Finite Group Databasebeta

← #44549447

Subgroups — #44549447

Order 32128.

240 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 53 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#30632402C211yesmaximal, characteristic, complemented
2#30632059C211yesmaximal, characteristic, complemented
3#30631879C211yesmaximal, characteristic, complemented
4#30631435C211yesmaximal, characteristic, complemented
5D4⋊(C251⋊C8)C211yesmaximal, characteristic, complemented
6#30631414C211yesmaximal, characteristic, complemented
7#459×C251C211yesmaximal, characteristic, complemented
8C2×#16860798—22no—
9C2×#16860795—21no—
10#22548715C2211yescharacteristic
11#22548701—22no—
12#22548689—21no—
13#22548686—21no—
14C4×D1004C2211yescharacteristic
15#22548674—21no—
16#22548673—21no—
17C4⋊(C251⋊C8)C2211yescharacteristic
18#22548636—21no—
19C4×C251⋊C8C2211yescharacteristic
20C2×D8×C251—21no—
21#127×C251C2211yescharacteristic
22C4×D4×C251C2211yescharacteristic
23C4⋊C8×C251C2211yescharacteristic
24#102×C251—21no—
25C2×Dic502D412yes—
26C2×D1004D411yescharacteristic
27C2×C22⋊D251—22no—
28C2×C4×D251—21no—
29#16860798—44no—
30#16860795—42no—
31C2×C251⋊C8—21no—
32C2×C251⋊C8—21no—
33C4⋊Dic251—21no—
34C4⋊Dic251—21no—
35C4⋊Dic251D411yescharacteristic
36C22⋊Dic251—22no—
37(C2×D251)⋊C4—22no—
38(C2×D251)⋊C4—21no—
39C4×Dic251—22no—
40C2×D4×C251—21no—
41C2×D4×C251—21no—
42C2×D4×C251—21no—
43C2×D4×C251D411yescharacteristic
44C22×C4×C251—21no—
45D8×C251—42no—
46C2×C8×C251—21no—
47C4⋊C4×C251D411yescharacteristic
48C22⋊C4×C251—21no—
49C42×C251C2311yescharacteristic
50C22×D251—21no—
51Dic502—42no—
52D1004—21no—
53D1004—21no—
54C22⋊D251—82no—
55C2×Dic251—21no—
56C2×Dic251—22no—
57C4×D251—41no—
58C4×D251—41no—
59C251⋊C8—22no—
60C251⋊C8—41no—
61C23×C251—21no—
62C23×C251—41no—
63D4×C251—21no—
64D4×C251—21no—
65D4×C251—42no—
66D4×C251—41no—
67D4×C251—41no—
68D4×C251—41no—
69D4×C251—41no—
70C2×C4×C251—21no—
71C2×C4×C251—41no—
72C2×C4×C251—41no—
73C2×C4×C251C2×D411yescharacteristic
74C2×C4×C251C2×D411yescharacteristic
75C2×C4×C251C2×D411yescharacteristic
76C8×C251—41no—
77C22×C251—41no—
78C22×C251—41no—
79C22×C251—41no—
80C22×C251—81no—
81C22×C251—81no—
82C22×C251—81no—
83C22×C251#12011yescharacteristic
84C2×D251—41no—
85C2×D251—41no—
86C2×D251—41no—
87C4×C251—21no—
88C4×C251—41no—
89C4×C251#13511yescharacteristic
90C4×C251#13511yescharacteristic
91C4×C251AGL(1,ℤ/8)11yescharacteristic
92C4×C251AGL(1,ℤ/8)11yescharacteristic
93Dic251—41no—
94Dic251—42no—
95C2×C251—81no—
96C2×C251—161no—
97C2×C251#44911yescharacteristic
98C2×C251#44911yescharacteristic
99C2×C251#45311yescharacteristic
100D251—81no—
101C251#162411yescharacteristic, complemented, Sylow
102#1624—2511nomaximal, Sylow
103#555—2511no—
104#492—2511no—
105#459—2511no—
106#459D25111yescharacteristic, complemented
107#333—2511no—
108D4⋊C8—2511no—
109D4⋊C8—2511no—
110C2×SD16—5022no—
111C2×D8—21no—
112C2×D8—5021no—
113C4⋊Q8—2511no—
114#127C2×D25111yescharacteristic
115#124—5022no—
116#121—5021no—
117#121—5021no—
118C4×D4—2511no—
119C4×D4C2×D25111yescharacteristic
120#117—5021no—
121#117—5021no—
122C4⋊C8—2511no—
123C4⋊C8C2×D25111yescharacteristic
124#102—21no—
125#102—5021no—
126C4×C8—2511no—
127C2×Q8—2512no—
128C2×D4—21no—
129C2×D4—21no—
130C2×D4—21no—
131C2×D4—5022no—
132C2×D4—2511no—
133C2×D4C22⋊D25111yescharacteristic
134C22×C4—21no—
135C22×C4—5021no—
136SD16—10044no—
137D8—42no—
138D8—10042no—
139C2×C8—21no—
140C2×C8—5021no—
141C2×C8—5021no—
142C4⋊C4—5021no—
143C4⋊C4—5021no—
144C4⋊C4—2511no—
145C4⋊C4C22⋊D25111yescharacteristic
146C22⋊C4—21no—
147C22⋊C4—5022no—
148C22⋊C4—5022no—
149C22⋊C4—5021no—
150C42—5022no—
151C42C22×D25111yescharacteristic
152C23—21no—
153C23—41no—
154C23—5021no—
155Q8—10042no—
156D4—21no—
157D4—21no—
158D4—42no—
159D4—41no—
160D4—41no—
161D4—41no—
162D4—41no—
163D4—20082no—
164D4—5021no—
165D4—5021no—
166C2×C4—21no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—10041no—
170C2×C4—10041no—
171C2×C4—5021no—
172C2×C4—5022no—
173C2×C4D4×D25111yescharacteristic
174C2×C4D4×D25111yescharacteristic
175C2×C4C2×C22⋊D25111yescharacteristic
176C8—41no—
177C8—10041no—
178C8—5022no—
179C22—41no—
180C22—41no—
181C22—41no—
182C22—81no—
183C22—81no—
184C22—81no—
185C22—10041no—
186C22—10041no—
187C22—10041no—
188C22#2254868311yescharacteristic
189C4—21no—
190C4—41no—
191C4—10041no—
192C4—10042no—
193C4#2254873411yescharacteristic
194C4#2254873511yescharacteristic
195C4#2254873911yescharacteristic
196C4#2254874111yescharacteristic
197C2—81no—
198C2—161no—
199C2—20081no—
200C2#3063182911yescharacteristic
201C2#3063183011yescharacteristic
202C2#3063185311yescharacteristic
subgroupG/H
centerC22#22548683
commutator, cosocleC42×C251C23
FrattiniC42C22×D251
Fitting#459×C251C2
socleC22×C251#120
hypercenter#459D251
hypocenterC251#1624
LCSG ▷ C42×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #30632402C2semidirect
2. #30632059C2semidirect
3. #30631879C2semidirect
4. #30631435C2semidirect
5. D4⋊(C251⋊C8)C2semidirect
6. #30631414C2semidirect
7. #459×C251C2semidirect
101. C251#1624semidirect
106. #459D251semidirect

Maximal subgroups

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

subgroupindex
#306324022
#306320592
#306318792
#306314352
D4⋊(C251⋊C8)2
#306314142
#459×C2512
#1624251