Finite Group Databasebeta

← #44560169

Subgroups — #44560169

Order 32128.

376 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 135 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#30632240C211yesmaximal, characteristic
2C2×#22548680C211yesmaximal, characteristic
3#30631997C212yesmaximal, complemented
4#30631993C212yesmaximal, complemented
5#30631981C212yesmaximal, complemented
6#30631971C212yesmaximal, complemented
7#30631727C211yesmaximal, characteristic
8#30631710C211yesmaximal, characteristic
9#30631697C211yesmaximal, characteristic
10#30631695C211yesmaximal, characteristic
11#428×C251C211yesmaximal, characteristic
12C22×Dic502C2211yescharacteristic
13C2×#16860794C2211yescharacteristic
14#22548714C2211yescharacteristic
15#22548714C2211yescharacteristic
16#22548706C2211yescharacteristic
17#22548706C2211yescharacteristic
18#22548692—21no—
19#22548692—21no—
20C4×Dic502—21no—
21C4×Dic502—21no—
22#22548680—21no—
23#22548680—21no—
24#22548680C2212yes—
25#22548680C2212yes—
26#22548674C2211yescharacteristic
27C2×Dic251⋊C4—21no—
28C2×C4×Dic251—21no—
29C2×C4×Dic251C2211yescharacteristic
30#22548648C2212yes—
31#22548646C2212yes—
32Dic251⋊C8C2212yes—
33Dic251⋊C8C2212yes—
34#22548640C2212yes—
35#22548640C2212yes—
36#22548638C2212yes—
37#22548638C2212yes—
38C2×M16×C251C2211yescharacteristic
39#117×C251C2211yescharacteristic
40C2×C4⋊C4×C251C2211yescharacteristic
41#107×C251C2212yes—
42#106×C251C2212yes—
43C2×Dic502—21no—
44C2×Dic502—21no—
45C2×Dic502—21no—
46C2×Dic502—22no—
47C2×Dic502Q814yescomplemented
48C2×Dic502C2312yes—
49C22×Dic251—21no—
50C22×Dic251D411yescharacteristic
51C22×Dic251D411yescharacteristic
52#16860794—22no—
53C2×C251⋊C8C2312yes—
54C4⋊Dic251—21no—
55C4⋊Dic251—21no—
56C4⋊Dic251C2311yescharacteristic
57C4⋊Dic251C2311yescharacteristic
58Dic251⋊C4—22no—
59Dic251⋊C4—22no—
60Dic251⋊C4—21no—
61Dic251⋊C4—21no—
62Dic251⋊C4—21no—
63Dic251⋊C4—21no—
64C22⋊Dic251—21no—
65C4×Dic251—22no—
66C4×Dic251—22no—
67C4×Dic251—21no—
68C4×Dic251D411yescharacteristic
69C4×Dic251D411yescharacteristic
70C4×Dic251C2311yescharacteristic
71C4×Dic251C2311yescharacteristic
72C22×C4×C251—21no—
73C22×C4×C251C2311yescharacteristic
74M16×C251—22no—
75C2×C8×C251C2312yes—
76C4⋊C4×C251—21no—
77C4⋊C4×C251C2311yescharacteristic
78C4⋊C4×C251C2312yes—
79C4⋊C4×C251C2311yescharacteristic
80C22⋊C4×C251—21no—
81C42×C251—21no—
82Dic502—24no—
83Dic502—22no—
84Dic502—42no—
85Dic502—42no—
86Dic502C2×Q814yes—
87C2×Dic251—24no—
88C2×Dic251—22no—
89C2×Dic251—22no—
90C2×Dic251—21no—
91C2×Dic251—21no—
92C2×Dic251—41no—
93C2×Dic251C2×D411yescharacteristic
94C2×Dic251C2×D411yescharacteristic
95C2×Dic251C2×D411yescharacteristic
96C2×Dic251C2×D411yescharacteristic
97C251⋊C8—22no—
98C23×C251C2×D411yescharacteristic
99C2×C4×C251—21no—
100C2×C4×C251—21no—
101C2×C4×C251—22no—
102C2×C4×C251—42no—
103C2×C4×C251C2×D411yescharacteristic
104C2×C4×C251C2×Q812yes—
105C2×C4×C251#4112yes—
106C2×C4×C251C2411yescharacteristic
107C8×C251—22no—
108C22×C251—21no—
109C22×C251—21no—
110C22×C251C22⋊Q812yes—
111C22×C251C22×D411yescharacteristic
112C4×C251—41no—
113C4×C251—42no—
114C4×C251—41no—
115C4×C251C22⋊Q812yes—
116C4×C251C22×Q811yescharacteristic
117C4×C251C2×#4111yescharacteristic
118Dic251—24no—
119Dic251—21no—
120Dic251—21no—
121Dic251—21no—
122Dic251—41no—
123Dic251—41no—
124Dic251#13712yes—
125C2×C251—22no—
126C2×C251C2×C22⋊Q811yescharacteristic
127C2×C251C2×#13711yescharacteristic
128C2×C251#57511yescharacteristic
129C251#307811yescharacteristic, complemented, Sylow
130#3078—2511nomaximal, Sylow
131#533—2511no—
132C2×C4×Q8—2511no—
133#479—2512no—
134#479—2512no—
135#477—2512no—
136#475—2512no—
137#428—2511no—
138#428D25111yescharacteristic
139#423—2511no—
140#419—2511no—
141#419—2511no—
142C22×Q8—2511no—
143C2×M16—2511no—
144C2×M16C2×D25111yescharacteristic
145C4⋊Q8—2511no—
146C4⋊Q8—2511no—
147#125—2511no—
148#125—2511no—
149C22⋊Q8—5021no—
150C22⋊Q8—5021no—
151C4×Q8—5021no—
152C4×Q8—5021no—
153C4×Q8—5021no—
154C4×Q8—5021no—
155C4×Q8—2512no—
156C4×Q8—2512no—
157#117—2511no—
158#117C2×D25111yescharacteristic
159C2×C4⋊C4—5021no—
160C2×C4⋊C4C2×D25111yescharacteristic
161C2×C42—5021no—
162C2×C42—2511no—
163#107—2512no—
164#107C2×D25112yes—
165#106—2512no—
166#106C2×D25112yes—
167C4⋊C8—2512no—
168C4⋊C8—2512no—
169#103—2512no—
170#103—2512no—
171#103—2512no—
172#103—2512no—
173C2×Q8—5021no—
174C2×Q8—5021no—
175C2×Q8—5021no—
176C2×Q8—5022no—
177C2×Q8—2514no—
178C2×Q8—2512no—
179C22×C4—21no—
180C22×C4—5021no—
181C22×C4—2511no—
182C22×C4—2511no—
183C22×C4C22×D25111yescharacteristic
184M16—22no—
185M16—5022no—
186C2×C8—2512no—
187C2×C8C22×D25112yes—
188C4⋊C4—21no—
189C4⋊C4—5022no—
190C4⋊C4—5022no—
191C4⋊C4—5021no—
192C4⋊C4—5021no—
193C4⋊C4—5021no—
194C4⋊C4—5021no—
195C4⋊C4—5021no—
196C4⋊C4—5021no—
197C4⋊C4—2511no—
198C4⋊C4—2511no—
199C4⋊C4C22×D25111yescharacteristic
200C4⋊C4C22×D25111yescharacteristic
201C4⋊C4C22×D25112yes—
202C22⋊C4—21no—
203C22⋊C4—5021no—
204C42—21no—
205C42—5022no—
206C42—5022no—
207C42—5021no—
208C42—2511no—
209C42—2511no—
210C42—2511no—
211C42—2511no—
212C23D4×D25111yescharacteristic
213Q8—10042no—
214Q8—10042no—
215Q8—5022no—
216Q8—5024no—
217Q8—2514no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—42no—
222C2×C4—10041no—
223C2×C4—5021no—
224C2×C4—5021no—
225C2×C4—5022no—
226C2×C4—5024no—
227C2×C4—5022no—
228C2×C4—2511no—
229C2×C4—2511no—
230C2×C4—2511no—
231C2×C4—2511no—
232C2×C4D4×D25111yescharacteristic
233C2×C4Q8×D25112yes—
234C2×C4#1686081012yes—
235C2×C4C23×D25111yescharacteristic
236C8—22no—
237C8—5022no—
238C22—21no—
239C22—21no—
240C22#2254869312yes—
241C22C2×D4×D25111yescharacteristic
242C4—41no—
243C4—42no—
244C4—41no—
245C4—10041no—
246C4—10041no—
247C4—5021no—
248C4—5021no—
249C4—5021no—
250C4—5024no—
251C4—2512no—
252C4#2254869312yes—
253C4C2×Q8×D25111yescharacteristic
254C4C2×#1686081011yescharacteristic
255C2—22no—
256C2C2×#2254869311yescharacteristic
257C2#137×D25111yescharacteristic
258C2#3063248311yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#428×C251C2
socleC22×C251C22×D4
prime radicalC23×C251C2×D4
hypercenter#428D251
hypocenterC251#3078
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
3. #30631997C2semidirect
4. #30631993C2semidirect
5. #30631981C2semidirect
6. #30631971C2semidirect
47. C2×Dic502Q8semidirect
129. C251#3078semidirect

Maximal subgroups

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

subgroupindex
#306322402
C2×#225486802
#306319972
#306319932
#306319812
#306319712
#306317272
#306317102
#306316972
#306316952
#428×C2512
#3078251