Finite Group Databasebeta

← #49130666

Subgroups — #49130666

Order 36224.

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

normal subgroups only

This group has 716 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
2#31827039C212yesmaximal, complemented
3#31826654C218yesmaximal, complemented
4#31826439C211yesmaximal, characteristic, complemented
5#31826438C212yesmaximal, complemented
6#536×C283C211yesmaximal, characteristic, complemented
7#22583658—21no—
8#22583657—22no—
9#22583656—22no—
10#22583655—21no—
11#22583655C2214yes—
12C2×#17985392—22no—
13C2×#17985392C2212yes—
14#41×D283—24no—
15#22583608—22no—
16#22583599—22no—
17#22583593C2218yescomplemented
18#22583586—24no—
19#22583569C2212yescomplemented
20#22583525C2214yescomplemented
21#22583524C2218yescomplemented
22C2×#41×C283C2211yescharacteristic
23C22×Q8×C283C2211yescharacteristic, complemented
24C4⋊Q8×C283—21no—
25#124×C283—21no—
26#123×C283C2214yescomplemented
27C22⋊Q8×C283—22no—
28#117×C283C2211yescharacteristic, complemented
29#17985393—24no—
30#17985392—24no—
31#17985392—24no—
32#17985392—22no—
33#17985392—22no—
34#17985392—24no—
35#17985392—22no—
36#17985392—22no—
37#17985391—24no—
38#17985391—28no—
39C2×Dic566—22no—
40C2×Dic566D412yes—
41Q8×D283—24no—
42C2×D1132—22no—
43C2×D1132D412yes—
44C2×C22⋊D283—24no—
45C2×C22⋊D283C2314yes—
46D4×D283—24no—
47D4×D283—28no—
48C2×C4×D283—24no—
49C2×C4×D283D414yes—
50C4⋊Dic283—22no—
51Dic283⋊C4—28no—
52C22⋊Dic283C2314yes—
53(C2×D283)⋊C4—28no—
54C4×Dic283—22no—
55#41×C283—22no—
56#41×C283—22no—
57C2×Q8×C283—22no—
58C2×Q8×C283—22no—
59C2×Q8×C283—21no—
60C2×Q8×C283—21no—
61C2×Q8×C283D411yescharacteristic
62C2×D4×C283D411yescharacteristic
63C2×D4×C283C2312yes—
64C22×C4×C283D412yes—
65C22×C4×C283C2312yes—
66C22×C4×C283C2311yescharacteristic
67C4⋊C4×C283—21no—
68C4⋊C4×C283—24no—
69C22⋊C4×C283—24no—
70C22⋊C4×C283C2312yes—
71C42×C283—21no—
72C22×D283—24no—
73C22×D283C2×D414yes—
74Dic566—24no—
75Dic566—22no—
76Dic566—22no—
77D1132—24no—
78D1132—22no—
79D1132—22no—
80C22⋊D283—28no—
81C22⋊D283—24no—
82C22⋊D283—24no—
83C2×Dic283—24no—
84C2×Dic283—24no—
85C2×Dic283C2×D414yes—
86C4×D283—28no—
87C4×D283—24no—
88C4×D283—24no—
89C23×C283C2×D412yes—
90C23×C283C2411yescharacteristic
91Q8×C283—22no—
92Q8×C283—41no—
93Q8×C283—41no—
94Q8×C283—42no—
95D4×C283—24no—
96D4×C283—22no—
97C2×C4×C283—22no—
98C2×C4×C283—22no—
99C2×C4×C283—22no—
100C2×C4×C283—22no—
101C2×C4×C283—24no—
102C2×C4×C283C2×D412yes—
103C2×C4×C283C2×D412yes—
104C2×C4×C283C2×D412yes—
105C2×C4×C283C2×D411yescharacteristic
106C2×C4×C283C2×D411yescharacteristic
107C22×C283—22no—
108C22×C283—41no—
109C22×C283—42no—
110C22×C283C22×D412yes—
111C22×C283C22×D411yescharacteristic
112C2×D283—24no—
113C2×D283—44no—
114C2×D283—84no—
115C4×C283—22no—
116C4×C283—42no—
117C4×C283—42no—
118C4×C283#12011yescharacteristic
119C4×C283#12011yescharacteristic
120C4×C283#12012yes—
121Dic283—24no—
122Dic283—44no—
123C2×C283—22no—
124C2×C283—21no—
125C2×C283—42no—
126C2×C283C2×#12011yescharacteristic
127D283—44no—
128C283#302811yescharacteristic, complemented, Sylow
129#3028—2831nomaximal, Sylow
130#585—2831no—
131#536—2832no—
132#536D28311yescharacteristic, complemented
133#458—2838no—
134#410—2831no—
135#410—2832no—
1362−1+4—5662no—
1372−1+4—5661no—
138H(5,𝔽2)—2834no—
139H(5,𝔽2)—5662no—
140H(5,𝔽2)—5661no—
141C2×#41—2832no—
142C2×#41—5664no—
143C2×#41—5662no—
144C2×#41C2×D28311yescharacteristic
145C22×Q8C2×D28311yescharacteristic, complemented
146C4⋊Q8—21no—
147C4⋊Q8—5662no—
148#124—21no—
149#124—5662no—
150#123—2838no—
151#123C2×D28314yescomplemented
152C22⋊Q8—22no—
153C22⋊Q8—5664no—
154#117—2832no—
155#117C2×D28311yescharacteristic, complemented
156#99—2834no—
157#99—2838no—
158#41—22no—
159#41—22no—
160#41—5664no—
161#41—5664no—
162#41—5668no—
163#41—5664no—
164#41—5664no—
165#41—5662no—
166#41—5662no—
167#41—5662no—
168#41—5662no—
169#41—5664no—
170C2×Q8—22no—
171C2×Q8—22no—
172C2×Q8—21no—
173C2×Q8—21no—
174C2×Q8—2832no—
175C2×Q8—5662no—
176C2×Q8—5664no—
177C2×Q8C22⋊D28311yescharacteristic
178C2×D4—2834no—
179C2×D4—2832no—
180C2×D4—5662no—
181C2×D4—5668no—
182C2×D4—5664no—
183C2×D4—5664no—
184C2×D4C22⋊D28311yescharacteristic
185C2×D4C22×D28312yes—
186C22×C4—2834no—
187C22×C4—5664no—
188C22×C4C22⋊D28312yes—
189C22×C4C22×D28312yes—
190C22×C4C22×D28311yescharacteristic
191C4⋊C4—21no—
192C4⋊C4—24no—
193C4⋊C4—5668no—
194C4⋊C4—5662no—
195C22⋊C4—24no—
196C22⋊C4—2834no—
197C22⋊C4—5668no—
198C22⋊C4C22×D28312yes—
199C42—21no—
200C42—5662no—
201C23—2834no—
202C23—5664no—
203C23C2×C22⋊D28312yes—
204C23C23×D28311yescharacteristic
205Q8—22no—
206Q8—41no—
207Q8—41no—
208Q8—42no—
209Q8—5664no—
210Q8—5662no—
211Q8—5662no—
212D4—24no—
213D4—22no—
214D4—5664no—
215D4—5668no—
216D4—5664no—
217D4—5664no—
218D4—5662no—
219D4—5662no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—24no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—2834no—
226C2×C4—5668no—
227C2×C4—5664no—
228C2×C4—5664no—
229C2×C4—5664no—
230C2×C4—5664no—
231C2×C4D4×D28312yes—
232C2×C4D4×D28312yes—
233C2×C4C2×C22⋊D28312yes—
234C2×C4C2×C22⋊D28311yescharacteristic
235C2×C4C2×C22⋊D28311yescharacteristic
236C22—22no—
237C22—41no—
238C22—42no—
239C22—5664no—
240C22—11324no—
241C22—22644no—
242C22C2×D4×D28312yes—
243C22C22×C22⋊D28311yescharacteristic
244C4—22no—
245C4—42no—
246C4—42no—
247C4—5664no—
248C4—11324no—
249C4#2258357811yescharacteristic
250C4#2258357811yescharacteristic
251C4#2258357812yes—
252C2—21no—
253C2—22no—
254C2—42no—
255C2—11324no—
256C2C2×#2258357811yescharacteristic
subgroupG/H
centerC2C2×#22583578
commutator, cosocleC23×C283C24
FrattiniC23C23×D283
Fitting#536×C283C2
socleC2×C283C2×#120
prime radical#31827301C2
hypercenter#536D283
hypocenterC283#3028
LCSG ▷ C23×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C2 ◁ C22×C4 ◁ #536

Complements

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

NH ≅ G/Ntype
2. #31827039C2semidirect
3. #31826654C2semidirect
4. #31826439C2semidirect
5. #31826438C2semidirect
6. #536×C283C2semidirect
17. #22583593C22semidirect
19. #22583569C22semidirect
20. #22583525C22semidirect
21. #22583524C22semidirect
23. C22×Q8×C283C22semidirect
26. #123×C283C22semidirect
28. #117×C283C22semidirect
128. C283#3028semidirect
132. #536D283semidirect
145. C22×Q8C2×D283semidirect
151. #123C2×D283semidirect
155. #117C2×D283semidirect

Maximal subgroups

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

subgroupindex
#318273012
#318270392
#318266542
#318264392
#318264382
#536×C2832
#3028283