Finite Group Databasebeta

← #44549492

Subgroups — #44549492

Order 32128.

256 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 61 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#30632059C211yesmaximal, characteristic, complemented
2#30631982C211yesmaximal, characteristic, complemented
3#30631977C211yesmaximal, characteristic, complemented
4#30631431C211yesmaximal, characteristic, complemented
5Q8⋊(C251⋊C8)C211yesmaximal, characteristic, complemented
6D4⋊(C251⋊C8)C211yesmaximal, characteristic, complemented
7#550×C251C211yesmaximal, characteristic
8C2×#16860798—22no—
9#22548715C2211yescharacteristic
10#22548648—21no—
11#22548646—21no—
12C4⋊(C251⋊C8)C2211yescharacteristic
13C4⋊(C251⋊C8)C2211yescharacteristic
14#22548639—21no—
15#22548636—21no—
16C4×C251⋊C8C2211yescharacteristic
17C2×#41×C251—21no—
18#127×C251—21no—
19#127×C251C2211yescharacteristic
20#121×C251—22no—
21#121×C251—21no—
22C4×Q8×C251C2211yescharacteristic
23C4×D4×C251—21no—
24C4×D4×C251C2211yescharacteristic
25C2×Dic502—21no—
26#16860798—44no—
27C2×C251⋊C8—21no—
28C2×C251⋊C8—21no—
29C2×C251⋊C8—21no—
30C4⋊Dic251—21no—
31C4⋊Dic251—21no—
32#41×C251—42no—
33#41×C251—42no—
34C2×Q8×C251D411yescharacteristic
35C2×D4×C251—22no—
36C2×D4×C251—22no—
37C2×D4×C251—21no—
38C2×D4×C251—21no—
39C2×D4×C251D411yescharacteristic
40C2×D4×C251D412yes—
41C22×C4×C251—22no—
42C22×C4×C251—21no—
43C4⋊C4×C251—21no—
44C4⋊C4×C251D411yescharacteristic
45C4⋊C4×C251D411yescharacteristic
46C22⋊C4×C251—22no—
47C22⋊C4×C251—21no—
48C42×C251—21no—
49C42×C251C2311yescharacteristic
50Dic502—42no—
51C2×Dic251—41no—
52C251⋊C8—22no—
53C251⋊C8—41no—
54C251⋊C8—41no—
55C23×C251—21no—
56C23×C251—22no—
57Q8×C251—21no—
58Q8×C251SD1612yescomplemented
59D4×C251—21no—
60D4×C251—21no—
61D4×C251—42no—
62D4×C251—44no—
63D4×C251—41no—
64D4×C251—41no—
65D4×C251—41no—
66D4×C251—41no—
67D4×C251—82no—
68C2×C4×C251—21no—
69C2×C4×C251—21no—
70C2×C4×C251—21no—
71C2×C4×C251—44no—
72C2×C4×C251—41no—
73C2×C4×C251—41no—
74C2×C4×C251C2×D411yescharacteristic
75C2×C4×C251C2×D411yescharacteristic
76C2×C4×C251C2×D411yescharacteristic
77C22×C251—42no—
78C22×C251—41no—
79C22×C251—41no—
80C22×C251—42no—
81C22×C251—42no—
82C22×C251—41no—
83C22×C251#12011yescharacteristic
84C4×C251—22no—
85C4×C251—21no—
86C4×C251—41no—
87C4×C251—41no—
88C4×C251C2×SD1611yescharacteristic
89C4×C251#13511yescharacteristic
90C4×C251AGL(1,ℤ/8)11yescharacteristic
91C4×C251#13711yescharacteristic
92Dic251—81no—
93C2×C251—81no—
94C2×C251—82no—
95C2×C251#45011yescharacteristic
96C2×C251#45211yescharacteristic
97C2×C251#45311yescharacteristic
98C251#163111yescharacteristic, complemented, Sylow
99#1631—2511nomaximal, Sylow
100#550D25111yescharacteristic
101#492—2511no—
102#477—2511no—
103#476—2511no—
104#331—2511no—
105Q8⋊C8—2511no—
106D4⋊C8—2511no—
107C2×#41—21no—
108C2×SD16—5022no—
109C4⋊Q8—2511no—
110#127—21no—
111#127C2×D25111yescharacteristic
112#121—22no—
113#121—21no—
114C4×Q8C2×D25111yescharacteristic
115C4×D4—21no—
116C4×D4C2×D25111yescharacteristic
117#107—5021no—
118#106—5021no—
119C4⋊C8—2511no—
120C4⋊C8—2511no—
121#103—5021no—
122#102—5021no—
123C4×C8—2511no—
124#41—42no—
125#41—42no—
126C2×Q8—5021no—
127C2×Q8C22⋊D25111yescharacteristic
128C2×D4—22no—
129C2×D4—22no—
130C2×D4—21no—
131C2×D4—21no—
132C2×D4C22⋊D25111yescharacteristic
133C2×D4C22⋊D25112yes—
134C22×C4—22no—
135C22×C4—21no—
136SD16—10044no—
137C2×C8—5021no—
138C2×C8—5021no—
139C2×C8—5021no—
140C4⋊C4—21no—
141C4⋊C4—5021no—
142C4⋊C4—5021no—
143C4⋊C4C22⋊D25111yescharacteristic
144C4⋊C4C22⋊D25111yescharacteristic
145C22⋊C4—22no—
146C22⋊C4—21no—
147C42—21no—
148C42C22×D25111yescharacteristic
149C23—21no—
150C23—22no—
151Q8—21no—
152Q8—10042no—
153Q8#1686079812yescomplemented
154D4—21no—
155D4—21no—
156D4—42no—
157D4—44no—
158D4—41no—
159D4—41no—
160D4—41no—
161D4—41no—
162D4—82no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—44no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—10041no—
170C2×C4C2×C22⋊D25111yescharacteristic
171C2×C4C2×C22⋊D25111yescharacteristic
172C2×C4C2×C22⋊D25111yescharacteristic
173C8—10041no—
174C8—10041no—
175C8—5022no—
176C22—42no—
177C22—41no—
178C22—41no—
179C22—42no—
180C22—42no—
181C22—41no—
182C22#2254868411yescharacteristic
183C4—22no—
184C4—21no—
185C4—41no—
186C4—41no—
187C4—20081no—
188C4C2×#1686079811yescharacteristic
189C4#2254873511yescharacteristic
190C4#2254874111yescharacteristic
191C4#2254874511yescharacteristic
192C2—81no—
193C2—82no—
194C2#3063183811yescharacteristic
195C2#3063184911yescharacteristic
196C2#3063185511yescharacteristic
subgroupG/H
centerC22#22548684
commutator, cosocleC42×C251C23
FrattiniC42C22×D251
Fitting, prime radical#550×C251C2
socleC22×C251#120
hypercenter#550D251
hypocenterC251#1631
LCSG ▷ C42×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #550

Complements

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

NH ≅ G/Ntype
1. #30632059C2semidirect
2. #30631982C2semidirect
3. #30631977C2semidirect
4. #30631431C2semidirect
5. Q8⋊(C251⋊C8)C2semidirect
6. D4⋊(C251⋊C8)C2semidirect
58. Q8×C251SD16semidirect
98. C251#1631semidirect
153. Q8#16860798semidirect

Maximal subgroups

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

subgroupindex
#306320592
#306319822
#306319772
#306314312
Q8⋊(C251⋊C8)2
D4⋊(C251⋊C8)2
#550×C2512
#1631251