Finite Group Databasebeta

← #45663438

Subgroups — #45663438

Order 32896.

314 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 59 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
1D4×D1028C211yesmaximal, characteristic
2#30657110C211yesmaximal, characteristic, complemented
3#30657028C211yesmaximal, characteristic, complemented
4#30657027C211yesmaximal, characteristic, complemented
5#30656480C211yesmaximal, characteristic, complemented
6#30656466C211yesmaximal, characteristic, complemented
7D4⋊C8×C257C211yesmaximal, characteristic, complemented
8C22×D1028—21no—
9C2×D4×D257—21no—
10C2×#16861607—22no—
11#22551592C2211yescharacteristic, complemented
12#22551589C2211yescharacteristic
13#22551567—21no—
14#22551566—21no—
15#22551562—22no—
16C4×D1028C2211yescharacteristic
17#22551524—21no—
18#22551523—21no—
19C4⋊#14523339C2211yescharacteristic, complemented
20#22551514—21no—
21#22551513—21no—
22C4×D4×C257C2211yescharacteristic
23C4⋊C8×C257C2211yescharacteristic, complemented
24C4×C8×C257C2211yescharacteristic, complemented
25C23×D257—22no—
26C2×Dic514—21no—
27C2×D1028—22no—
28C2×D1028—21no—
29C2×D1028—21no—
30C2×D1028—42no—
31C2×D1028—42no—
32C2×D1028D411yescharacteristic
33C2×D1028D412yes—
34C2×C22⋊D257—22no—
35D4×D257—42no—
36D4×D257—42no—
37C2×C4×D257—21no—
38#16861607—44no—
39C2×#14523339—21no—
40C4⋊Dic257—21no—
41C4⋊Dic257—21no—
42C4⋊Dic257D411yescharacteristic
43#16861595—22no—
44#16861595—21no—
45C2×D4×C257D411yescharacteristic
46C22×C4×C257—21no—
47C2×C8×C257—21no—
48C2×C8×C257—21no—
49C4⋊C4×C257D411yescharacteristic
50C22⋊C4×C257—21no—
51C42×C257C2311yescharacteristic
52C22×D257—21no—
53C22×D257—22no—
54C22×D257—44no—
55C22×D257—44no—
56C22×D257—82no—
57Dic514—42no—
58D1028—21no—
59D1028—42no—
60D1028—44no—
61D1028—41no—
62D1028—41no—
63D1028—41no—
64D1028—41no—
65D1028SD1612yescomplemented
66C22⋊D257—44no—
67C2×Dic257—21no—
68C2×Dic257—41no—
69C4×D257—41no—
70C4×D257—41no—
71#14523339—41no—
72C23×C257—21no—
73D4×C257—21no—
74D4×C257SD1612yescomplemented
75C2×C4×C257—21no—
76C2×C4×C257—41no—
77C2×C4×C257—41no—
78C2×C4×C257C2×D411yescharacteristic
79C2×C4×C257C2×D411yescharacteristic
80C2×C4×C257C2×D411yescharacteristic
81C8×C257—22no—
82C8×C257—41no—
83C22×C257—22no—
84C22×C257—41no—
85C22×C257—41no—
86C22×C257#12011yescharacteristic
87C2×D257—22no—
88C2×D257—41no—
89C2×D257—41no—
90C2×D257—42no—
91C2×D257—42no—
92C2×D257—42no—
93C2×D257—88no—
94C4×C257—21no—
95C4×C257—41no—
96C4×C257C2×SD1611yescharacteristic
97C4×C257C2×SD1611yescharacteristic
98C4×C257AGL(1,ℤ/8)11yescharacteristic
99C4×C257AGL(1,ℤ/8)11yescharacteristic
100Dic257—41no—
101Dic257—81no—
102C2×C257—42no—
103C2×C257#45011yescharacteristic
104C2×C257#45011yescharacteristic
105C2×C257#45311yescharacteristic
106D257—42no—
107D257—82no—
108C257#162911yescharacteristic, complemented, Sylow
109#1629—2571nomaximal, Sylow
110D42—2571no—
111#492—2571no—
112#476—2571no—
113#476—2571no—
114#331—2571no—
115D4⋊C8—2571no—
116D4⋊C8D25711yescharacteristic, complemented
117C22×D4—5141no—
118C22×D4—5141no—
119C2×SD16—5142no—
120C4⋊Q8—2571no—
121#127—2571no—
122#121—5141no—
123#121—5141no—
124#120—5142no—
125C4×D4—2571no—
126C4×D4C2×D25711yescharacteristic
127#106—5141no—
128#106—5141no—
129C4⋊C8—2571no—
130C4⋊C8C2×D25711yescharacteristic, complemented
131#102—5141no—
132#102—5141no—
133C4×C8C2×D25711yescharacteristic, complemented
134C24—5142no—
135C2×Q8—5141no—
136C2×D4—2571no—
137C2×D4—2572no—
138C2×D4—5142no—
139C2×D4—5142no—
140C2×D4—5141no—
141C2×D4—5141no—
142C2×D4—10282no—
143C2×D4—10282no—
144C2×D4—10282no—
145C2×D4—10282no—
146C2×D4D102811yescharacteristic
147C22×C4—21no—
148C22×C4—5141no—
149SD16—10284no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—5141no—
153C4⋊C4—2571no—
154C4⋊C4—5141no—
155C4⋊C4—5141no—
156C4⋊C4D102811yescharacteristic
157C22⋊C4—21no—
158C22⋊C4—5142no—
159C22⋊C4—5141no—
160C42C22×D25711yescharacteristic
161C23—21no—
162C23—5141no—
163C23—5142no—
164C23—10284no—
165C23—10284no—
166C23—20562no—
167Q8—10282no—
168D4—2572no—
169D4—21no—
170D4—5141no—
171D4—10282no—
172D4—10284no—
173D4—10284no—
174D4—10281no—
175D4—10281no—
176D4—10281no—
177D4—10281no—
178D4#1686160712yescomplemented
179C2×C4—21no—
180C2×C4—5141no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—10281no—
184C2×C4—10281no—
185C2×C4—10281no—
186C2×C4D4×D25711yescharacteristic
187C2×C4D4×D25711yescharacteristic
188C2×C4C2×D102811yescharacteristic
189C8—22no—
190C8—41no—
191C8—10281no—
192C22—22no—
193C22—5142no—
194C22—41no—
195C22—41no—
196C22—10281no—
197C22—10281no—
198C22—10282no—
199C22—10282no—
200C22—10282no—
201C22—20568no—
202C22#2255156211yescharacteristic
203C4—21no—
204C4—41no—
205C4—10281no—
206C4—20561no—
207C4SD16×D25711yescharacteristic
208C4C2×#1686160711yescharacteristic
209C4#2255161611yescharacteristic
210C4#2255162011yescharacteristic
211C2—42no—
212C2—10282no—
213C2—20562no—
214C2#3065688911yescharacteristic
215C2#3065689011yescharacteristic
216C2#3065690611yescharacteristic
subgroupG/H
centerC22#22551562
commutator, cosocleC42×C257C23
FrattiniC42C22×D257
FittingD4⋊C8×C257C2
socleC22×C257#120
prime radicalD4×D1028C2
hypercenterD4⋊C8D257
hypocenterC257#1629
LCSG ▷ C42×C257 ▷ C22×C257 ▷ C257
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #30657110C2semidirect
3. #30657028C2semidirect
4. #30657027C2semidirect
5. #30656480C2semidirect
6. #30656466C2semidirect
7. D4⋊C8×C257C2semidirect
11. #22551592C22semidirect
19. C4⋊#14523339C22semidirect
23. C4⋊C8×C257C22semidirect
24. C4×C8×C257C22semidirect
65. D1028SD16semidirect
74. D4×C257SD16semidirect
108. C257#1629semidirect
116. D4⋊C8D257semidirect
130. C4⋊C8C2×D257semidirect
133. C4×C8C2×D257semidirect
178. D4#16861607semidirect

Maximal subgroups

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

subgroupindex
D4×D10282
#306571102
#306570282
#306570272
#306564802
#306564662
D4⋊C8×C2572
#1629257