Finite Group Databasebeta

← #44549568

Subgroups — #44549568

Order 32128.

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

normal subgroups only

This group has 250 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#30632034C211yesmaximal, characteristic, complemented
3#30631972C211yesmaximal, characteristic, complemented
4#30631893C211yesmaximal, characteristic, complemented
5#30631430C211yesmaximal, characteristic, complemented
6#30631416C211yesmaximal, characteristic, complemented
7Q8⋊C8×C251C211yesmaximal, characteristic, complemented
8C2×#16860811—21no—
9C2×#16860799—21no—
10#22548715C2211yescharacteristic
11#22548712—21no—
12#22548712C2211yescharacteristic
13#22548690—22no—
14#22548690—21no—
15C4×D1004—21no—
16C4×D1004C2211yescharacteristic
17#22548646—21no—
18C4⋊(C251⋊C8)C2211yescharacteristic
19#22548640—21no—
20#22548639—21no—
21#22548637—22no—
22C4×Q8×C251C2211yescharacteristic
23C4⋊C8×C251C2211yescharacteristic
24C4×C8×C251C2211yescharacteristic
25#16860811—42no—
26#16860811—42no—
27C2×Dic502—21no—
28C2×D1004—22no—
29C2×D1004—22no—
30C2×D1004—21no—
31C2×D1004—21no—
32C2×D1004D411yescharacteristic
33C2×D1004D412yes—
34C2×C4×D251—22no—
35C2×C4×D251—21no—
36#16860799—42no—
37C2×C251⋊C8—21no—
38C4⋊Dic251—21no—
39C4⋊Dic251—21no—
40C4⋊Dic251D411yescharacteristic
41(C2×D251)⋊C4—22no—
42(C2×D251)⋊C4—21no—
43C2×Q8×C251D411yescharacteristic
44C2×C8×C251—21no—
45C2×C8×C251—21no—
46C4⋊C4×C251—21no—
47C4⋊C4×C251D411yescharacteristic
48C42×C251—21no—
49C42×C251C2311yescharacteristic
50C22×D251—21no—
51C22×D251—22no—
52Dic502—41no—
53Dic502—41no—
54D1004—21no—
55D1004—21no—
56D1004—42no—
57D1004—44no—
58D1004—41no—
59D1004—41no—
60D1004—41no—
61D1004—41no—
62D1004—82no—
63C2×Dic251—21no—
64C2×Dic251—41no—
65C4×D251—44no—
66C4×D251—41no—
67C4×D251—41no—
68C251⋊C8—41no—
69Q8×C251—21no—
70Q8×C251SD1612yescomplemented
71C2×C4×C251—21no—
72C2×C4×C251—21no—
73C2×C4×C251C2×D411yescharacteristic
74C2×C4×C251C2×D411yescharacteristic
75C2×C4×C251C2×D411yescharacteristic
76C8×C251—41no—
77C8×C251—41no—
78C22×C251#12011yescharacteristic
79C2×D251—41no—
80C2×D251—42no—
81C2×D251—42no—
82C2×D251—41no—
83C2×D251—42no—
84C2×D251—41no—
85C4×C251—22no—
86C4×C251—21no—
87C4×C251—41no—
88C4×C251C2×SD1611yescharacteristic
89C4×C251#13511yescharacteristic
90C4×C251AGL(1,ℤ/8)11yescharacteristic
91C4×C251AGL(1,ℤ/8)11yescharacteristic
92Dic251—41no—
93Dic251—81no—
94C2×C251#44911yescharacteristic
95C2×C251#45011yescharacteristic
96C2×C251#45511yescharacteristic
97D251—81no—
98D251—82no—
99C251#164211yescharacteristic, complemented, Sylow
100#1642—2511nomaximal, Sylow
101#550—2511no—
102#486—2511no—
103#475—2511no—
104#461—2511no—
105#331—2511no—
106Q8⋊C8D25111yescharacteristic, complemented
107D4⋊C8—2511no—
108C2×#41—5021no—
109C2×SD16—5021no—
110C4⋊Q8—2511no—
111#127—5021no—
112#127—2511no—
113#121—5022no—
114#121—5021no—
115C4×Q8C2×D25111yescharacteristic
116C4×D4—5021no—
117C4×D4—2511no—
118#106—5021no—
119C4⋊C8—2511no—
120C4⋊C8C2×D25111yescharacteristic
121#103—5021no—
122#103—5021no—
123#102—5022no—
124C4×C8C2×D25111yescharacteristic
125#41—10042no—
126#41—10042no—
127C2×Q8—5021no—
128C2×Q8D100411yescharacteristic
129C2×D4—5022no—
130C2×D4—5022no—
131C2×D4—5021no—
132C2×D4—5021no—
133C2×D4—2512no—
134C2×D4—2511no—
135C22×C4—5022no—
136C22×C4—5021no—
137SD16—10042no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—5021no—
141C4⋊C4—21no—
142C4⋊C4—5021no—
143C4⋊C4—5021no—
144C4⋊C4—2511no—
145C4⋊C4D100411yescharacteristic
146C22⋊C4—5022no—
147C22⋊C4—5021no—
148C42—21no—
149C42C22×D25111yescharacteristic
150C23—5022no—
151C23—5021no—
152Q8—21no—
153Q8—10041no—
154Q8—10041no—
155Q8#1686079912yescomplemented
156D4—20082no—
157D4—10042no—
158D4—10044no—
159D4—10041no—
160D4—10041no—
161D4—10041no—
162D4—10041no—
163D4—5021no—
164D4—5021no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—10044no—
168C2×C4—10041no—
169C2×C4—10041no—
170C2×C4—10041no—
171C2×C4—5021no—
172C2×C4D4×D25111yescharacteristic
173C2×C4D4×D25111yescharacteristic
174C2×C4C2×D100411yescharacteristic
175C8—41no—
176C8—41no—
177C8—10041no—
178C22—10042no—
179C22—10041no—
180C22—10042no—
181C22—10041no—
182C22—10041no—
183C22—10042no—
184C22#2254868511yescharacteristic
185C4—22no—
186C4—21no—
187C4—41no—
188C4—20081no—
189C4—10041no—
190C4C2×#1686079911yescharacteristic
191C4#2254873811yescharacteristic
192C4#2254874211yescharacteristic
193C4#2254874311yescharacteristic
194C2—20082no—
195C2—20081no—
196C2#3063183511yescharacteristic
197C2#3063184011yescharacteristic
198C2#3063186611yescharacteristic
subgroupG/H
centerC22#22548685
commutator, cosocleC42×C251C23
FrattiniC42C22×D251
FittingQ8⋊C8×C251C2
socleC22×C251#120
prime radical#30632359C2
hypercenterQ8⋊C8D251
hypocenterC251#1642
LCSG ▷ C42×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
2. #30632034C2semidirect
3. #30631972C2semidirect
4. #30631893C2semidirect
5. #30631430C2semidirect
6. #30631416C2semidirect
7. Q8⋊C8×C251C2semidirect
70. Q8×C251SD16semidirect
99. C251#1642semidirect
106. Q8⋊C8D251semidirect
155. Q8#16860799semidirect

Maximal subgroups

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

subgroupindex
#306323592
#306320342
#306319722
#306318932
#306314302
#306314162
Q8⋊C8×C2512
#1642251