Finite Group Databasebeta

← #49124525

Subgroups — #49124525

Order 36224.

560 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 147 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×#22583572C211yesmaximal, characteristic
2C2×#22583568C211yesmaximal, characteristic, complemented
3#31826404C211yesmaximal, characteristic, complemented
4#31826402C211yesmaximal, characteristic, complemented
5#31826392C212yesmaximal, complemented
6#31826391C211yesmaximal, characteristic, complemented
7#31826382C212yesmaximal
8C4⋊C4×Dic283C212yesmaximal, complemented
9#31826312C211yesmaximal, characteristic, complemented
10#31826312C212yesmaximal, complemented
11#398×C283C211yesmaximal, characteristic, complemented
12C22×D1132C2211yescharacteristic
13C22×C4×D283C2211yescharacteristic
14C22×C4×D283C2211yescharacteristic
15#22583572—24no—
16#22583568—24no—
17C2×C4⋊Dic283C2212yes—
18C2×C4⋊Dic283C2211yescharacteristic
19C2×(C2×D283)⋊C4C2212yes—
20C2×(C2×D283)⋊C4C2212yes—
21C2×(C2×D283)⋊C4C2212yes—
22C2×(C2×D283)⋊C4C2212yes—
23C2×(C2×D283)⋊C4C2211yescharacteristic
24C2×(C2×D283)⋊C4C2211yescharacteristic
25C2×C4×Dic283C2212yes—
26C2×C4×Dic283C2212yes—
27C2×C4×Dic283C2211yescharacteristic
28C283⋊#95C2211yescharacteristic
29C283⋊#95C2212yes—
30C283⋊#95C2212yes—
31C283⋊#95C2212yes—
32C2×C4⋊C4×C283C2212yes—
33C2×C4⋊C4×C283C2211yescharacteristic
34C2×C4⋊C4×C283C2211yescharacteristic
35#95×C283C2212yes—
36#95×C283C2211yescharacteristic
37C23×D283C2311yescharacteristic
38C23×D283C2311yescharacteristic
39C2×D1132—22no—
40C2×D1132—22no—
41C2×D1132—22no—
42C22×Dic283C2311yescharacteristic
43C22×Dic283C2312yes—
44C22×Dic283C2312yes—
45C22×Dic283C2311yescharacteristic
46C2×C4×D283—24no—
47C2×C4×D283—22no—
48C2×C4×D283—24no—
49C2×C4×D283D414yes—
50C4⋊Dic283—22no—
51C4⋊Dic283—24no—
52(C2×D283)⋊C4—24no—
53(C2×D283)⋊C4—22no—
54(C2×D283)⋊C4—24no—
55(C2×D283)⋊C4—22no—
56(C2×D283)⋊C4—24no—
57(C2×D283)⋊C4—24no—
58C4×Dic283—24no—
59C4×Dic283—22no—
60C4×Dic283—24no—
61C22×C4×C283C2312yes—
62C22×C4×C283C2311yescharacteristic
63C22×C4×C283C2311yescharacteristic
64C22×C4×C283C2312yes—
65C22×C4×C283C2311yescharacteristic
66C4⋊C4×C283—22no—
67C4⋊C4×C283—22no—
68C4⋊C4×C283—24no—
69C22×D283—21no—
70C22×D283—22no—
71C22×D283—22no—
72C22×D283—22no—
73C22×D283—22no—
74C22×D283—21no—
75C22×D283—21no—
76C22×D283—21no—
77C22×D283C2×D412yes—
78C22×D283#4112yes—
79D1132—28no—
80C2×Dic283—21no—
81C2×Dic283—22no—
82C2×Dic283—22no—
83C2×Dic283—22no—
84C2×Dic283—22no—
85C2×Dic283—22no—
86C2×Dic283—22no—
87C2×Dic283—22no—
88C2×Dic283C2×D412yes—
89C2×Dic283#4114yes—
90C4×D283—28no—
91C4×D283—44no—
92C23×C283C2411yescharacteristic
93C2×C4×C283—21no—
94C2×C4×C283—22no—
95C2×C4×C283—22no—
96C2×C4×C283—22no—
97C2×C4×C283—21no—
98C2×C4×C283—22no—
99C2×C4×C283—21no—
100C2×C4×C283—22no—
101C2×C4×C283—22no—
102C2×C4×C283—22no—
103C2×C4×C283C2×D412yes—
104C2×C4×C283#4112yes—
105C2×C4×C283#4114yes—
106C22×C283C22×D411yescharacteristic
107C22×C283C2×#4112yes—
108C22×C283C2×#4111yescharacteristic
109C22×C283C2×#4111yescharacteristic
110C22×C283C2×#4112yes—
111C2×D283—22no—
112C2×D283—22no—
113C2×D283—22no—
114C2×D283—24no—
115C2×D283—22no—
116C2×D283—41no—
117C2×D283—42no—
118C2×D283—42no—
119C2×D283—42no—
120C2×D283—41no—
121C4×C283—22no—
122C4×C283—22no—
123C4×C283—24no—
124C4×C283—41no—
125C4×C283—42no—
126Dic283—42no—
127Dic283—41no—
128Dic283—42no—
129Dic283#12314yes—
130C2×C283C2×#12311yescharacteristic
131C2×C283#52511yescharacteristic
132C2×C283#52912yes—
133C2×C283#52912yes—
134C2×C283#53211yescharacteristic
135D283—42no—
136D283—42no—
137C283#239511yescharacteristic, complemented, Sylow
138#2395—2831nomaximal, Sylow
139C2×C4×D4—2831no—
140C2×#117—2831no—
141#400—2831no—
142#399—2831no—
143#398D28311yescharacteristic, complemented
144#397—2831no—
145#397—2832no—
146#394—2832no—
147C4×C4⋊C4—2832no—
148C4×C22⋊C4—2832no—
149C4×C22⋊C4—2831no—
150C22×D4—2831no—
151C23×C4—2831no—
152C23×C4—2831no—
153C4×D4—5664no—
154#117—5664no—
155C2×C4⋊C4—2831no—
156C2×C4⋊C4—2832no—
157C2×C4⋊C4C2×D28311yescharacteristic
158C2×C4⋊C4C2×D28311yescharacteristic
159C2×C4⋊C4C2×D28312yes—
160C2×C22⋊C4—2832no—
161C2×C22⋊C4—2831no—
162C2×C22⋊C4—2831no—
163C2×C22⋊C4—2832no—
164C2×C22⋊C4—2832no—
165C2×C22⋊C4—2832no—
166C2×C42—2832no—
167C2×C42—2831no—
168C2×C42—2832no—
169#95—2831no—
170#95—2832no—
171#95—2832no—
172#95—2832no—
173#95C2×D28312yes—
174#95C2×D28311yescharacteristic
175C24—2831no—
176C24—2831no—
177C2×D4—5662no—
178C2×D4—5662no—
179C2×D4—5662no—
180C22×C4—2831no—
181C22×C4—2834no—
182C22×C4—2832no—
183C22×C4—2831no—
184C22×C4—2832no—
185C22×C4—5664no—
186C22×C4—5662no—
187C22×C4—5664no—
188C22×C4C22×D28311yescharacteristic
189C22×C4C22×D28312yes—
190C22×C4C22×D28311yescharacteristic
191C22×C4C22×D28311yescharacteristic
192C22×C4C22×D28312yes—
193C4⋊C4—22no—
194C4⋊C4—22no—
195C4⋊C4—24no—
196C4⋊C4—5662no—
197C4⋊C4—5664no—
198C22⋊C4—5664no—
199C22⋊C4—5664no—
200C22⋊C4—5662no—
201C22⋊C4—5662no—
202C22⋊C4—5664no—
203C22⋊C4—5664no—
204C42—5664no—
205C42—5664no—
206C42—5662no—
207C23—2832no—
208C23—2832no—
209C23—5661no—
210C23—5662no—
211C23—5662no—
212C23—5661no—
213C23—5661no—
214C23—5661no—
215C23—5662no—
216C23—5662no—
217C23C23×D28311yescharacteristic
218D4—5668no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—2832no—
230C2×C4—2834no—
231C2×C4—5668no—
232C2×C4—5661no—
233C2×C4—5662no—
234C2×C4—5662no—
235C2×C4—5662no—
236C2×C4—5662no—
237C2×C4—5662no—
238C2×C4—5662no—
239C2×C4—5662no—
240C2×C4—11324no—
241C2×C4D4×D28312yes—
242C2×C4#1798539112yes—
243C2×C4#1798539314yes—
244C22—5662no—
245C22—5662no—
246C22—5662no—
247C22—5664no—
248C22—5662no—
249C22—11321no—
250C22—11322no—
251C22—11322no—
252C22—11322no—
253C22—11321no—
254C22C2×D4×D28311yescharacteristic
255C22#41×D28312yes—
256C22#41×D28311yescharacteristic
257C22C2×#1798539111yescharacteristic
258C22C2×#1798539312yes—
259C4—22no—
260C4—22no—
261C4—24no—
262C4—41no—
263C4—42no—
264C4—2834no—
265C4—11322no—
266C4—11321no—
267C4—11322no—
268C2—11322no—
269C2—11322no—
270C2#123×D28311yescharacteristic
271C2#3182697511yescharacteristic
272C2#3182700012yes—
273C2#3182700112yes—
274C2#3182701611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D283
commutator, socle, cosocleC23×C283C24
Fitting#398×C283C2
prime radicalC22×D1132C22
hypercenter#398D283
hypocenterC283#2395
LCSG ▷ C23×C283 ▷ C283
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
2. C2×#22583568C2semidirect
3. #31826404C2semidirect
4. #31826402C2semidirect
5. #31826392C2semidirect
6. #31826391C2semidirect
8. C4⋊C4×Dic283C2semidirect
9. #31826312C2semidirect
10. #31826312C2semidirect
11. #398×C283C2semidirect
137. C283#2395semidirect
143. #398D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225835722
C2×#225835682
#318264042
#318264022
#318263922
#318263912
#318263822
C4⋊C4×Dic2832
#318263122
#398×C2832
#2395283