Finite Group Databasebeta

← #44560464

Subgroups — #44560464

Order 32128.

444 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 133 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#30632244C211yesmaximal, characteristic, complemented
2#30632022C212yesmaximal, complemented
3#30631967C212yesmaximal, complemented
4#30631892C212yesmaximal, complemented
5#30631838C212yesmaximal
6#30631787C214yesmaximal, complemented
7#30631744C211yesmaximal, characteristic, complemented
8C2×C4×D4×C251C211yesmaximal, characteristic
9C2×#16860798C2214yes—
10#22548715C2211yescharacteristic, complemented
11#22548715C2211yescharacteristic, complemented
12#22548707—21no—
13#22548694—21no—
14#22548694C2212yes—
15C4×Dic502—21no—
16C4×Dic502C2212yescomplemented
17#22548674—21no—
18#22548648C2214yescomplemented
19C4⋊(C251⋊C8)C2212yescomplemented
20#22548638C2214yescomplemented
21#22548636C2214yes—
22C22⋊(C251⋊C8)C2212yes—
23#22548626C2212yescomplemented
24C22×D4×C251C2211yescharacteristic
25C23×C4×C251—21no—
26C4×D4×C251—21no—
27C4×D4×C251—21no—
28C4×D4×C251C2212yes—
29C4×D4×C251C2212yes—
30C2×C4⋊C4×C251C2211yescharacteristic
31C2×C22⋊C4×C251—21no—
32C2×C42×C251C2211yescharacteristic
33C2×Dic502—21no—
34C2×Dic502C2312yes—
35#16860798—28no—
36C2×C251⋊C8C2314yes—
37C4⋊Dic251—21no—
38C4⋊Dic251—21no—
39C4⋊Dic251—21no—
40C4⋊Dic251C2312yes—
41Dic251⋊C4—22no—
42Dic251⋊C4—22no—
43C22⋊Dic251—22no—
44C4×Dic251—22no—
45C24×C251—21no—
46C2×D4×C251—24no—
47C2×D4×C251—21no—
48C2×D4×C251C2312yes—
49C22×C4×C251—21no—
50C22×C4×C251—22no—
51C22×C4×C251—22no—
52C22×C4×C251—44no—
53C22×C4×C251D411yescharacteristic
54C22×C4×C251D411yescharacteristic
55C22×C4×C251C2311yescharacteristic
56C4⋊C4×C251—21no—
57C4⋊C4×C251C2312yes—
58C22⋊C4×C251—22no—
59C22⋊C4×C251—22no—
60C42×C251D411yescharacteristic
61C42×C251D411yescharacteristic
62C42×C251C2311yescharacteristic
63C42×C251C2311yescharacteristic
64Dic502—24no—
65Dic502—42no—
66C2×Dic251—22no—
67C2×Dic251—22no—
68C251⋊C8—24no—
69C23×C251—24no—
70C23×C251—22no—
71C23×C251—42no—
72C23×C251—42no—
73C23×C251C2×D411yescharacteristic
74D4×C251—24no—
75D4×C251—22no—
76D4×C251#4114yes—
77C2×C4×C251—22no—
78C2×C4×C251—22no—
79C2×C4×C251—22no—
80C2×C4×C251—22no—
81C2×C4×C251—44no—
82C2×C4×C251—42no—
83C2×C4×C251—44no—
84C2×C4×C251C2×D411yescharacteristic
85C2×C4×C251C2×D411yescharacteristic
86C2×C4×C251C2×D411yescharacteristic
87C2×C4×C251C2×D411yescharacteristic
88C2×C4×C251C2×D411yescharacteristic
89C2×C4×C251C2411yescharacteristic
90C22×C251—21no—
91C22×C251—21no—
92C22×C251—24no—
93C22×C251—48no—
94C22×C251—42no—
95C22×C251—42no—
96C22×C251#13712yes—
97C22×C251C22×D411yescharacteristic
98C4×C251—21no—
99C4×C251—21no—
100C4×C251—21no—
101C4×C251—21no—
102C4×C251—42no—
103C4×C251AGL(1,ℤ/8)12yes—
104C4×C251C2×#4112yes—
105Dic251—42no—
106Dic251—42no—
107C2×C251—22no—
108C2×C251—44no—
109C2×C251#52511yescharacteristic
110C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
111C2×C251C2×#13711yescharacteristic
112C251#311011yescharacteristic, complemented, Sylow
113#3110—2511nomaximal, Sylow
114#533—2511no—
115C2×C4×D4D25111yescharacteristic
116#484—2512no—
117#474—2512no—
118#461—2512no—
119#450—2512no—
120#440—2514no—
121#432—2511no—
122C22×D4C2×D25111yescharacteristic
123C23×C4—21no—
124C2×SD16—2514no—
125C4⋊Q8—2511no—
126C4⋊Q8—2511no—
127#125—5021no—
128C22⋊Q8—5021no—
129C22⋊Q8—2512no—
130C4×Q8—5021no—
131C4×Q8—2512no—
132C4×D4—21no—
133C4×D4—21no—
134C4×D4C2×D25112yes—
135C4×D4C2×D25112yes—
136#117—5021no—
137C2×C4⋊C4C2×D25111yescharacteristic
138C2×C22⋊C4—21no—
139C2×C42C2×D25111yescharacteristic
140#107—2514no—
141C4⋊C8—2512no—
142#103—2514no—
143#102—2514no—
144C22⋊C8—2512no—
145#97—2512no—
146C24—21no—
147C2×Q8—5021no—
148C2×Q8—2512no—
149C2×D4—24no—
150C2×D4—21no—
151C2×D4C22×D25112yes—
152C22×C4—21no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—44no—
156C22×C4C22⋊D25111yescharacteristic
157C22×C4C22⋊D25111yescharacteristic
158C22×C4C22×D25111yescharacteristic
159SD16—5028no—
160C2×C8—2514no—
161C4⋊C4—21no—
162C4⋊C4—5021no—
163C4⋊C4—5021no—
164C4⋊C4—5022no—
165C4⋊C4—5022no—
166C4⋊C4—5021no—
167C4⋊C4—2512no—
168C4⋊C4C22×D25112yes—
169C22⋊C4—22no—
170C22⋊C4—22no—
171C22⋊C4—5022no—
172C42—5022no—
173C42C22⋊D25111yescharacteristic
174C42C22⋊D25111yescharacteristic
175C42C22×D25111yescharacteristic
176C42C22×D25111yescharacteristic
177C23—24no—
178C23—22no—
179C23—42no—
180C23—42no—
181C23C2×C22⋊D25111yescharacteristic
182Q8—10042no—
183Q8—5024no—
184D4—24no—
185D4—22no—
186D4#1686081014yes—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—44no—
192C2×C4—42no—
193C2×C4—44no—
194C2×C4—5022no—
195C2×C4—5022no—
196C2×C4C2×C22⋊D25111yescharacteristic
197C2×C4C2×C22⋊D25111yescharacteristic
198C2×C4C2×C22⋊D25111yescharacteristic
199C2×C4C2×C22⋊D25111yescharacteristic
200C2×C4C2×C22⋊D25111yescharacteristic
201C2×C4C23×D25111yescharacteristic
202C8—5024no—
203C22—21no—
204C22—21no—
205C22—24no—
206C22—48no—
207C22—42no—
208C22—42no—
209C22#2254874512yes—
210C22C22×C22⋊D25111yescharacteristic
211C4—21no—
212C4—21no—
213C4—21no—
214C4—21no—
215C4—42no—
216C4—10042no—
217C4—10042no—
218C4#2254874112yes—
219C4C2×#1686081012yes—
220C2—22no—
221C2—44no—
222C2#3063219811yescharacteristic
223C2C2×#2254874111yescharacteristic
224C2C2×#2254874511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC2×C4×D4×C251C2
socleC22×C251C22×D4
prime radicalC22×D4×C251C22
hypercenterC2×C4×D4D251
hypocenterC251#3110
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. #30632244C2semidirect
2. #30632022C2semidirect
3. #30631967C2semidirect
4. #30631892C2semidirect
6. #30631787C2semidirect
7. #30631744C2semidirect
10. #22548715C22semidirect
11. #22548715C22semidirect
16. C4×Dic502C22semidirect
18. #22548648C22semidirect
19. C4⋊(C251⋊C8)C22semidirect
20. #22548638C22semidirect
23. #22548626C22semidirect
112. C251#3110semidirect

Maximal subgroups

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

subgroupindex
#306322442
#306320222
#306319672
#306318922
#306318382
#306317872
#306317442
C2×C4×D4×C2512
#3110251