no—
93C2×C4×C283—22no—
94C2×C4×C283—22no—
95C2×C4×C283—44no—
96C2×C4×C283C22×C412yes—
97C22×C283—22no—
98C22×C283—22no—
99C22×C283—24no—
100C22×C283—24no—
101C22×C283—41no—
102C22×C283—41no—
103C22×C283—42no—
104C22×C283—42no—
105C22×C283—41no—
106C22×C283—41no—
107C22×C283C23×C411yescharacteristic
108C22×C283H(5,𝔽2)11yescharacteristic
109C22×C283H(5,𝔽2)11yescharacteristic
110C22×C283H(5,𝔽2)12yes—
111C22×C2832−1+412yes—
112C4×C283—22no—
113C4×C283—42no—
114C4×C283—42no—
115Dic283—24no—
116Dic283—24no—
117Dic283—42no—
118Dic283—42no—
119C2×C283—42no—
120C2×C283—42no—
121C2×C283#51811yescharacteristic
122C2×C283#52012yes—
123C2×C283#56011yescharacteristic
124C2×C283#56212yes—
125C2×C283#56311yescharacteristic
126C283#237611yescharacteristic, complemented, Sylow
127#2376—2831nomaximal, Sylow
128C2×#123—2831no—
129#390—2831no—
130#389—2832no—
131#388—2832no—
132#388—2832no—
133#387—2832no—
134#386D28311yescharacteristic
135#385—2832no—
136#380—2832no—
137C22×D4C2×D28311yescharacteristic
138C23×C4—2831no—
139C23×C4C2×D28311yescharacteristic
140#123—2838no—
141C2×C4⋊C4—2832no—
142C2×C4⋊C4—2832no—
143C2×C4⋊C4—2832no—
144C2×C4⋊C4—2832no—
145C2×C4⋊C4—2831no—
146C2×C22⋊C4—2832no—
147C2×C22⋊C4—2831no—
148C2×C22⋊C4—2832no—
149C2×C22⋊C4—2832no—
150C2×C22⋊C4C2×D28312yes—
151C2×C22⋊C4C2×D28311yescharacteristic
152C2×C42—2831no—
153C2×C42—2832no—
154#95—2832no—
155#95—2832no—
156#95—2832no—
157#95—2832no—
158#95C2×D28312yes—
159C24C22×D28311yescharacteristic
160C24C22×D28311yescharacteristic
161C2×D4—41no—
162C2×D4—41no—
163C2×D4C4×D28314yes—
164C22×C4—42no—
165C22×C4—41no—
166C22×C4—2832no—
167C22×C4—2832no—
168C22×C4—2834no—
169C22×C4—2832no—
170C22×C4—2832no—
171C22×C4—11321no—
172C22×C4—11321no—
173C22×C4C22×D28312yes—
174C22×C4C22×D28312yes—
175C22×C4C22×D28311yescharacteristic
176C4⋊C4—2838no—
177C4⋊C4—11322no—
178C4⋊C4—11322no—
179C4⋊C4—11322no—
180C4⋊C4—11321no—
181C22⋊C4—42no—
182C22⋊C4—41no—
183C22⋊C4—2838no—
184C22⋊C4—2834no—
185C22⋊C4—11322no—
186C22⋊C4—11322no—
187C42—11321no—
188C42—11322no—
189C23—22no—
190C23—21no—
191C23—22no—
192C23—21no—
193C23—21no—
194C23—21no—
195C23—22no—
196C23—22no—
197C23C2×C4×D28312yes—
198C23C2×C4×D28312yes—
199C23C23×D28311yescharacteristic
200D4—44no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—44no—
210C2×C4—2834no—
211C2×C4—2834no—
212C2×C4—5662no—
213C2×C4—5662no—
214C2×C4—5662no—
215C2×C4—5662no—
216C2×C4—5662no—
217C2×C4—5662no—
218C2×C4—5662no—
219C2×C4—5662no—
220C2×C4—5662no—
221C2×C4—5662no—
222C2×C4—11324no—
223C2×C4C2×C4×D28312yes—
224C22—24no—
225C22—24no—
226C22—22no—
227C22—22no—
228C22—41no—
229C22—41no—
230C22—42no—
231C22—42no—
232C22—41no—
233C22—41no—
234C22C22×C4×D28311yescharacteristic
235C22#2258365512yes—
236C22#2258365511yescharacteristic
237C22#2258365511yescharacteristic
238C22#2258365712yes—
239C4—22no—
240C4—42no—
241C4—42no—
242C4—5664no—
243C4—5664no—
244C4—11322no—
245C4—11322no—
246C2—42no—
247C2—42no—
248C2#3182693811yescharacteristic
249C2#3182694412yes—
250C2#3182720411yescharacteristic
251C2#3182721012yes—
252C2#3182721811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D283
commutatorC22×C283C23×C4
Fitting#386×C283C2
socle, cosocleC23×C283C24
prime radicalC22×D4×C283C22
hypercenter#386D283
hypocenterC283#2376
LCSG ▷ C22×C283 ▷ C283
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
3. #31826362C2semidirect
4. #31826359C2semidirect
5. #31826353C2semidirect
6. #31826347C2semidirect
7. #31826338C2semidirect
8. #31826323C2semidirect
11. #22583591C4semidirect
34. C22×Dic283C2×C4semidirect
43. Dic283⋊C4C2×C4semidirect
47. C22⋊Dic283C2×C4semidirect
126. C283#2376semidirect

Maximal subgroups

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

subgroupindex
C2×#225835912
#318263682
#318263622
#318263592
#318263532
#318263472
#318263382
#318263232
#386×C2832
#2376283