Finite Group Databasebeta

← #45664841

Subgroups — #45664841

Order 32896.

312 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 79 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#30657503C211yesmaximal, characteristic
2#30657313C211yesmaximal, characteristic
3#30656747C211yesmaximal, characteristic, complemented
4#30656745C211yesmaximal, characteristic, complemented
5#30656710C211yesmaximal, characteristic
6#30656493C211yesmaximal, characteristic, complemented
7H(3,ℤ/4)×C257C211yesmaximal, characteristic, complemented
8C2×#16861618C2211yescharacteristic
9C22×Dic514C2211yescharacteristic
10#22551600—22no—
11#22551598—22no—
12C2×#16861602C2211yescharacteristic
13M16×D257—22no—
14#22551592C412yescomplemented
15#22551580C412yescomplemented
16#22551576—22no—
17#22551572—22no—
18#22551551C2211yescharacteristic
19#22551517—22no—
20#22551515—22no—
21#22551506—22no—
22C2×M16×C257C2211yescharacteristic
23#117×C257C2211yescharacteristic
24#117×C257C2211yescharacteristic
25#16861618—21no—
26#16861618—21no—
27#16861618—21no—
28#16861618—21no—
29C2×Dic514—21no—
30C2×Dic514—21no—
31C2×Dic514—21no—
32C2×Dic514—21no—
33C2×Dic514—21no—
34C2×Dic514C2×C412yes—
35C2×Dic514C2×C411yescharacteristic
36C2×D1028C2×C411yescharacteristic
37C2×C22⋊D257D412yes—
38C22×Dic257—21no—
39C2×C4×D257D412yes—
40#16861602—21no—
41#16861602—21no—
42#16861601—24no—
43C2×#14523339D412yes—
44C8×D257—24no—
45C4⋊Dic257—24no—
46C4⋊Dic257—21no—
47C22⋊Dic257—21no—
48#16861595—24no—
49C4×Dic257—21no—
50C22×C4×C257C2311yescharacteristic
51M16×C257—21no—
52M16×C257—21no—
53C2×C8×C257D412yes—
54C4⋊C4×C257—21no—
55C4⋊C4×C257—21no—
56C22⋊C4×C257—21no—
57C22⋊C4×C257—21no—
58C42×C257—21no—
59C42×C257C2×C412yes—
60C22×D257C22⋊C412yes—
61Dic514—21no—
62Dic514—21no—
63Dic514—22no—
64Dic514—22no—
65Dic514—41no—
66Dic514—41no—
67D1028—21no—
68D1028—21no—
69C22⋊D257—22no—
70C22⋊D257—22no—
71C2×Dic257—22no—
72C2×Dic257—41no—
73C2×Dic257—41no—
74C2×Dic257—41no—
75C2×Dic257C22⋊C412yes—
76C4×D257—22no—
77C4×D257—22no—
78#14523339—22no—
79C23×C257C2×D411yescharacteristic
80C2×C4×C257—22no—
81C2×C4×C257—41no—
82C2×C4×C257C22×C411yescharacteristic
83C2×C4×C257C2×D411yescharacteristic
84C2×C4×C257C2×D411yescharacteristic
85C2×C4×C257C2×D411yescharacteristic
86C2×C4×C257#4111yescharacteristic
87C2×C4×C257#4111yescharacteristic
88C8×C257—22no—
89C22×C257—41no—
90C22×C257C2×C22⋊C411yescharacteristic
91C22×C257#12111yescharacteristic
92C22×C257#12111yescharacteristic
93C2×D257—22no—
94C2×D257—42no—
95C4×C257—42no—
96C4×C257—81no—
97C4×C257C4×D411yescharacteristic
98C4×C257C4×D411yescharacteristic
99C4×C257#12011yescharacteristic
100C4×C257#12311yescharacteristic
101Dic257—22no—
102Dic257—42no—
103Dic257—81no—
104C2×C257—21no—
105C2×C257—21no—
106C2×C257—21no—
107C2×C257#38611yescharacteristic
108D257—42no—
109C257#190911yescharacteristic, complemented, Sylow
110#1909—2571nomaximal, Sylow
111#568—2571no—
112#536—2571no—
113#419—2571no—
114#419—2571no—
115#410—2571no—
116H(3,ℤ/4)—2571no—
117H(3,ℤ/4)D25711yescharacteristic, complemented
118C2×#41—2571no—
119C22×Q8—2571no—
120#131—5142no—
121#131—5142no—
122C2×M16—2571no—
123C2×M16—5142no—
124C2×M16C2×D25711yescharacteristic
125C4⋊Q8—2572no—
126#124—2572no—
127#123—5142no—
128C22⋊Q8—5142no—
129#117—2571no—
130#117C2×D25711yescharacteristic
131#117C2×D25711yescharacteristic
132#103—5142no—
133#103—5142no—
134#99—5142no—
135#41—5141no—
136#41—5141no—
137#41—5141no—
138#41—5141no—
139C2×Q8—2571no—
140C2×Q8—2572no—
141C2×Q8—5141no—
142C2×Q8—5141no—
143C2×Q8—5141no—
144C2×Q8—5141no—
145C2×Q8—5141no—
146C2×D4—2572no—
147C2×D4—2571no—
148C22×C4—2572no—
149C22×C4—5141no—
150C22×C4C22×D25711yescharacteristic
151M16—21no—
152M16—21no—
153M16—5144no—
154M16—5141no—
155M16—5141no—
156C2×C8—2572no—
157C2×C8—5144no—
158C2×C8D102812yes—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—5144no—
162C4⋊C4—5141no—
163C22⋊C4—21no—
164C22⋊C4—21no—
165C22⋊C4—5144no—
166C22⋊C4—5141no—
167C42—21no—
168C42—5141no—
169C42C4×D25712yes—
170C23—2572no—
171C23D4×D25711yescharacteristic
172Q8—5141no—
173Q8—5141no—
174Q8—5142no—
175Q8—5142no—
176Q8—10281no—
177Q8—10281no—
178D4—5142no—
179D4—5142no—
180D4—5141no—
181D4—5141no—
182C2×C4—2572no—
183C2×C4—22no—
184C2×C4—5142no—
185C2×C4—5142no—
186C2×C4—5142no—
187C2×C4—41no—
188C2×C4—10281no—
189C2×C4—10281no—
190C2×C4—10281no—
191C2×C4C2×C4×D25711yescharacteristic
192C2×C4D4×D25711yescharacteristic
193C2×C4D4×D25711yescharacteristic
194C2×C4C2×D102811yescharacteristic
195C2×C4#1686161811yescharacteristic
196C2×C4#1686161911yescharacteristic
197C8—22no—
198C8—5142no—
199C22—5142no—
200C22—41no—
201C22—10282no—
202C22C22⋊C4×D25711yescharacteristic
203C22#2255156511yescharacteristic
204C22#2255156711yescharacteristic
205C4—5142no—
206C4—42no—
207C4—10282no—
208C4—81no—
209C4—20561no—
210C4#2255155411yescharacteristic
211C4C4×D102811yescharacteristic
212C4#2255156211yescharacteristic
213C4#2255157711yescharacteristic
214C2—21no—
215C2—21no—
216C2—21no—
217C2—10282no—
218C2#3065661511yescharacteristic
subgroupG/H
centerC2#30656615
commutatorC2×C4×C257C22×C4
FrattiniC22×C4C22×D257
FittingH(3,ℤ/4)×C257C2
socleC2×C257#386
prime radicalC2×#16861618C22
cosocleC22×C4×C257C23
hypercenterH(3,ℤ/4)D257
hypocenterC257#1909
LCSG ▷ C2×C4×C257 ▷ C2×C257 ▷ C257
UCSC1 ◁ C2 ◁ C22×C4 ◁ H(3,ℤ/4)

Complements

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

NH ≅ G/Ntype
3. #30656747C2semidirect
4. #30656745C2semidirect
6. #30656493C2semidirect
7. H(3,ℤ/4)×C257C2semidirect
14. #22551592C4semidirect
15. #22551580C4semidirect
109. C257#1909semidirect
117. H(3,ℤ/4)D257semidirect

Maximal subgroups

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

subgroupindex
#306575032
#306573132
#306567472
#306567452
#306567102
#306564932
H(3,ℤ/4)×C2572
#1909257