Finite Group Databasebeta

← #44560495

Subgroups — #44560495

Order 32128.

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

normal subgroups only

This group has 380 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
2#30632017C212yesmaximal
3#30631972C212yesmaximal, complemented
4#30631897C212yesmaximal, complemented
5#30631843C212yesmaximal
6#30631792C214yesmaximal, complemented
7#30631744C211yesmaximal, characteristic
8C2×C4×Q8×C251C211yesmaximal, characteristic
9C2×#16860800C2214yes—
10#22548715C2211yescharacteristic
11#22548715C2211yescharacteristic
12#22548707—21no—
13#22548694—21no—
14#22548694C2212yes—
15C4×Dic502—21no—
16C4×Dic502C2212yes—
17#22548674—21no—
18#22548646C2214yes—
19C4⋊(C251⋊C8)C2212yes—
20#22548639C2214yes—
21#22548638C2214yes—
22C22⋊(C251⋊C8)C2212yes—
23#22548626C2212yes—
24C22×Q8×C251C2211yescharacteristic
25C4×Q8×C251—21no—
26C4×Q8×C251—21no—
27C4×Q8×C251C2212yes—
28C4×Q8×C251C2212yes—
29C2×C4⋊C4×C251—21no—
30C2×C4⋊C4×C251C2211yescharacteristic
31C2×C42×C251—21no—
32C2×C42×C251C2211yescharacteristic
33C2×Dic502—21no—
34C2×Dic502C2312yes—
35#16860800—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—
45C2×Q8×C251—24no—
46C2×Q8×C251—21no—
47C2×Q8×C251C2312yes—
48C22×C4×C251—21no—
49C22×C4×C251—21no—
50C22×C4×C251D411yescharacteristic
51C22×C4×C251D411yescharacteristic
52C22×C4×C251C2311yescharacteristic
53C4⋊C4×C251—22no—
54C4⋊C4×C251—22no—
55C4⋊C4×C251—21no—
56C4⋊C4×C251C2312yes—
57C42×C251—22no—
58C42×C251—22no—
59C42×C251D411yescharacteristic
60C42×C251D411yescharacteristic
61C42×C251C2311yescharacteristic
62C42×C251C2311yescharacteristic
63Dic502—24no—
64Dic502—42no—
65C2×Dic251—22no—
66C2×Dic251—22no—
67C251⋊C8—24no—
68C23×C251C2×D411yescharacteristic
69Q8×C251—24no—
70Q8×C251—22no—
71Q8×C251#4114yes—
72C2×C4×C251—24no—
73C2×C4×C251—22no—
74C2×C4×C251—22no—
75C2×C4×C251—22no—
76C2×C4×C251—22no—
77C2×C4×C251—22no—
78C2×C4×C251—42no—
79C2×C4×C251C2×D411yescharacteristic
80C2×C4×C251C2×D411yescharacteristic
81C2×C4×C251C2×D411yescharacteristic
82C2×C4×C251C2×D411yescharacteristic
83C2×C4×C251C2×D411yescharacteristic
84C2×C4×C251C2411yescharacteristic
85C22×C251—21no—
86C22×C251—21no—
87C22×C251#13712yes—
88C22×C251C22×D411yescharacteristic
89C4×C251—24no—
90C4×C251—21no—
91C4×C251—21no—
92C4×C251—21no—
93C4×C251—21no—
94C4×C251—42no—
95C4×C251#13712yes—
96C4×C251C2×#4112yes—
97Dic251—42no—
98Dic251—42no—
99C2×C251—22no—
100C2×C251#52511yescharacteristic
101C2×C251C2×#13711yescharacteristic
102C2×C251C2×#13711yescharacteristic
103C251#311311yescharacteristic, complemented, Sylow
104#3113—2511nomaximal, Sylow
105#533—2511no—
106C2×C4×Q8D25111yescharacteristic
107#483—2512no—
108#475—2512no—
109#462—2512no—
110#451—2512no—
111#441—2514no—
112#432—2511no—
113C22×Q8C2×D25111yescharacteristic
114C2×Q16—2514no—
115C4⋊Q8—2511no—
116C4⋊Q8—2511no—
117#125—5021no—
118C22⋊Q8—5021no—
119C22⋊Q8—2512no—
120C4×Q8—21no—
121C4×Q8—21no—
122C4×Q8—5021no—
123C4×Q8—2512no—
124C4×Q8C2×D25112yes—
125C4×Q8C2×D25112yes—
126#117—5021no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4C2×D25111yescharacteristic
129C2×C42—21no—
130C2×C42C2×D25111yescharacteristic
131#106—2514no—
132C4⋊C8—2512no—
133#103—2514no—
134#103—2514no—
135C22⋊C8—2512no—
136#97—2512no—
137C2×Q8—24no—
138C2×Q8—21no—
139C2×Q8—5021no—
140C2×Q8—2512no—
141C2×Q8C22×D25112yes—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4C22⋊D25111yescharacteristic
145C22×C4C22⋊D25111yescharacteristic
146C22×C4C22×D25111yescharacteristic
147Q16—5028no—
148C2×C8—2514no—
149C4⋊C4—22no—
150C4⋊C4—22no—
151C4⋊C4—21no—
152C4⋊C4—5021no—
153C4⋊C4—5021no—
154C4⋊C4—5022no—
155C4⋊C4—5022no—
156C4⋊C4—5021no—
157C4⋊C4—2512no—
158C4⋊C4C22×D25112yes—
159C22⋊C4—5022no—
160C42—22no—
161C42—22no—
162C42—5022no—
163C42C22⋊D25111yescharacteristic
164C42C22⋊D25111yescharacteristic
165C42C22×D25111yescharacteristic
166C42C22×D25111yescharacteristic
167C23C2×C22⋊D25111yescharacteristic
168Q8—24no—
169Q8—22no—
170Q8—10042no—
171Q8—5024no—
172Q8#1686081014yes—
173C2×C4—24no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—42no—
180C2×C4—5022no—
181C2×C4—5022no—
182C2×C4C2×C22⋊D25111yescharacteristic
183C2×C4C2×C22⋊D25111yescharacteristic
184C2×C4C2×C22⋊D25111yescharacteristic
185C2×C4C2×C22⋊D25111yescharacteristic
186C2×C4C2×C22⋊D25111yescharacteristic
187C2×C4C23×D25111yescharacteristic
188C8—5024no—
189C22—21no—
190C22—21no—
191C22#2254874512yes—
192C22C22×C22⋊D25111yescharacteristic
193C4—24no—
194C4—21no—
195C4—21no—
196C4—21no—
197C4—21no—
198C4—42no—
199C4—10042no—
200C4—10042no—
201C4#2254874612yes—
202C4C2×#1686081012yes—
203C2—22no—
204C2#3063219811yescharacteristic
205C2C2×#2254874511yescharacteristic
206C2C2×#2254874611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC2×C4×Q8×C251C2
socleC22×C251C22×D4
prime radicalC23×C251C2×D4
hypercenterC2×C4×Q8D251
hypocenterC251#3113
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
3. #30631972C2semidirect
4. #30631897C2semidirect
6. #30631792C2semidirect
103. C251#3113semidirect

Maximal subgroups

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

subgroupindex
#306322442
#306320172
#306319722
#306318972
#306318432
#306317922
#306317442
C2×C4×Q8×C2512
#3113251