Finite Group Databasebeta

← #45663520

Subgroups — #45663520

Order 32896.

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

normal subgroups only

This group has 250 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#30657083C211yesmaximal, characteristic, complemented
2#30657022C211yesmaximal, characteristic, complemented
3#30656942C211yesmaximal, characteristic, complemented
4#30656481C211yesmaximal, characteristic, complemented
5Q8⋊#14523339C211yesmaximal, characteristic, complemented
6D4⋊#14523339C211yesmaximal, characteristic, complemented
7#550×C257C211yesmaximal, characteristic
8C2×#16861606—21no—
9#22551592C2211yescharacteristic
10#22551523—21no—
11C4⋊#14523339C2211yescharacteristic
12C4⋊#14523339C2211yescharacteristic
13#22551516—21no—
14#22551515—21no—
15#22551513—22no—
16C4×#14523339C2211yescharacteristic
17C2×#41×C257—21no—
18#127×C257—21no—
19#127×C257C2211yescharacteristic
20#121×C257—22no—
21#121×C257—21no—
22C4×Q8×C257C2211yescharacteristic
23C4×D4×C257—21no—
24C4×D4×C257C2211yescharacteristic
25C2×Dic514—21no—
26#16861606—42no—
27C2×#14523339—21no—
28C2×#14523339—21no—
29C2×#14523339—21no—
30C4⋊Dic257—21no—
31C4⋊Dic257—21no—
32#41×C257—42no—
33#41×C257—42no—
34C2×Q8×C257D411yescharacteristic
35C2×D4×C257—22no—
36C2×D4×C257—22no—
37C2×D4×C257—21no—
38C2×D4×C257—21no—
39C2×D4×C257D411yescharacteristic
40C2×D4×C257D412yes—
41C22×C4×C257—22no—
42C22×C4×C257—21no—
43C4⋊C4×C257—21no—
44C4⋊C4×C257D411yescharacteristic
45C4⋊C4×C257D411yescharacteristic
46C22⋊C4×C257—22no—
47C22⋊C4×C257—21no—
48C42×C257—21no—
49C42×C257C2311yescharacteristic
50Dic514—41no—
51Dic514—41no—
52C2×Dic257—41no—
53#14523339—41no—
54#14523339—41no—
55#14523339—41no—
56C23×C257—21no—
57C23×C257—22no—
58Q8×C257—21no—
59Q8×C257SD1612yescomplemented
60D4×C257—21no—
61D4×C257—21no—
62D4×C257—42no—
63D4×C257—44no—
64D4×C257—41no—
65D4×C257—41no—
66D4×C257—41no—
67D4×C257—41no—
68D4×C257—82no—
69C2×C4×C257—21no—
70C2×C4×C257—21no—
71C2×C4×C257—21no—
72C2×C4×C257—44no—
73C2×C4×C257—41no—
74C2×C4×C257—41no—
75C2×C4×C257C2×D411yescharacteristic
76C2×C4×C257C2×D411yescharacteristic
77C2×C4×C257C2×D411yescharacteristic
78C22×C257—41no—
79C22×C257—42no—
80C22×C257—42no—
81C22×C257—41no—
82C22×C257—42no—
83C22×C257—41no—
84C22×C257#12011yescharacteristic
85C4×C257—22no—
86C4×C257—21no—
87C4×C257—41no—
88C4×C257—41no—
89C4×C257C2×SD1611yescharacteristic
90C4×C257#13511yescharacteristic
91C4×C257AGL(1,ℤ/8)11yescharacteristic
92C4×C257AGL(1,ℤ/8)11yescharacteristic
93Dic257—81no—
94C2×C257—81no—
95C2×C257—82no—
96C2×C257#44911yescharacteristic
97C2×C257#45011yescharacteristic
98C2×C257#45511yescharacteristic
99C257#164211yescharacteristic, complemented, Sylow
100#1642—2571nomaximal, Sylow
101#550D25711yescharacteristic
102#486—2571no—
103#475—2571no—
104#461—2571no—
105#331—2571no—
106Q8⋊C8—2571no—
107D4⋊C8—2571no—
108C2×#41—21no—
109C2×SD16—5141no—
110C4⋊Q8—2571no—
111#127—21no—
112#127C2×D25711yescharacteristic
113#121—22no—
114#121—21no—
115C4×Q8C2×D25711yescharacteristic
116C4×D4—21no—
117C4×D4C2×D25711yescharacteristic
118#106—5141no—
119C4⋊C8—2571no—
120C4⋊C8—2571no—
121#103—5141no—
122#103—5141no—
123#102—5142no—
124C4×C8—2571no—
125#41—42no—
126#41—42no—
127C2×Q8—5141no—
128C2×Q8C22⋊D25711yescharacteristic
129C2×D4—22no—
130C2×D4—22no—
131C2×D4—21no—
132C2×D4—21no—
133C2×D4C22⋊D25711yescharacteristic
134C2×D4C22⋊D25712yes—
135C22×C4—22no—
136C22×C4—21no—
137SD16—10282no—
138C2×C8—5141no—
139C2×C8—5141no—
140C2×C8—5141no—
141C4⋊C4—21no—
142C4⋊C4—5141no—
143C4⋊C4—5141no—
144C4⋊C4C22⋊D25711yescharacteristic
145C4⋊C4C22⋊D25711yescharacteristic
146C22⋊C4—22no—
147C22⋊C4—21no—
148C42—21no—
149C42C22×D25711yescharacteristic
150C23—21no—
151C23—22no—
152Q8—21no—
153Q8—10281no—
154Q8—10281no—
155Q8#1686160612yescomplemented
156D4—21no—
157D4—21no—
158D4—42no—
159D4—44no—
160D4—41no—
161D4—41no—
162D4—41no—
163D4—41no—
164D4—82no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—44no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—10281no—
172C2×C4C2×C22⋊D25711yescharacteristic
173C2×C4C2×C22⋊D25711yescharacteristic
174C2×C4C2×C22⋊D25711yescharacteristic
175C8—10281no—
176C8—10281no—
177C8—10281no—
178C22—41no—
179C22—42no—
180C22—42no—
181C22—41no—
182C22—42no—
183C22—41no—
184C22#2255156111yescharacteristic
185C4—22no—
186C4—21no—
187C4—41no—
188C4—41no—
189C4—20561no—
190C4C2×#1686160611yescharacteristic
191C4#2255161211yescharacteristic
192C4#2255161711yescharacteristic
193C4#2255161811yescharacteristic
194C2—81no—
195C2—82no—
196C2#3065688211yescharacteristic
197C2#3065688811yescharacteristic
198C2#3065691511yescharacteristic
subgroupG/H
centerC22#22551561
commutator, cosocleC42×C257C23
FrattiniC42C22×D257
Fitting, prime radical#550×C257C2
socleC22×C257#120
hypercenter#550D257
hypocenterC257#1642
LCSG ▷ C42×C257 ▷ C22×C257 ▷ C257
UCSC1 ◁ C22 ◁ C42 ◁ #550

Complements

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

NH ≅ G/Ntype
1. #30657083C2semidirect
2. #30657022C2semidirect
3. #30656942C2semidirect
4. #30656481C2semidirect
5. Q8⋊#14523339C2semidirect
6. D4⋊#14523339C2semidirect
59. Q8×C257SD16semidirect
99. C257#1642semidirect
155. Q8#16861606semidirect

Maximal subgroups

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

subgroupindex
#306570832
#306570222
#306569422
#306564812
Q8⋊#145233392
D4⋊#145233392
#550×C2572
#1642257