Finite Group Databasebeta

← #44549490

Subgroups — #44549490

Order 32128.

256 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 57 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#30632359C211yesmaximal, characteristic
2#30632061C211yesmaximal, characteristic, complemented
3#30631974C211yesmaximal, characteristic, complemented
4#30631430C211yesmaximal, characteristic, complemented
5Q8⋊(C251⋊C8)C211yesmaximal, characteristic, complemented
6#30631414C211yesmaximal, characteristic, complemented
7#477×C251C211yesmaximal, characteristic, complemented
8C2×#16860811—21no—
9C2×#16860797—22no—
10#22548712—21no—
11#22548712C2211yescharacteristic
12#22548690—22no—
13#22548690—21no—
14C4×D1004—21no—
15C4×D1004C2211yescharacteristic
16#22548646—21no—
17C4⋊(C251⋊C8)C2211yescharacteristic
18#22548635—21no—
19C4×C251⋊C8C2211yescharacteristic
20C4⋊Q8×C251C2211yescharacteristic
21C4×Q8×C251C2211yescharacteristic
22#107×C251—21no—
23C4⋊C8×C251C2211yescharacteristic
24#103×C251—21no—
25#16860811—42no—
26#16860811—42no—
27C2×D1004—22no—
28C2×D1004—22no—
29C2×D1004—21no—
30C2×D1004—21no—
31C2×D1004D411yescharacteristic
32C2×D1004D412yes—
33C2×C4×D251—22no—
34C2×C4×D251—21no—
35#16860797—44no—
36C2×C251⋊C8—21no—
37C2×C251⋊C8—21no—
38C4⋊Dic251D411yescharacteristic
39(C2×D251)⋊C4—22no—
40(C2×D251)⋊C4—21no—
41C2×Q8×C251—21no—
42C2×Q8×C251D411yescharacteristic
43C2×C8×C251—21no—
44C4⋊C4×C251—21no—
45C4⋊C4×C251—21no—
46C4⋊C4×C251—21no—
47C4⋊C4×C251D411yescharacteristic
48C42×C251—21no—
49C42×C251C2311yescharacteristic
50C22×D251—21no—
51C22×D251—22no—
52D1004—21no—
53D1004—21no—
54D1004—42no—
55D1004—44no—
56D1004—41no—
57D1004—41no—
58D1004—41no—
59D1004—41no—
60D1004—82no—
61C2×Dic251—21no—
62C4×D251—44no—
63C4×D251—41no—
64C4×D251—41no—
65C251⋊C8—22no—
66C251⋊C8—41no—
67Q8×C251—21no—
68Q8×C251—42no—
69Q8×C251SD1612yescomplemented
70C2×C4×C251—21no—
71C2×C4×C251—21no—
72C2×C4×C251—41no—
73C2×C4×C251C2×D411yescharacteristic
74C2×C4×C251C2×D411yescharacteristic
75C2×C4×C251C2×D411yescharacteristic
76C8×C251—41no—
77C22×C251#12011yescharacteristic
78C2×D251—42no—
79C2×D251—41no—
80C2×D251—41no—
81C2×D251—42no—
82C2×D251—42no—
83C2×D251—41no—
84C4×C251—22no—
85C4×C251—21no—
86C4×C251—41no—
87C4×C251—81no—
88C4×C251C2×SD1611yescharacteristic
89C4×C251#13511yescharacteristic
90C4×C251AGL(1,ℤ/8)11yescharacteristic
91C4×C251#13711yescharacteristic
92Dic251—41no—
93C2×C251#45011yescharacteristic
94C2×C251#45211yescharacteristic
95C2×C251#45311yescharacteristic
96D251—81no—
97D251—82no—
98C251#163111yescharacteristic, complemented, Sylow
99#1631—2511nomaximal, Sylow
100#550—2511no—
101#492—2511no—
102#477D25111yescharacteristic, complemented
103#476—2511no—
104#331—2511no—
105Q8⋊C8—2511no—
106D4⋊C8—2511no—
107C2×#41—5021no—
108C2×SD16—5022no—
109C4⋊Q8C2×D25111yescharacteristic
110#127—5021no—
111#127—2511no—
112#121—5022no—
113#121—5021no—
114C4×Q8C2×D25111yescharacteristic
115C4×D4—5021no—
116C4×D4—2511no—
117#107—21no—
118#106—5021no—
119C4⋊C8—2511no—
120C4⋊C8C2×D25111yescharacteristic
121#103—21no—
122#102—5021no—
123C4×C8—2511no—
124#41—10042no—
125#41—10042no—
126C2×Q8—21no—
127C2×Q8C22⋊D25111yescharacteristic
128C2×D4—5022no—
129C2×D4—5022no—
130C2×D4—5021no—
131C2×D4—5021no—
132C2×D4—2512no—
133C2×D4—2511no—
134C22×C4—5022no—
135C22×C4—5021no—
136SD16—10044no—
137C2×C8—21no—
138C2×C8—5021no—
139C2×C8—5021no—
140C4⋊C4—21no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—2511no—
144C4⋊C4C22⋊D25111yescharacteristic
145C22⋊C4—5022no—
146C22⋊C4—5021no—
147C42—21no—
148C42C22×D25111yescharacteristic
149C23—5022no—
150C23—5021no—
151Q8—21no—
152Q8—42no—
153Q8#1686079712yescomplemented
154D4—20082no—
155D4—10042no—
156D4—10044no—
157D4—10041no—
158D4—10041no—
159D4—10041no—
160D4—10041no—
161D4—5021no—
162D4—5021no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—41no—
166C2×C4—10044no—
167C2×C4—10041no—
168C2×C4—10041no—
169C2×C4—5021no—
170C2×C4D4×D25111yescharacteristic
171C2×C4D4×D25111yescharacteristic
172C2×C4C2×C22⋊D25111yescharacteristic
173C8—41no—
174C8—10041no—
175C8—5022no—
176C22—10042no—
177C22—10041no—
178C22—10042no—
179C22—10041no—
180C22—10041no—
181C22—10042no—
182C22#2254868311yescharacteristic
183C4—22no—
184C4—21no—
185C4—41no—
186C4—81no—
187C4—10041no—
188C4C2×#1686079711yescharacteristic
189C4#2254873811yescharacteristic
190C4#2254874011yescharacteristic
191C4#2254874711yescharacteristic
192C2—20082no—
193C2—20081no—
194C2#3063183711yescharacteristic
195C2#3063184811yescharacteristic
196C2#3063185311yescharacteristic
subgroupG/H
centerC22#22548683
commutator, cosocleC42×C251C23
FrattiniC42C22×D251
Fitting#477×C251C2
socleC22×C251#120
prime radical#30632359C2
hypercenter#477D251
hypocenterC251#1631
LCSG ▷ C42×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
2. #30632061C2semidirect
3. #30631974C2semidirect
4. #30631430C2semidirect
5. Q8⋊(C251⋊C8)C2semidirect
6. #30631414C2semidirect
7. #477×C251C2semidirect
69. Q8×C251SD16semidirect
98. C251#1631semidirect
102. #477D251semidirect
153. Q8#16860797semidirect

Maximal subgroups

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

subgroupindex
#306323592
#306320612
#306319742
#306314302
Q8⋊(C251⋊C8)2
#306314142
#477×C2512
#1631251