Finite Group Databasebeta

← #49131220

Subgroups — #49131220

Order 36224.

418 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 133 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
1#31827029C211yesmaximal, characteristic
2#31826948C211yesmaximal, characteristic, complemented
3#31826790C212yesmaximal, complemented
4#31826790C212yesmaximal, complemented
5#31826784C212yesmaximal, complemented
6#31826779C212yesmaximal, complemented
7#31826507C211yesmaximal, characteristic, complemented
8#31826503C211yesmaximal, characteristic, complemented
9#31826471C211yesmaximal, characteristic, complemented
10C2×#22583532C211yesmaximal, characteristic, complemented
11#408×C283C211yesmaximal, characteristic, complemented
12C22×D1132C2211yescharacteristic
13#22583606—21no—
14#22583599—21no—
15#22583584C2212yes—
16#22583584C2212yes—
17#22583578—22no—
18Q8×Dic283—21no—
19D4×Dic283—22no—
20D4×Dic283—21no—
21#22583569C2211yescharacteristic
22#22583569C2211yescharacteristic
23#22583569C2212yescomplemented
24C2×C4⋊Dic283C2211yescharacteristic
25C2×C4⋊Dic283C2212yescomplemented
26#22583544C2211yescharacteristic, complemented
27#22583544C2211yescharacteristic, complemented
28#22583544C2212yescomplemented
29#22583542C2211yescharacteristic, complemented
30#22583542C2211yescharacteristic, complemented
31#22583542C2212yescomplemented
32#22583532C2212yes—
33#22583532C2212yes—
34#22583532C2212yes—
35#22583532C2212yes—
36C2×#41×C283C2211yescharacteristic
37C2×M16×C283C2211yescharacteristic
38C22×C8×C283C2211yescharacteristic
39C22⋊C8×C283C2212yescomplemented
40C22⋊C8×C283C2212yescomplemented
41C23×D283—21no—
42C2×D1132—21no—
43C2×D1132—22no—
44C2×D1132—22no—
45C2×D1132C2312yes—
46C2×C22⋊D283—22no—
47C2×C22⋊D283—22no—
48C22×Dic283—21no—
49C22×Dic283—22no—
50C4⋊Dic283—21no—
51C4⋊Dic283—22no—
52C4⋊Dic283C2312yes—
53C4⋊Dic283C2311yescharacteristic
54C4⋊Dic283C2311yescharacteristic
55C4⋊Dic283C2311yescharacteristic
56C4⋊Dic283C2311yescharacteristic
57C22⋊Dic283—21no—
58C22⋊Dic283—21no—
59C22⋊Dic283—22no—
60(C2×D283)⋊C4—22no—
61(C2×D283)⋊C4—22no—
62C4×Dic283—21no—
63C4×Dic283—21no—
64C4×Dic283—22no—
65#41×C283—22no—
66#41×C283—22no—
67C2×Q8×C283D411yescharacteristic
68C2×D4×C283D412yes—
69C2×D4×C283D411yescharacteristic
70C22×C4×C283C2312yes—
71C22×C4×C283C2311yescharacteristic
72M16×C283—22no—
73C2×C8×C283—22no—
74C2×C8×C283C2311yescharacteristic
75C2×C8×C283C2311yescharacteristic
76C2×C8×C283C2312yes—
77C22×D283—22no—
78C22×D283—42no—
79C22×D283—81no—
80C22×D283—81no—
81D1132—22no—
82D1132—22no—
83D1132—42no—
84C22⋊D283—82no—
85C22⋊D283—82no—
86C2×Dic283—21no—
87C2×Dic283—21no—
88C2×Dic283—21no—
89C2×Dic283—21no—
90C2×Dic283—22no—
91C2×Dic283—81no—
92C2×Dic283—82no—
93C23×C283C2×D412yes—
94C23×C283C2×D411yescharacteristic
95Q8×C283—22no—
96D4×C283—24no—
97D4×C283—22no—
98C2×C4×C283—22no—
99C2×C4×C283—22no—
100C2×C4×C283C2×D412yes—
101C2×C4×C283C2×D411yescharacteristic
102C2×C4×C283#4112yes—
103C2×C4×C283#4112yes—
104C2×C4×C283C2411yescharacteristic
105C8×C283—21no—
106C8×C283—21no—
107C8×C283—22no—
108C22×C283—21no—
109C22×C283—21no—
110C22×C283—22no—
111C22×C283—22no—
112C22×C283—22no—
113C22×C283#12312yes—
114C22×C283C22×D411yescharacteristic
115C2×D283—42no—
116C2×D283—42no—
117C2×D283—42no—
118C2×D283—84no—
119C4×C283—22no—
120C4×C283#12312yes—
121C4×C283C2×#4111yescharacteristic
122C4×C283C2×#4111yescharacteristic
123Dic283—41no—
124Dic283—42no—
125Dic283—41no—
126Dic283—41no—
127Dic283—41no—
128C2×C283—22no—
129C2×C283—42no—
130C2×C283C2×#12311yescharacteristic
131C2×C283#57611yescharacteristic
132C2×C283#57711yescharacteristic
133D283—82no—
134C283#309611yescharacteristic, complemented, Sylow
135#3096—2831nomaximal, Sylow
136#535—2831no—
137#520—2831no—
138#482—2832no—
139#482—2832no—
140#481—2832no—
141#480—2832no—
142#428—2831no—
143#427—2831no—
144#418—2831no—
145C2×#102—2831no—
146#408D28311yescharacteristic, complemented
147C2×#41C2×D28311yescharacteristic
148C22×D4—2831no—
149C2×M16C2×D28311yescharacteristic
150C22×C8C2×D28311yescharacteristic
151#127—5661no—
152#124—5661no—
153#121—2832no—
154#121—2832no—
155#120—5662no—
156C4×Q8—5661no—
157C4×D4—5662no—
158C4×D4—5661no—
159#117—2831no—
160#117—2832no—
161#117—2831no—
162C2×C4⋊C4—2831no—
163C2×C4⋊C4—2832no—
164#107—2831no—
165#107—2831no—
166#107—2832no—
167#106—2831no—
168#106—2831no—
169#106—2832no—
170#102—2832no—
171#102—2832no—
172#102—2832no—
173#102—2832no—
174C22⋊C8C2×D28312yescomplemented
175C22⋊C8C2×D28312yescomplemented
176C24—5661no—
177#41—22no—
178#41—22no—
179C2×Q8D113211yescharacteristic
180C2×D4—2832no—
181C2×D4—5662no—
182C2×D4—5662no—
183C2×D4—5661no—
184C2×D4—5662no—
185C2×D4—5662no—
186C2×D4D113212yes—
187C2×D4D113211yescharacteristic
188C22×C4—5662no—
189C22×C4—5661no—
190C22×C4C22×D28312yes—
191C22×C4C22×D28311yescharacteristic
192M16—22no—
193C2×C8—22no—
194C2×C8C22×D28311yescharacteristic
195C2×C8C22×D28312yes—
196C2×C8C22×D28311yescharacteristic
197C4⋊C4—2832no—
198C4⋊C4—2831no—
199C4⋊C4—2831no—
200C4⋊C4—2831no—
201C4⋊C4—2831no—
202C4⋊C4—5662no—
203C4⋊C4—5661no—
204C22⋊C4—5662no—
205C22⋊C4—5661no—
206C22⋊C4—5661no—
207C22⋊C4—5662no—
208C22⋊C4—5662no—
209C42—5662no—
210C42—5661no—
211C42—5661no—
212C23—5662no—
213C23—11322no—
214C23—22641no—
215C23—22641no—
216C23C2×D113212yes—
217C23C2×D113211yescharacteristic
218Q8—22no—
219D4—24no—
220D4—22no—
221D4—5662no—
222D4—5662no—
223D4—11322no—
224D4—22642no—
225D4—22642no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—5661no—
229C2×C4—5661no—
230C2×C4—5661no—
231C2×C4—5662no—
232C2×C4—5661no—
233C2×C4—22641no—
234C2×C4—22642no—
235C2×C4C2×D113212yes—
236C2×C4C2×D113211yescharacteristic
237C2×C4#1798539112yes—
238C2×C4#1798539112yes—
239C2×C4C23×D28311yescharacteristic
240C8—21no—
241C8—21no—
242C8—22no—
243C22—21no—
244C22—21no—
245C22—22no—
246C22—22no—
247C22—22no—
248C22—11322no—
249C22—11322no—
250C22—11322no—
251C22—22644no—
252C22#2258359412yes—
253C22C22×D113211yescharacteristic
254C4—22no—
255C4—11322no—
256C4—11321no—
257C4—11321no—
258C4—11321no—
259C4—11321no—
260C4#2258359412yes—
261C4C2×#1798539111yescharacteristic
262C4C2×#1798539111yescharacteristic
263C2—22no—
264C2—42no—
265C2—22642no—
266C2C2×#2258359411yescharacteristic
267C2#3182727011yescharacteristic
268C2#3182727711yescharacteristic
subgroupG/H
centerC22C22×D1132
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#408×C283C2
socleC22×C283C22×D4
prime radical#31827029C2
hypercenter#408D283
hypocenterC283#3096
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
2. #31826948C2semidirect
3. #31826790C2semidirect
4. #31826790C2semidirect
5. #31826784C2semidirect
6. #31826779C2semidirect
7. #31826507C2semidirect
8. #31826503C2semidirect
9. #31826471C2semidirect
10. C2×#22583532C2semidirect
11. #408×C283C2semidirect
23. #22583569C22semidirect
25. C2×C4⋊Dic283C22semidirect
26. #22583544C22semidirect
27. #22583544C22semidirect
28. #22583544C22semidirect
29. #22583542C22semidirect
30. #22583542C22semidirect
31. #22583542C22semidirect
39. C22⋊C8×C283C22semidirect
40. C22⋊C8×C283C22semidirect
134. C283#3096semidirect
146. #408D283semidirect
174. C22⋊C8C2×D283semidirect
175. C22⋊C8C2×D283semidirect

Maximal subgroups

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

subgroupindex
#318270292
#318269482
#318267902
#318267842
#318267792
#318265072
#318265032
#318264712
C2×#225835322
#408×C2832
#3096283