Finite Group Databasebeta

← #49123883

Subgroups — #49123883

Order 36224.

600 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22583568C211yesmaximal, characteristic
2C2×#22583567C211yesmaximal, characteristic
3C2×C4⋊C4×D283C211yesmaximal, characteristic
4#31826364C211yesmaximal, characteristic, complemented
5#31826363C211yesmaximal, characteristic, complemented
6#31826361C211yesmaximal, characteristic, complemented
7#31826339C212yesmaximal
8#31826336C212yesmaximal
9#31826334C212yesmaximal, complemented
10#31826329C212yesmaximal, complemented
11#389×C283C211yesmaximal, characteristic, complemented
12C22×C4×D283C2212yes—
13C22×C4×D283C2211yescharacteristic
14#22583568C418yescomplemented
15#22583567—24no—
16C2×C4⋊Dic283C2211yescharacteristic
17C2×C4⋊Dic283C2212yes—
18C2×C4⋊Dic283C2211yescharacteristic
19C2×Dic283⋊C4C2211yescharacteristic
20C2×Dic283⋊C4C2212yes—
21C2×Dic283⋊C4C2211yescharacteristic
22C2×Dic283⋊C4C2212yes—
23C2×Dic283⋊C4C2212yes—
24C4⋊C4×D283—24no—
25C2×(C2×D283)⋊C4C2211yescharacteristic
26C2×(C2×D283)⋊C4C2211yescharacteristic
27C2×(C2×D283)⋊C4C2212yes—
28C2×C4×Dic283C2211yescharacteristic
29C2×C4×Dic283C2212yes—
30C283⋊#95C2212yes—
31C283⋊#95C2212yes—
32C283⋊#95C2212yes—
33C2×C4⋊C4×C283C2211yescharacteristic
34C2×C4⋊C4×C283C2212yes—
35C2×C4⋊C4×C283C2211yescharacteristic
36C2×C42×C283C2211yescharacteristic
37#95×C283C2212yes—
38C23×D283C2311yescharacteristic
39C22×Dic283C2311yescharacteristic
40C22×Dic283C2311yescharacteristic
41C22×Dic283C2312yes—
42C22×Dic283C2312yes—
43C22×Dic283C2311yescharacteristic
44C2×C4×D283—24no—
45C2×C4×D283—42no—
46C2×C4×D283—42no—
47C2×C4×D283C2×C414yes—
48C2×C4×D283D414yes—
49C2×C4×D283Q814yes—
50C4⋊Dic283—22no—
51C4⋊Dic283—42no—
52C4⋊Dic283C2×C414yescomplemented
53Dic283⋊C4—24no—
54Dic283⋊C4—22no—
55Dic283⋊C4—22no—
56Dic283⋊C4—42no—
57Dic283⋊C4—42no—
58(C2×D283)⋊C4—22no—
59(C2×D283)⋊C4—22no—
60(C2×D283)⋊C4C2×C418yes—
61C4×Dic283—22no—
62C4×Dic283C2×C418yescomplemented
63C22×C4×C283C2311yescharacteristic
64C22×C4×C283C2311yescharacteristic
65C22×C4×C283C2312yes—
66C22×C4×C283C2312yes—
67C22×C4×C283C2311yescharacteristic
68C4⋊C4×C283—22no—
69C4⋊C4×C283—42no—
70C4⋊C4×C283C2×C414yescomplemented
71C42×C283—22no—
72C22×D283—21no—
73C22×D283—21no—
74C22×D283—21no—
75C22×D283—21no—
76C22×D283C22×C412yes—
77C22×D283C2×D412yes—
78C22×D283C2×Q812yes—
79C2×Dic283—22no—
80C2×Dic283—22no—
81C2×Dic283—21no—
82C2×Dic283—21no—
83C2×Dic283—22no—
84C2×Dic283—21no—
85C2×Dic283—22no—
86C2×Dic283—21no—
87C2×Dic283—22no—
88C2×Dic283—21no—
89C2×Dic283—21no—
90C2×Dic283C22×C414yes—
91C2×Dic283C22×C412yes—
92C2×Dic283C2×D412yes—
93C2×Dic283C2×Q812yes—
94C4×D283—48no—
95C4×D283C4⋊C4116yescomplemented
96C23×C283C2411yescharacteristic
97C2×C4×C283—22no—
98C2×C4×C283—22no—
99C2×C4×C283—21no—
100C2×C4×C283—21no—
101C2×C4×C283—22no—
102C2×C4×C283—21no—
103C2×C4×C283—22no—
104C2×C4×C283—21no—
105C2×C4×C283—22no—
106C2×C4×C283—21no—
107C2×C4×C283—21no—
108C2×C4×C283C22×C414yes—
109C2×C4×C283C22×C412yes—
110C2×C4×C283C2×D412yes—
111C2×C4×C283C2×Q812yes—
112C22×C283C23×C411yescharacteristic
113C22×C283C22×D411yescharacteristic
114C22×C283C22×Q811yescharacteristic
115C22×C283H(5,𝔽2)11yescharacteristic
116C22×C283H(5,𝔽2)11yescharacteristic
117C22×C2832−1+411yescharacteristic
118C22×C2832−1+411yescharacteristic
119C2×D283—22no—
120C2×D283—22no—
121C2×D283—22no—
122C2×D283—22no—
123C2×D283—22no—
124C2×D283—22no—
125C2×D283C2×C4⋊C414yes—
126C4×C283—24no—
127C4×C283—41no—
128C4×C283—42no—
129C4×C283—41no—
130C4×C283C2×C4⋊C414yes—
131Dic283—24no—
132Dic283—42no—
133Dic283—41no—
134Dic283—41no—
135Dic283C2×C4⋊C414yes—
136C2×C283C22×C4⋊C411yescharacteristic
137C2×C283#52011yescharacteristic
138C2×C283#52011yescharacteristic
139C2×C283#53511yescharacteristic
140C2×C283#53611yescharacteristic
141C2×C283#54411yescharacteristic
142C2×C283#54411yescharacteristic
143D283—24no—
144C283#231711yescharacteristic, complemented, Sylow
145#2317—2831nomaximal, Sylow
146C2×#117—2831no—
147C2×#117—2831no—
148C22×C4⋊C4—2831no—
149#389—2831no—
150#389—2831no—
151#389—2831no—
152#389D28311yescharacteristic, complemented
153#385—2832no—
154#385—2832no—
155#384—2832no—
156#382—2832no—
157C23×C4—2831no—
158C23×C4—2832no—
159#117—2838no—
160#117—5664no—
161C2×C4⋊C4—2831no—
162C2×C4⋊C4—2832no—
163C2×C4⋊C4—2831no—
164C2×C4⋊C4—2831no—
165C2×C4⋊C4—2831no—
166C2×C4⋊C4—2832no—
167C2×C4⋊C4—2832no—
168C2×C4⋊C4—2832no—
169C2×C4⋊C4—5664no—
170C2×C4⋊C4C2×D28311yescharacteristic
171C2×C4⋊C4C2×D28312yes—
172C2×C4⋊C4C2×D28311yescharacteristic
173C2×C22⋊C4—2832no—
174C2×C22⋊C4—2831no—
175C2×C22⋊C4—2831no—
176C2×C42—2832no—
177C2×C42—2831no—
178C2×C42C2×D28311yescharacteristic
179#95—2832no—
180#95—2832no—
181#95—2832no—
182#95C2×D28312yes—
183C24—2831no—
184C22×C4—2831no—
185C22×C4—2834no—
186C22×C4—2834no—
187C22×C4—2834no—
188C22×C4—2832no—
189C22×C4—2831no—
190C22×C4—2832no—
191C22×C4—2831no—
192C22×C4—5664no—
193C22×C4—11322no—
194C22×C4—11322no—
195C22×C4C22×D28311yescharacteristic
196C22×C4C22×D28312yes—
197C22×C4C22×D28311yescharacteristic
198C22×C4C22×D28312yes—
199C22×C4C22×D28311yescharacteristic
200C4⋊C4—22no—
201C4⋊C4—42no—
202C4⋊C4—2834no—
203C4⋊C4—5664no—
204C4⋊C4—5662no—
205C4⋊C4—5662no—
206C4⋊C4—5662no—
207C4⋊C4—11322no—
208C4⋊C4—11322no—
209C4⋊C4—11322no—
210C4⋊C4C4×D28314yescomplemented
211C22⋊C4—2838no—
212C22⋊C4—5662no—
213C22⋊C4—5662no—
214C42—22no—
215C42—2838no—
216C42—5662no—
217C23—2832no—
218C23—2832no—
219C23—2832no—
220C23—5661no—
221C23—5661no—
222C23—5661no—
223C23—5661no—
224C23C23×D28311yescharacteristic
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—2832no—
237C2×C4—2832no—
238C2×C4—2832no—
239C2×C4—28316no—
240C2×C4—2834no—
241C2×C4—5662no—
242C2×C4—5662no—
243C2×C4—5661no—
244C2×C4—5661no—
245C2×C4—5662no—
246C2×C4—5661no—
247C2×C4—5662no—
248C2×C4—5661no—
249C2×C4—5662no—
250C2×C4—5661no—
251C2×C4—5661no—
252C2×C4—11328no—
253C2×C4C2×C4×D28314yes—
254C2×C4C2×C4×D28312yes—
255C2×C4D4×D28312yes—
256C2×C4Q8×D28312yes—
257C22—2834no—
258C22—5662no—
259C22—5662no—
260C22—5662no—
261C22—5662no—
262C22—5662no—
263C22—5662no—
264C22C22×C4×D28311yescharacteristic
265C22C2×D4×D28311yescharacteristic
266C22C2×Q8×D28311yescharacteristic
267C22#2258365511yescharacteristic
268C22#2258365611yescharacteristic
269C22#2258365711yescharacteristic
270C22#2258365811yescharacteristic
271C4—24no—
272C4—42no—
273C4—41no—
274C4—41no—
275C4—2834no—
276C4—5664no—
277C4—11322no—
278C4—11321no—
279C4—11321no—
280C4C4⋊C4×D28314yes—
281C2—5664no—
282C2C2×C4⋊C4×D28311yescharacteristic
283C2#3182694411yescharacteristic
284C2#3182694711yescharacteristic
285C2#3182703111yescharacteristic
286C2#3182703811yescharacteristic
287C2#3182709211yescharacteristic
288C2#3182709511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D283
commutatorC22×C283C23×C4
Fitting#389×C283C2
socle, cosocleC23×C283C24
prime radicalC23×D283C23
hypercenter#389D283
hypocenterC283#2317
LCSG ▷ C22×C283 ▷ C283
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
4. #31826364C2semidirect
5. #31826363C2semidirect
6. #31826361C2semidirect
9. #31826334C2semidirect
10. #31826329C2semidirect
11. #389×C283C2semidirect
14. #22583568C4semidirect
52. C4⋊Dic283C2×C4semidirect
62. C4×Dic283C2×C4semidirect
70. C4⋊C4×C283C2×C4semidirect
95. C4×D283C4⋊C4semidirect
144. C283#2317semidirect
152. #389D283semidirect
210. C4⋊C4C4×D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225835682
C2×#225835672
C2×C4⋊C4×D2832
#318263642
#318263632
#318263612
#318263392
#318263362
#318263342
#318263292
#389×C2832
#2317283