Finite Group Databasebeta

← #44559770

Subgroups — #44559770

Order 32128.

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#30632523C211yesmaximal, characteristic
2#30632261C212yesmaximal, complemented
3#30631876C218yesmaximal, complemented
4#30631661C211yesmaximal, characteristic, complemented
5#30631660C212yesmaximal, complemented
6#536×C251C211yesmaximal, characteristic, complemented
7#22548763—21no—
8#22548762—22no—
9#22548761—22no—
10#22548760—21no—
11#22548760C2214yes—
12C2×#16860810—22no—
13C2×#16860810C2212yes—
14#41×D251—24no—
15#22548713—22no—
16#22548704—22no—
17#22548698C2218yescomplemented
18#22548691—24no—
19#22548674C2212yescomplemented
20#22548630C2214yescomplemented
21#22548629C2218yescomplemented
22C2×#41×C251C2211yescharacteristic
23C22×Q8×C251C2211yescharacteristic, complemented
24C4⋊Q8×C251—21no—
25#124×C251—21no—
26#123×C251C2214yescomplemented
27C22⋊Q8×C251—22no—
28#117×C251C2211yescharacteristic, complemented
29#16860811—24no—
30#16860810—24no—
31#16860810—24no—
32#16860810—22no—
33#16860810—22no—
34#16860810—24no—
35#16860810—22no—
36#16860810—22no—
37#16860809—24no—
38#16860809—28no—
39C2×Dic502—22no—
40C2×Dic502D412yes—
41Q8×D251—24no—
42C2×D1004—22no—
43C2×D1004D412yes—
44C2×C22⋊D251—24no—
45C2×C22⋊D251C2314yes—
46D4×D251—24no—
47D4×D251—28no—
48C2×C4×D251—24no—
49C2×C4×D251D414yes—
50C4⋊Dic251—22no—
51Dic251⋊C4—28no—
52C22⋊Dic251C2314yes—
53(C2×D251)⋊C4—28no—
54C4×Dic251—22no—
55#41×C251—22no—
56#41×C251—22no—
57C2×Q8×C251—21no—
58C2×Q8×C251—21no—
59C2×Q8×C251—22no—
60C2×Q8×C251—22no—
61C2×Q8×C251D411yescharacteristic
62C2×D4×C251D411yescharacteristic
63C2×D4×C251C2312yes—
64C22×C4×C251D412yes—
65C22×C4×C251C2312yes—
66C22×C4×C251C2311yescharacteristic
67C4⋊C4×C251—21no—
68C4⋊C4×C251—24no—
69C22⋊C4×C251—24no—
70C22⋊C4×C251C2312yes—
71C42×C251—21no—
72C22×D251—24no—
73C22×D251C2×D414yes—
74Dic502—24no—
75Dic502—22no—
76Dic502—22no—
77D1004—24no—
78D1004—22no—
79D1004—22no—
80C22⋊D251—28no—
81C22⋊D251—24no—
82C22⋊D251—24no—
83C2×Dic251—24no—
84C2×Dic251—24no—
85C2×Dic251C2×D414yes—
86C4×D251—28no—
87C4×D251—24no—
88C4×D251—24no—
89C23×C251C2×D412yes—
90C23×C251C2411yescharacteristic
91Q8×C251—22no—
92Q8×C251—41no—
93Q8×C251—41no—
94Q8×C251—42no—
95D4×C251—24no—
96D4×C251—22no—
97C2×C4×C251—22no—
98C2×C4×C251—22no—
99C2×C4×C251—22no—
100C2×C4×C251—22no—
101C2×C4×C251—24no—
102C2×C4×C251C2×D412yes—
103C2×C4×C251C2×D412yes—
104C2×C4×C251C2×D412yes—
105C2×C4×C251C2×D411yescharacteristic
106C2×C4×C251C2×D411yescharacteristic
107C22×C251—22no—
108C22×C251—41no—
109C22×C251—42no—
110C22×C251C22×D412yes—
111C22×C251C22×D411yescharacteristic
112C2×D251—24no—
113C2×D251—44no—
114C2×D251—84no—
115C4×C251—22no—
116C4×C251—42no—
117C4×C251—42no—
118C4×C251#12011yescharacteristic
119C4×C251#12011yescharacteristic
120C4×C251#12012yes—
121Dic251—24no—
122Dic251—44no—
123C2×C251—22no—
124C2×C251—21no—
125C2×C251—42no—
126C2×C251C2×#12011yescharacteristic
127D251—44no—
128C251#302811yescharacteristic, complemented, Sylow
129#3028—2511nomaximal, Sylow
130#585—2511no—
131#536—2512no—
132#536D25111yescharacteristic, complemented
133#458—2518no—
134#410—2511no—
135#410—2512no—
1362−1+4—5022no—
1372−1+4—5021no—
138H(5,𝔽2)—5022no—
139H(5,𝔽2)—5021no—
140H(5,𝔽2)—2514no—
141C2×#41—5024no—
142C2×#41—5022no—
143C2×#41—2512no—
144C2×#41C2×D25111yescharacteristic
145C22×Q8C2×D25111yescharacteristic, complemented
146C4⋊Q8—21no—
147C4⋊Q8—5022no—
148#124—21no—
149#124—5022no—
150#123—2518no—
151#123C2×D25114yescomplemented
152C22⋊Q8—22no—
153C22⋊Q8—5024no—
154#117—2512no—
155#117C2×D25111yescharacteristic, complemented
156#99—2514no—
157#99—2518no—
158#41—22no—
159#41—22no—
160#41—5024no—
161#41—5024no—
162#41—5024no—
163#41—5024no—
164#41—5028no—
165#41—5022no—
166#41—5022no—
167#41—5022no—
168#41—5024no—
169#41—5022no—
170C2×Q8—22no—
171C2×Q8—22no—
172C2×Q8—21no—
173C2×Q8—21no—
174C2×Q8—5022no—
175C2×Q8—5024no—
176C2×Q8—2512no—
177C2×Q8C22⋊D25111yescharacteristic
178C2×D4—5022no—
179C2×D4—5024no—
180C2×D4—5028no—
181C2×D4—5024no—
182C2×D4—2514no—
183C2×D4—2512no—
184C2×D4C22⋊D25111yescharacteristic
185C2×D4C22×D25112yes—
186C22×C4—5024no—
187C22×C4—2514no—
188C22×C4C22⋊D25112yes—
189C22×C4C22×D25112yes—
190C22×C4C22×D25111yescharacteristic
191C4⋊C4—21no—
192C4⋊C4—24no—
193C4⋊C4—5028no—
194C4⋊C4—5022no—
195C22⋊C4—24no—
196C22⋊C4—5028no—
197C22⋊C4—2514no—
198C22⋊C4C22×D25112yes—
199C42—21no—
200C42—5022no—
201C23—5024no—
202C23—2514no—
203C23C2×C22⋊D25112yes—
204C23C23×D25111yescharacteristic
205Q8—22no—
206Q8—41no—
207Q8—41no—
208Q8—42no—
209Q8—5024no—
210Q8—5022no—
211Q8—5022no—
212D4—24no—
213D4—22no—
214D4—5024no—
215D4—5028no—
216D4—5024no—
217D4—5024no—
218D4—5022no—
219D4—5022no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—24no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—5028no—
226C2×C4—5024no—
227C2×C4—5024no—
228C2×C4—5024no—
229C2×C4—5024no—
230C2×C4—2514no—
231C2×C4D4×D25112yes—
232C2×C4D4×D25112yes—
233C2×C4C2×C22⋊D25112yes—
234C2×C4C2×C22⋊D25111yescharacteristic
235C2×C4C2×C22⋊D25111yescharacteristic
236C22—22no—
237C22—41no—
238C22—42no—
239C22—20084no—
240C22—10044no—
241C22—5024no—
242C22C2×D4×D25112yes—
243C22C22×C22⋊D25111yescharacteristic
244C4—22no—
245C4—42no—
246C4—42no—
247C4—10044no—
248C4—5024no—
249C4#2254868311yescharacteristic
250C4#2254868311yescharacteristic
251C4#2254868312yes—
252C2—21no—
253C2—22no—
254C2—42no—
255C2—10044no—
256C2C2×#2254868311yescharacteristic
subgroupG/H
centerC2C2×#22548683
commutator, cosocleC23×C251C24
FrattiniC23C23×D251
Fitting#536×C251C2
socleC2×C251C2×#120
prime radical#30632523C2
hypercenter#536D251
hypocenterC251#3028
LCSG ▷ C23×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2 ◁ C22×C4 ◁ #536

Complements

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

NH ≅ G/Ntype
2. #30632261C2semidirect
3. #30631876C2semidirect
4. #30631661C2semidirect
5. #30631660C2semidirect
6. #536×C251C2semidirect
17. #22548698C22semidirect
19. #22548674C22semidirect
20. #22548630C22semidirect
21. #22548629C22semidirect
23. C22×Q8×C251C22semidirect
26. #123×C251C22semidirect
28. #117×C251C22semidirect
128. C251#3028semidirect
132. #536D251semidirect
145. C22×Q8C2×D251semidirect
151. #123C2×D251semidirect
155. #117C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306325232
#306322612
#306318762
#306316612
#306316602
#536×C2512
#3028251