Finite Group Databasebeta

← #44560401

Subgroups — #44560401

Order 32128.

384 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 125 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#30632239C211yesmaximal, characteristic
2C2×#22548680C211yesmaximal, characteristic
3#30632022C211yesmaximal, characteristic
4#30631981C212yesmaximal, complemented
5#30631896C212yesmaximal, complemented
6#30631845C211yesmaximal, characteristic
7#30631842C211yesmaximal, characteristic
8#30631781C212yesmaximal, complemented
9#30631779C212yesmaximal, complemented
10#30631737C211yesmaximal, characteristic
11#484×C251C211yesmaximal, characteristic
12C22×Dic502C2211yescharacteristic
13C2×Dic1004C2212yes—
14C2×#16860800C2212yes—
15#22548714C2211yescharacteristic
16#22548713C2211yescharacteristic
17#22548705—21no—
18#22548694C2211yescharacteristic
19#22548692—21no—
20Q8×Dic251C2211yescharacteristic
21#22548680—21no—
22#22548680—21no—
23#22548680—21no—
24#22548680C2212yes—
25#22548680C2211yescharacteristic
26#22548680C2212yes—
27#22548671—21no—
28C2×Dic251⋊C4—21no—
29C2×C4×Dic251—21no—
30C2×C4×Dic251C2211yescharacteristic
31#22548648C2212yes—
32Dic251⋊C8C2212yes—
33#22548640C2212yes—
34#22548638C2212yes—
35#22548638C2212yes—
36C22⋊(C251⋊C8)C2211yescharacteristic
37C8×Dic251C2212yes—
38C22⋊Q8×C251C2211yescharacteristic
39C2×C4⋊C4×C251C2211yescharacteristic
40#107×C251C2212yes—
41#103×C251C2212yes—
42C22⋊C8×C251C2211yescharacteristic
43C2×Dic502—24no—
44C2×Dic502—21no—
45C2×Dic502—21no—
46C2×Dic502C2311yescharacteristic
47C2×Dic502C2312yes—
48C22×Dic251—21no—
49C22×Dic251D411yescharacteristic
50C22×Dic251D411yescharacteristic
51Dic1004—24no—
52#16860800—24no—
53C2×C251⋊C8C2312yes—
54C4⋊Dic251—21no—
55C4⋊Dic251C2311yescharacteristic
56Dic251⋊C4—22no—
57Dic251⋊C4—22no—
58Dic251⋊C4—21no—
59Dic251⋊C4—21no—
60Dic251⋊C4—21no—
61Dic251⋊C4—21no—
62Dic251⋊C4—21no—
63C22⋊Dic251—21no—
64C4×Dic251—22no—
65C4×Dic251—22no—
66C4×Dic251—21no—
67C4×Dic251—21no—
68C4×Dic251D411yescharacteristic
69C4×Dic251D411yescharacteristic
70C4×Dic251C2311yescharacteristic
71C4×Dic251C2311yescharacteristic
72C2×Q8×C251C2311yescharacteristic
73C22×C4×C251—21no—
74C22×C4×C251C2311yescharacteristic
75C2×C8×C251C2312yes—
76C4⋊C4×C251—21no—
77C4⋊C4×C251—21no—
78C4⋊C4×C251C2312yes—
79C4⋊C4×C251C2311yescharacteristic
80C22⋊C4×C251—21no—
81Dic502—24no—
82Dic502—22no—
83Dic502—22no—
84Dic502—42no—
85Dic502#4114yes—
86C2×Dic251—24no—
87C2×Dic251—22no—
88C2×Dic251—21no—
89C2×Dic251—21no—
90C2×Dic251—41no—
91C2×Dic251Q1614yescomplemented
92C2×Dic251C2×D411yescharacteristic
93C2×Dic251C2×D411yescharacteristic
94C2×Dic251C2×D411yescharacteristic
95C2×Dic251C2×D411yescharacteristic
96C251⋊C8—22no—
97C23×C251C2×D411yescharacteristic
98Q8×C251—22no—
99C2×C4×C251—22no—
100C2×C4×C251—21no—
101C2×C4×C251—21no—
102C2×C4×C251—22no—
103C2×C4×C251—42no—
104C2×C4×C251C2×D411yescharacteristic
105C2×C4×C251C2411yescharacteristic
106C8×C251—22no—
107C22×C251—21no—
108C22×C251—21no—
109C22×C251C2×Q1612yes—
110C22×C251C22×D411yescharacteristic
111C4×C251—21no—
112C4×C251—41no—
113C4×C251—42no—
114C4×C251—41no—
115C4×C251C2×#4112yes—
116Dic251—24no—
117Dic251—21no—
118Dic251—21no—
119Dic251—41no—
120Dic251—41no—
121Dic251C2×Q1612yes—
122Dic251C2×Q1612yes—
123C2×C251—22no—
124C2×C251#52511yescharacteristic
125C2×C251C22×Q1611yescharacteristic
126C2×C251#57511yescharacteristic
127C251#310411yescharacteristic, complemented, Sylow
128#3104—2511nomaximal, Sylow
129#533—2511no—
130C2×C4×Q8—2511no—
131#484—2511no—
132#484D25111yescharacteristic
133#477—2512no—
134#462—2512no—
135#451—2511no—
136#451—2511no—
137C4×Q16—2512no—
138C4×Q16—2512no—
139#431—2511no—
140C22×Q8—2511no—
141C2×Q16—2512no—
142C2×Q16—2512no—
143C4⋊Q8—2511no—
144C4⋊Q8—2511no—
145#125—5021no—
146C22⋊Q8—5021no—
147C22⋊Q8—2511no—
148C22⋊Q8C2×D25111yescharacteristic
149C4×Q8—5021no—
150C4×Q8—5021no—
151C4×Q8—5021no—
152C4×Q8—2511no—
153C4×Q8—2512no—
154C4×Q8—2512no—
155C4×Q8—2511no—
156#117—5021no—
157C2×C4⋊C4—5021no—
158C2×C4⋊C4C2×D25111yescharacteristic
159C2×C42—5021no—
160C2×C42—2511no—
161#107—2512no—
162#107C2×D25112yes—
163C4⋊C8—2512no—
164#103—2512no—
165#103—2512no—
166#103—2512no—
167#103C2×D25112yes—
168C22⋊C8—2511no—
169C22⋊C8C2×D25111yescharacteristic
170C4×C8—2512no—
171C2×Q8—5024no—
172C2×Q8—5021no—
173C2×Q8—5021no—
174C2×Q8—2511no—
175C2×Q8—2512no—
176C2×Q8C22×D25111yescharacteristic
177C22×C4—21no—
178C22×C4—5021no—
179C22×C4—2511no—
180C22×C4—2511no—
181C22×C4C22×D25111yescharacteristic
182Q16—5024no—
183Q16—5024no—
184C2×C8—2512no—
185C2×C8C22×D25112yes—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—5022no—
189C4⋊C4—5022no—
190C4⋊C4—5021no—
191C4⋊C4—5021no—
192C4⋊C4—5021no—
193C4⋊C4—5021no—
194C4⋊C4—5021no—
195C4⋊C4—5021no—
196C4⋊C4—2511no—
197C4⋊C4C22×D25111yescharacteristic
198C4⋊C4C22×D25112yes—
199C22⋊C4—21no—
200C22⋊C4—5021no—
201C42—5022no—
202C42—5022no—
203C42—5021no—
204C42—5021no—
205C42—2511no—
206C42—2511no—
207C42—2511no—
208C42—2511no—
209C23D4×D25111yescharacteristic
210Q8—22no—
211Q8—10042no—
212Q8—5024no—
213Q8—5022no—
214Q8—5022no—
215Q8—2514no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—42no—
221C2×C4—10041no—
222C2×C4—5021no—
223C2×C4—5021no—
224C2×C4—5024no—
225C2×C4—5022no—
226C2×C4—2511no—
227C2×C4—2511no—
228C2×C4—2511no—
229C2×C4—2511no—
230C2×C4—2514no—
231C2×C4D4×D25111yescharacteristic
232C2×C4C23×D25111yescharacteristic
233C8—22no—
234C8—5022no—
235C22—21no—
236C22—21no—
237C22Q16×D25112yes—
238C22C2×D4×D25111yescharacteristic
239C4—21no—
240C4—41no—
241C4—42no—
242C4—41no—
243C4—10041no—
244C4—10041no—
245C4—5024no—
246C4—5021no—
247C4—5021no—
248C4—2512no—
249C4—2512no—
250C4#41×D25112yes—
251C2—22no—
252C2#3063219511yescharacteristic
253C2C2×Q16×D25111yescharacteristic
254C2#3063248311yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#484×C251C2
socleC22×C251C22×D4
prime radicalC23×C251C2×D4
hypercenter#484D251
hypocenterC251#3104
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
4. #30631981C2semidirect
5. #30631896C2semidirect
8. #30631781C2semidirect
9. #30631779C2semidirect
91. C2×Dic251Q16semidirect
127. C251#3104semidirect

Maximal subgroups

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

subgroupindex
#306322392
C2×#225486802
#306320222
#306319812
#306318962
#306318452
#306318422
#306317812
#306317792
#306317372
#484×C2512
#3104251