Finite Group Databasebeta

← #44549566

Subgroups — #44549566

Order 32128.

250 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 55 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#30632357C211yesmaximal, characteristic
2#30632033C211yesmaximal, characteristic, complemented
3#30631973C211yesmaximal, characteristic, complemented
4#30631893C211yesmaximal, characteristic, complemented
5#30631417C211yesmaximal, characteristic, complemented
6#30631414C211yesmaximal, characteristic, complemented
7#331×C251C211yesmaximal, characteristic, complemented
8C2×#16860809—21no—
9C2×#16860799—21no—
10#22548715C2211yescharacteristic
11#22548711—21no—
12#22548687—22no—
13#22548686—21no—
14C4×Dic502C2211yescharacteristic
15C4×D1004C2211yescharacteristic
16D4×Dic251—21no—
17#22548647—21no—
18#22548640—21no—
19#22548640—21no—
20#22548636—22no—
21C4×C251⋊C8C2211yescharacteristic
22#127×C251C2211yescharacteristic
23C4⋊C8×C251C2211yescharacteristic
24C4⋊C8×C251C2211yescharacteristic
25#16860809—42no—
26#16860809—42no—
27C2×Dic502—21no—
28C2×Dic502D411yescharacteristic
29C2×D1004D411yescharacteristic
30C2×C22⋊D251—22no—
31C2×C22⋊D251—22no—
32C22×Dic251—22no—
33C2×C4×D251—21no—
34#16860799—42no—
35C2×C251⋊C8—21no—
36C4⋊Dic251—21no—
37C4⋊Dic251—21no—
38C4⋊Dic251D411yescharacteristic
39C4⋊Dic251D411yescharacteristic
40Dic251⋊C4—21no—
41C22⋊Dic251—22no—
42(C2×D251)⋊C4—21no—
43C4×Dic251—21no—
44C2×D4×C251—21no—
45C2×D4×C251—21no—
46C2×D4×C251D412yes—
47C2×C8×C251—21no—
48C2×C8×C251—21no—
49C42×C251C2311yescharacteristic
50C22×D251—21no—
51Dic502—21no—
52Dic502—41no—
53Dic502—41no—
54Dic502SD1612yescomplemented
55D1004—21no—
56D1004—21no—
57C22⋊D251—44no—
58C22⋊D251—82no—
59C2×Dic251—21no—
60C2×Dic251—21no—
61C2×Dic251—21no—
62C2×Dic251—44no—
63C2×Dic251—41no—
64C4×D251—41no—
65C4×D251—41no—
66C251⋊C8—41no—
67C23×C251—22no—
68D4×C251—42no—
69D4×C251—41no—
70D4×C251—41no—
71D4×C251—41no—
72D4×C251—41no—
73C2×C4×C251C2×D411yescharacteristic
74C2×C4×C251C2×D411yescharacteristic
75C2×C4×C251C2×D411yescharacteristic
76C8×C251—41no—
77C8×C251—41no—
78C22×C251—42no—
79C22×C251—42no—
80C22×C251—42no—
81C22×C251#12011yescharacteristic
82C2×D251—41no—
83C2×D251—41no—
84C2×D251—41no—
85C4×C251—21no—
86C4×C251C2×SD1611yescharacteristic
87C4×C251#13511yescharacteristic
88C4×C251AGL(1,ℤ/8)11yescharacteristic
89C4×C251AGL(1,ℤ/8)11yescharacteristic
90Dic251—22no—
91Dic251—41no—
92Dic251—41no—
93Dic251—81no—
94C2×C251—82no—
95C2×C251#44911yescharacteristic
96C2×C251#45011yescharacteristic
97C2×C251#45511yescharacteristic
98D251—81no—
99C251#164211yescharacteristic, complemented, Sylow
100#1642—2511nomaximal, Sylow
101#550—2511no—
102#486—2511no—
103#475—2511no—
104#461—2511no—
105#331D25111yescharacteristic, complemented
106Q8⋊C8—2511no—
107D4⋊C8—2511no—
108C2×#41—5021no—
109C2×SD16—5021no—
110C4⋊Q8—2511no—
111#127—5021no—
112#127C2×D25111yescharacteristic
113#121—5022no—
114#121—5021no—
115C4×Q8—2511no—
116C4×D4—5021no—
117C4×D4—2511no—
118#106—5021no—
119C4⋊C8C2×D25111yescharacteristic
120C4⋊C8C2×D25111yescharacteristic
121#103—5021no—
122#103—5021no—
123#102—5022no—
124C4×C8—2511no—
125#41—10042no—
126#41—10042no—
127C2×Q8—5021no—
128C2×Q8—2511no—
129C2×D4—21no—
130C2×D4—21no—
131C2×D4—5022no—
132C2×D4—5022no—
133C2×D4—2511no—
134C2×D4D100412yes—
135C22×C4—5022no—
136C22×C4—5021no—
137SD16—10042no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—5021no—
141C4⋊C4—5021no—
142C4⋊C4—5021no—
143C4⋊C4—5021no—
144C4⋊C4—2511no—
145C4⋊C4—2511no—
146C22⋊C4—5022no—
147C22⋊C4—5021no—
148C42—5021no—
149C42C22×D25111yescharacteristic
150C23—22no—
151C23—5021no—
152Q8—10041no—
153Q8—10041no—
154Q8—5021no—
155Q8—2512no—
156D4—42no—
157D4—41no—
158D4—41no—
159D4—41no—
160D4—41no—
161D4—20082no—
162D4—10044no—
163D4—5021no—
164D4—5021no—
165C2×C4—10041no—
166C2×C4—10044no—
167C2×C4—10041no—
168C2×C4—10041no—
169C2×C4—5021no—
170C2×C4—5021no—
171C2×C4—5021no—
172C2×C4D4×D25111yescharacteristic
173C2×C4D4×D25111yescharacteristic
174C2×C4C2×D100411yescharacteristic
175C8—41no—
176C8—41no—
177C8—10041no—
178C22—42no—
179C22—42no—
180C22—42no—
181C22—10041no—
182C22—10041no—
183C22—10041no—
184C22#2254868511yescharacteristic
185C4—21no—
186C4—20081no—
187C4—10041no—
188C4—10041no—
189C4—5022no—
190C4SD16×D25111yescharacteristic
191C4#2254873611yescharacteristic
192C4#2254873911yescharacteristic
193C4#2254873911yescharacteristic
194C2—82no—
195C2—20081no—
196C2#3063183311yescharacteristic
197C2#3063183911yescharacteristic
198C2#3063186711yescharacteristic
subgroupG/H
centerC22#22548685
commutator, cosocleC42×C251C23
FrattiniC42C22×D251
Fitting#331×C251C2
socleC22×C251#120
prime radical#30632357C2
hypercenter#331D251
hypocenterC251#1642
LCSG ▷ C42×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #30632033C2semidirect
3. #30631973C2semidirect
4. #30631893C2semidirect
5. #30631417C2semidirect
6. #30631414C2semidirect
7. #331×C251C2semidirect
54. Dic502SD16semidirect
99. C251#1642semidirect
105. #331D251semidirect

Maximal subgroups

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

subgroupindex
#306323572
#306320332
#306319732
#306318932
#306314172
#306314142
#331×C2512
#1642251