Finite Group Databasebeta

← #49123880

Subgroups — #49123880

Order 36224.

600 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22583566C211yesmaximal, characteristic
2C2×#22583566C211yesmaximal, characteristic
3C22×Dic283⋊C4C211yesmaximal, characteristic
4#31826362C212yesmaximal, complemented
5#31826360C212yesmaximal, complemented
6#31826340C211yesmaximal, characteristic
7#31826338C211yesmaximal, characteristic
8#31826337C211yesmaximal, characteristic
9#31826334C212yesmaximal, complemented
10#31826329C212yesmaximal, complemented
11#385×C283C211yesmaximal, characteristic
12C23×Dic283C2211yescharacteristic
13C23×Dic283C2211yescharacteristic
14#22583566—24no—
15#22583566C418yescomplemented
16C2×C4⋊Dic283C2212yes—
17C2×Dic283⋊C4—24no—
18C2×Dic283⋊C4C2211yescharacteristic
19C2×Dic283⋊C4C2212yes—
20C2×Dic283⋊C4C2211yescharacteristic
21C2×Dic283⋊C4C2212yes—
22C2×Dic283⋊C4C2212yes—
23C2×Dic283⋊C4C2212yes—
24C2×Dic283⋊C4C2212yes—
25C2×C22⋊Dic283C2211yescharacteristic
26C2×C22⋊Dic283C2211yescharacteristic
27C2×C4×Dic283C2211yescharacteristic
28C2×C4×Dic283C2211yescharacteristic
29C2×C4×Dic283C2212yes—
30C283⋊#95C2212yes—
31C283⋊#95C2212yes—
32C283⋊#95C2212yes—
33C23×C4×C283C2211yescharacteristic
34C2×C4⋊C4×C283C2212yes—
35C2×C22⋊C4×C283C2211yescharacteristic
36C2×C22⋊C4×C283C2211yescharacteristic
37#95×C283C2212yes—
38C22×Dic283—22no—
39C22×Dic283—42no—
40C22×Dic283C2×C414yes—
41C22×Dic283D414yes—
42C22×Dic283Q814yes—
43C22×Dic283C2311yescharacteristic
44C22×Dic283C2311yescharacteristic
45C22×Dic283C2312yes—
46C22×Dic283C2312yes—
47C22×Dic283C2311yescharacteristic
48C22×Dic283C2311yescharacteristic
49C4⋊Dic283—42no—
50Dic283⋊C4—24no—
51Dic283⋊C4—24no—
52Dic283⋊C4—22no—
53Dic283⋊C4—22no—
54Dic283⋊C4—42no—
55Dic283⋊C4—42no—
56Dic283⋊C4C2×C418yescomplemented
57C22⋊Dic283—22no—
58C22⋊Dic283C2×C414yes—
59C4×Dic283—22no—
60C4×Dic283—22no—
61C4×Dic283C2×C418yescomplemented
62C24×C283C2311yescharacteristic
63C22×C4×C283—22no—
64C22×C4×C283—42no—
65C22×C4×C283C2311yescharacteristic
66C22×C4×C283C2311yescharacteristic
67C22×C4×C283C2312yes—
68C22×C4×C283C2312yes—
69C4⋊C4×C283—42no—
70C22⋊C4×C283—22no—
71C22⋊C4×C283C2×C414yes—
72C2×Dic283—22no—
73C2×Dic283—22no—
74C2×Dic283—21no—
75C2×Dic283—21no—
76C2×Dic283—22no—
77C2×Dic283—22no—
78C2×Dic283—22no—
79C2×Dic283—22no—
80C2×Dic283—22no—
81C2×Dic283—44no—
82C2×Dic283C4⋊C4116yescomplemented
83C2×Dic283C22×C414yes—
84C2×Dic283C22×C412yes—
85C2×Dic283C22×C412yes—
86C2×Dic283C2×D412yes—
87C2×Dic283C2×D412yes—
88C2×Dic283C2×Q812yes—
89C2×Dic283C2×Q812yes—
90C23×C283—22no—
91C23×C283—22no—
92C23×C283C22×C412yes—
93C23×C283C2×D412yes—
94C23×C283C2×Q812yes—
95C23×C283C2411yescharacteristic
96C2×C4×C283—22no—
97C2×C4×C283—22no—
98C2×C4×C283—21no—
99C2×C4×C283—21no—
100C2×C4×C283—22no—
101C2×C4×C283—22no—
102C2×C4×C283—22no—
103C2×C4×C283—22no—
104C2×C4×C283—22no—
105C2×C4×C283—44no—
106C2×C4×C283C22×C414yes—
107C22×C283—24no—
108C22×C283—22no—
109C22×C283—22no—
110C22×C283—24no—
111C22×C283C2×C4⋊C414yes—
112C22×C283C23×C411yescharacteristic
113C22×C283C22×D411yescharacteristic
114C22×C283C22×Q811yescharacteristic
115C22×C283H(5,𝔽2)12yes—
116C22×C2832−1+412yes—
117C4×C283—24no—
118C4×C283—41no—
119C4×C283—42no—
120C4×C283—41no—
121Dic283—24no—
122Dic283—41no—
123Dic283—42no—
124Dic283—41no—
125Dic283C2×C4⋊C414yes—
126Dic283C2×C4⋊C414yes—
127C2×C283—24no—
128C2×C283C22×C4⋊C411yescharacteristic
129C2×C283#52012yes—
130C2×C283#53511yescharacteristic
131C2×C283#53611yescharacteristic
132C2×C283#54412yes—
133C283#231711yescharacteristic, complemented, Sylow
134#2317—2831nomaximal, Sylow
135C2×#117—2831no—
136C2×#117—2831no—
137C22×C4⋊C4—2831no—
138#389—2832no—
139#389—2832no—
140#385—2831no—
141#385—2831no—
142#385—2831no—
143#385D28311yescharacteristic
144#384—2832no—
145#382—2832no—
146C23×C4—2831no—
147C23×C4—2831no—
148C23×C4C2×D28311yescharacteristic
149#117—2838no—
150#117—5664no—
151C2×C4⋊C4—2832no—
152C2×C4⋊C4—2832no—
153C2×C4⋊C4—2832no—
154C2×C4⋊C4—2831no—
155C2×C4⋊C4—2831no—
156C2×C4⋊C4—2832no—
157C2×C4⋊C4—2832no—
158C2×C4⋊C4—2832no—
159C2×C4⋊C4—5664no—
160C2×C4⋊C4C2×D28312yes—
161C2×C22⋊C4—2831no—
162C2×C22⋊C4—2831no—
163C2×C22⋊C4C2×D28311yescharacteristic
164C2×C22⋊C4C2×D28311yescharacteristic
165C2×C42—2832no—
166C2×C42—2831no—
167C2×C42—2831no—
168#95—2832no—
169#95—2832no—
170#95—2832no—
171#95C2×D28312yes—
172C24C22×D28311yescharacteristic
173C22×C4—22no—
174C22×C4—42no—
175C22×C4—2834no—
176C22×C4—2834no—
177C22×C4—2834no—
178C22×C4—2831no—
179C22×C4—2831no—
180C22×C4—2832no—
181C22×C4—2831no—
182C22×C4—2832no—
183C22×C4—2831no—
184C22×C4—5662no—
185C22×C4—11322no—
186C22×C4C22×D28311yescharacteristic
187C22×C4C22×D28312yes—
188C22×C4C22×D28311yescharacteristic
189C22×C4C22×D28312yes—
190C4⋊C4—42no—
191C4⋊C4—2838no—
192C4⋊C4—5664no—
193C4⋊C4—5664no—
194C4⋊C4—5662no—
195C4⋊C4—5662no—
196C4⋊C4—11322no—
197C4⋊C4—11322no—
198C4⋊C4—11322no—
199C22⋊C4—22no—
200C22⋊C4—2834no—
201C22⋊C4—5662no—
202C22⋊C4C4×D28314yes—
203C42—2838no—
204C42—5662no—
205C42—5662no—
206C23—22no—
207C23—22no—
208C23C2×C4×D28312yes—
209C23D4×D28312yes—
210C23Q8×D28312yes—
211C23C23×D28311yescharacteristic
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—44no—
222C2×C4—28316no—
223C2×C4—2832no—
224C2×C4—2832no—
225C2×C4—2832no—
226C2×C4—2832no—
227C2×C4—2832no—
228C2×C4—2832no—
229C2×C4—2834no—
230C2×C4—5662no—
231C2×C4—5662no—
232C2×C4—5661no—
233C2×C4—5661no—
234C2×C4—5662no—
235C2×C4—5662no—
236C2×C4—5662no—
237C2×C4—5662no—
238C2×C4—5662no—
239C2×C4—11324no—
240C2×C4C2×C4×D28314yes—
241C22—24no—
242C22—22no—
243C22—22no—
244C22—24no—
245C22C4⋊C4×D28314yes—
246C22C22×C4×D28311yescharacteristic
247C22C2×D4×D28311yescharacteristic
248C22C2×Q8×D28311yescharacteristic
249C22#2258365512yes—
250C22#2258365712yes—
251C4—24no—
252C4—42no—
253C4—41no—
254C4—41no—
255C4—2834no—
256C4—2834no—
257C4—5664no—
258C4—11322no—
259C4—11321no—
260C4—11321no—
261C2—24no—
262C2C2×C4⋊C4×D28311yescharacteristic
263C2#3182694412yes—
264C2#3182703011yescharacteristic
265C2#3182703711yescharacteristic
266C2#3182709412yes—
subgroupG/H
center, FrattiniC23C23×D283
commutatorC22×C283C23×C4
Fitting#385×C283C2
socle, cosocleC23×C283C24
prime radicalC24×C283C23
hypercenter#385D283
hypocenterC283#2317
LCSG ▷ C22×C283 ▷ C283
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
4. #31826362C2semidirect
5. #31826360C2semidirect
9. #31826334C2semidirect
10. #31826329C2semidirect
15. #22583566C4semidirect
56. Dic283⋊C4C2×C4semidirect
61. C4×Dic283C2×C4semidirect
82. C2×Dic283C4⋊C4semidirect
133. C283#2317semidirect

Maximal subgroups

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

subgroupindex
C2×#225835662
C22×Dic283⋊C42
#318263622
#318263602
#318263402
#318263382
#318263372
#318263342
#318263292
#385×C2832
#2317283