Finite Group Databasebeta

← #44560916

Subgroups — #44560916

Order 32128.

348 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 137 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#30632242C211yesmaximal, characteristic
2#125×D251C211yesmaximal, characteristic
3#30632097C211yesmaximal, characteristic, complemented
4#30632096C211yesmaximal, characteristic, complemented
5#30632093C211yesmaximal, characteristic, complemented
6#30632088C212yesmaximal, complemented
7#30632087C212yesmaximal, complemented
8#30631728C212yesmaximal
9#30631726C212yesmaximal
10#30631639C211yesmaximal, characteristic
11#501×C251C211yesmaximal, characteristic, complemented
12C2×#16860793C2212yes—
13#22548715C2211yescharacteristic
14#22548707C2211yescharacteristic
15#22548706C2211yescharacteristic
16#22548706C2211yescharacteristic
17#22548706C2212yes—
18#22548705—21no—
19#22548705—21no—
20#22548695—21no—
21#22548691—21no—
22Q8×Dic251—21no—
23#22548680—21no—
24#22548673C2212yes—
25C4⋊C4×D251—21no—
26C4⋊C4×D251—21no—
27C4⋊C4×D251C2211yescharacteristic
28C4⋊C4×D251C2211yescharacteristic
29C42×D251C2211yescharacteristic
30#22548649C2212yes—
31#22548648C2212yes—
32#22548648C2212yes—
33#22548647C2212yes—
34#22548646C2212yes—
35#22548646C2212yes—
36#22548626C2211yescharacteristic
37C8×Dic251C2212yes—
38C4⋊Q8×C251C2211yescharacteristic
39#125×C251C2211yescharacteristic
40#107×C251C2212yes—
41#106×C251C2212yes—
42#97×C251C2211yescharacteristic
43C2×Dic502—21no—
44C2×C4×D251—21no—
45C2×C4×D251—21no—
46C2×C4×D251D411yescharacteristic
47C2×C4×D251D411yescharacteristic
48C2×C4×D251C2311yescharacteristic
49#16860793Q818yescomplemented
50C2×C251⋊C8C2312yes—
51C4⋊Dic251—21no—
52C4⋊Dic251—21no—
53C4⋊Dic251—22no—
54C4⋊Dic251—21no—
55C4⋊Dic251C2311yescharacteristic
56C4⋊Dic251C2312yes—
57C4⋊Dic251C2311yescharacteristic
58Dic251⋊C4—21no—
59Dic251⋊C4—21no—
60Dic251⋊C4—21no—
61Dic251⋊C4—21no—
62Dic251⋊C4—22no—
63Dic251⋊C4—21no—
64Dic251⋊C4—21no—
65(C2×D251)⋊C4—22no—
66C4×Dic251—22no—
67C4×Dic251D411yescharacteristic
68C4×Dic251D411yescharacteristic
69C4×Dic251C2311yescharacteristic
70C2×Q8×C251—21no—
71C2×C8×C251C2312yes—
72C4⋊C4×C251—21no—
73C4⋊C4×C251—21no—
74C4⋊C4×C251—21no—
75C4⋊C4×C251C2311yescharacteristic
76C4⋊C4×C251C2312yes—
77C4⋊C4×C251C2311yescharacteristic
78C42×C251C2311yescharacteristic
79C22×D251C2×D411yescharacteristic
80Dic502—42no—
81C2×Dic251—22no—
82C2×Dic251—21no—
83C2×Dic251—21no—
84C2×Dic251C2×D411yescharacteristic
85C2×Dic251C2×D411yescharacteristic
86C2×Dic251C2×D411yescharacteristic
87C4×D251—41no—
88C4×D251—41no—
89C4×D251—42no—
90C4×D251—42no—
91C4×D251C2×Q814yes—
92C251⋊C8C2×Q814yes—
93Q8×C251—42no—
94C2×C4×C251—21no—
95C2×C4×C251—21no—
96C2×C4×C251—22no—
97C2×C4×C251C2×D411yescharacteristic
98C2×C4×C251C2×D411yescharacteristic
99C2×C4×C251C2411yescharacteristic
100C8×C251C2×Q814yes—
101C22×C251C22×D411yescharacteristic
102C2×D251—21no—
103C2×D251—21no—
104C2×D251C4⋊Q812yes—
105C4×C251—21no—
106C4×C251—21no—
107C4×C251—42no—
108C4×C251—41no—
109C4×C251—41no—
110C4×C251C22×Q812yes—
111Dic251—21no—
112Dic251—21no—
113Dic251—41no—
114Dic251—41no—
115Dic251—42no—
116Dic251C4⋊Q812yes—
117C2×C251C2×C4⋊Q811yescharacteristic
118C2×C251#57511yescharacteristic
119C2×C251#57511yescharacteristic
120D251—22no—
121C251#316611yescharacteristic, complemented, Sylow
122#3166—2511nomaximal, Sylow
123#533—2511no—
124C2×#125—2511no—
125#501—2511no—
126#501—2511no—
127#501—2511no—
128#501D25111yescharacteristic, complemented
129#499—2512no—
130#499—2512no—
131#428—2512no—
132#428—2512no—
133C4×M16—2511no—
134C2×M16—2512no—
135C4⋊Q8—2511no—
136C4⋊Q8C2×D25111yescharacteristic
137#125—5021no—
138#125—5021no—
139#125—2511no—
140#125—2511no—
141#125—2512no—
142#125—2511no—
143#125C2×D25111yescharacteristic
144C22⋊Q8—5021no—
145C22⋊Q8—5021no—
146C4×Q8—5021no—
147C4×Q8—5021no—
148#117—2512no—
149C2×C4⋊C4—5021no—
150C2×C4⋊C4—5021no—
151C2×C4⋊C4—2511no—
152C2×C4⋊C4—2511no—
153C2×C42—2511no—
154#107—2512no—
155#107—2512no—
156#107—2512no—
157#107C2×D25112yes—
158#106—2512no—
159#106—2512no—
160#106—2512no—
161#106C2×D25112yes—
162#97—2511no—
163#97C2×D25111yescharacteristic
164C4×C8—2512no—
165C2×Q8—21no—
166C2×Q8—5021no—
167C22×C4—5021no—
168C22×C4—5021no—
169C22×C4—2511no—
170C22×C4—2511no—
171C22×C4—2511no—
172M16—2518no—
173C2×C8—2512no—
174C2×C8C22×D25112yes—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—5021no—
179C4⋊C4—5021no—
180C4⋊C4—5021no—
181C4⋊C4—5021no—
182C4⋊C4—5021no—
183C4⋊C4—5021no—
184C4⋊C4—5022no—
185C4⋊C4—5022no—
186C4⋊C4—5021no—
187C4⋊C4—5021no—
188C4⋊C4—5021no—
189C4⋊C4—2511no—
190C4⋊C4—2511no—
191C4⋊C4—2512no—
192C4⋊C4C22×D25111yescharacteristic
193C4⋊C4C22×D25112yes—
194C4⋊C4C22×D25111yescharacteristic
195C22⋊C4—5022no—
196C42—5022no—
197C42—2511no—
198C42—2511no—
199C42—2511no—
200C42C22×D25111yescharacteristic
201C23—2511no—
202Q8—42no—
203Q8—10042no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—10041no—
208C2×C4—10041no—
209C2×C4—10042no—
210C2×C4—10042no—
211C2×C4—5022no—
212C2×C4—5021no—
213C2×C4—5021no—
214C2×C4—2511no—
215C2×C4—2511no—
216C2×C4—2511no—
217C2×C4—2514no—
218C2×C4D4×D25111yescharacteristic
219C2×C4D4×D25111yescharacteristic
220C2×C4C23×D25111yescharacteristic
221C8—2514no—
222C8Q8×D25114yes—
223C22—5021no—
224C22—5021no—
225C22—2512no—
226C22C2×D4×D25111yescharacteristic
227C4—21no—
228C4—21no—
229C4—41no—
230C4—42no—
231C4—41no—
232C4—10041no—
233C4—10041no—
234C4—10042no—
235C4—5021no—
236C4—5021no—
237C4—2512no—
238C4C2×Q8×D25112yes—
239C2—5022no—
240C2C4⋊Q8×D25111yescharacteristic
241C2#3063248311yescharacteristic
242C2#3063248611yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#501×C251C2
socleC22×C251C22×D4
prime radicalC22×D251C2×D4
hypercenter#501D251
hypocenterC251#3166
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #501

Complements

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

NH ≅ G/Ntype
3. #30632097C2semidirect
4. #30632096C2semidirect
5. #30632093C2semidirect
6. #30632088C2semidirect
7. #30632087C2semidirect
11. #501×C251C2semidirect
49. #16860793Q8semidirect
121. C251#3166semidirect
128. #501D251semidirect

Maximal subgroups

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

subgroupindex
#306322422
#125×D2512
#306320972
#306320962
#306320932
#306320882
#306320872
#306317282
#306317262
#306316392
#501×C2512
#3166251