Finite Group Databasebeta

← #49130642

Subgroups — #49130642

Order 36224.

704 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 145 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#31827301C211yesmaximal, characteristic
2C2×#22583643C212yesmaximal, complemented
3#31827036C211yesmaximal, characteristic, complemented
4#31826650C214yesmaximal, complemented
5#31826645C214yesmaximal, complemented
6#31826481C212yesmaximal, complemented
7#413×C283C211yesmaximal, characteristic, complemented
8#22583658C2212yes—
9#22583657—22no—
10#22583656—22no—
11#22583655—22no—
12#22583655C2212yes—
13C2×#17985392—22no—
14C2×#17985392C2212yes—
15#41×D283—24no—
16C22×Dic566C2211yescharacteristic, complemented
17#22583643—24no—
18#22583643C2218yescomplemented
19C2×Dic1132—22no—
20C2×#17985381—22no—
21#22583608C2212yescomplemented
22#22583597C2212yescomplemented
23#22583591—22no—
24#22583589—22no—
25#22583569C2211yescharacteristic, complemented
26#22583536C2218yescomplemented
27C2×#41×C283C2211yescharacteristic
28C2×M16×C283C2212yescomplemented
29#101×C283C2212yescomplemented
30#100×C283C2212yescomplemented
31#17985393—24no—
32#17985392—24no—
33#17985392—24no—
34#17985392—22no—
35#17985392—24no—
36#17985392—22no—
37#17985392D414yes—
38#17985392C2314yes—
39#17985391—28no—
40#17985391—24no—
41C2×Dic566—22no—
42C2×Dic566—21no—
43C2×Dic566—22no—
44C2×Dic566—22no—
45C2×Dic566D412yes—
46C2×Dic566C2312yes—
47Q8×D283—24no—
48C2×D1132—22no—
49C2×D1132D412yes—
50C2×C22⋊D283—24no—
51C2×C22⋊D283—22no—
52D4×D283—28no—
53D4×D283—24no—
54C22×Dic283—21no—
55C2×C4×D283—24no—
56C2×C4×D283—22no—
57Dic1132—28no—
58#17985381—28no—
59C4⋊Dic283—21no—
60Dic283⋊C4—24no—
61C22⋊Dic283—21no—
62C22⋊Dic283—24no—
63C4×Dic283C2312yes—
64#41×C283—22no—
65#41×C283—22no—
66C2×Q8×C283C2311yescharacteristic
67C2×D4×C283D412yes—
68C2×D4×C283C2311yescharacteristic
69C22×C4×C283D412yes—
70C22×C4×C283C2311yescharacteristic
71M16×C283—22no—
72M16×C283C2314yes—
73C2×C8×C283—22no—
74C22×D283—24no—
75C22×D283—22no—
76Dic566—24no—
77Dic566—22no—
78Dic566—24no—
79Dic566—41no—
80Dic566—41no—
81Dic566C2×D414yes—
82D1132—24no—
83D1132—22no—
84D1132C2×D414yes—
85C22⋊D283—28no—
86C22⋊D283—24no—
87C22⋊D283—24no—
88C2×Dic283—24no—
89C2×Dic283—22no—
90C2×Dic283—22no—
91C2×Dic283—22no—
92C2×Dic283—42no—
93C4×D283—28no—
94C4×D283—24no—
95C4×D283—24no—
96C23×C283C2×D412yes—
97C23×C283C2×D411yescharacteristic
98Q8×C283—22no—
99D4×C283—24no—
100D4×C283—22no—
101C2×C4×C283—22no—
102C2×C4×C283—22no—
103C2×C4×C283C2×D412yes—
104C2×C4×C283C2×D412yes—
105C2×C4×C283C2×D412yes—
106C2×C4×C283C2×D411yescharacteristic
107C2×C4×C283C2411yescharacteristic
108C8×C283—24no—
109C22×C283—22no—
110C22×C283—41no—
111C22×C283—42no—
112C22×C283#12012yes—
113C22×C283C22×D411yescharacteristic
114C2×D283—24no—
115C2×D283—82no—
116C2×D283—84no—
117C4×C283—22no—
118C4×C283#12011yescharacteristic
119C4×C283#12011yescharacteristic
120C4×C283C22×D412yes—
121Dic283—24no—
122Dic283—42no—
123Dic283—42no—
124C2×C283—21no—
125C2×C283—22no—
126C2×C283—42no—
127C2×C283C2×#12011yescharacteristic
128D283—44no—
129C283#302211yescharacteristic, complemented, Sylow
130#3022—2831nomaximal, Sylow
131#585—2831no—
132C2×#137—2832no—
133#536—2831no—
134#456—2834no—
135#455—2834no—
136#421—2832no—
137#413D28311yescharacteristic, complemented
1382−1+4—2832no—
1392−1+4—5662no—
140H(5,𝔽2)—2832no—
141H(5,𝔽2)—5662no—
142H(5,𝔽2)—5662no—
143C2×#41—2832no—
144C2×#41—5664no—
145C2×#41—5662no—
146C2×#41C2×D28311yescharacteristic
147C22×Q8—2831no—
148#137—2838no—
149#137—5664no—
150C2×Q16—5662no—
151C2×SD16—5662no—
152C2×M16C2×D28312yescomplemented
153C4⋊Q8—2832no—
154#124—2832no—
155#123—5662no—
156C22⋊Q8—5662no—
157#117—2831no—
158#104—2838no—
159#101C2×D28312yescomplemented
160#100C2×D28312yescomplemented
161#41—22no—
162#41—22no—
163#41—2834no—
164#41—2834no—
165#41—5664no—
166#41—5664no—
167#41—5664no—
168#41—5668no—
169#41—5664no—
170#41—5662no—
171#41—5662no—
172#41—5664no—
173C2×Q8—2832no—
174C2×Q8—2832no—
175C2×Q8—5662no—
176C2×Q8—5664no—
177C2×Q8—5661no—
178C2×Q8—5662no—
179C2×Q8—5662no—
180C2×Q8C22×D28311yescharacteristic
181C2×D4—2832no—
182C2×D4—5662no—
183C2×D4—5664no—
184C2×D4—5668no—
185C2×D4—5662no—
186C2×D4—5664no—
187C2×D4D113212yes—
188C2×D4C22×D28311yescharacteristic
189C22×C4—5664no—
190C22×C4—5662no—
191C22×C4—5661no—
192C22×C4D113212yes—
193C22×C4C22×D28311yescharacteristic
194Q16—5668no—
195SD16—5668no—
196M16—22no—
197M16C22×D28314yes—
198C2×C8—22no—
199C4⋊C4—5664no—
200C4⋊C4—5661no—
201C22⋊C4—5661no—
202C22⋊C4—5664no—
203C42—2832no—
204C23—5664no—
205C23—5662no—
206C23C2×D113212yes—
207C23C2×D113211yescharacteristic
208Q8—22no—
209Q8—2834no—
210Q8—5664no—
211Q8—5662no—
212Q8—5664no—
213Q8—11321no—
214Q8—11321no—
215D4—24no—
216D4—22no—
217D4—2834no—
218D4—5664no—
219D4—5668no—
220D4—5664no—
221D4—5664no—
222D4—5662no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—5668no—
226C2×C4—5664no—
227C2×C4—5662no—
228C2×C4—5664no—
229C2×C4—5664no—
230C2×C4—5662no—
231C2×C4—5662no—
232C2×C4—11322no—
233C2×C4D4×D28312yes—
234C2×C4D4×D28312yes—
235C2×C4C2×D113212yes—
236C2×C4C2×D113211yescharacteristic
237C2×C4C23×D28311yescharacteristic
238C8—24no—
239C22—22no—
240C22—41no—
241C22—42no—
242C22—5664no—
243C22—22642no—
244C22—22644no—
245C22#2258358012yes—
246C22C22×D113211yescharacteristic
247C4—22no—
248C4—5664no—
249C4—11322no—
250C4—11322no—
251C4#2258358011yescharacteristic
252C4#2258358011yescharacteristic
253C4C2×D4×D28312yes—
254C2—21no—
255C2—22no—
256C2—42no—
257C2—11324no—
258C2C2×#2258358011yescharacteristic
subgroupG/H
centerC2C2×#22583580
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#413×C283C2
socleC2×C283C2×#120
prime radical#31827301C2
hypercenter#413D283
hypocenterC283#3022
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C2 ◁ C22×C4 ◁ #413

Complements

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

NH ≅ G/Ntype
2. C2×#22583643C2semidirect
3. #31827036C2semidirect
4. #31826650C2semidirect
5. #31826645C2semidirect
6. #31826481C2semidirect
7. #413×C283C2semidirect
16. C22×Dic566C22semidirect
18. #22583643C22semidirect
21. #22583608C22semidirect
22. #22583597C22semidirect
25. #22583569C22semidirect
26. #22583536C22semidirect
28. C2×M16×C283C22semidirect
29. #101×C283C22semidirect
30. #100×C283C22semidirect
129. C283#3022semidirect
137. #413D283semidirect
152. C2×M16C2×D283semidirect
159. #101C2×D283semidirect
160. #100C2×D283semidirect

Maximal subgroups

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

subgroupindex
#318273012
C2×#225836432
#318270362
#318266502
#318266452
#318264812
#413×C2832
#3022283