Finite Group Databasebeta

← #45665460

Subgroups — #45665460

Order 32896.

290 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 87 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
1C2×#22551591C211yesmaximal, characteristic
2C2×#22551517C211yesmaximal, characteristic
3C2×#22551515C211yesmaximal, characteristic
4#30656642C211yesmaximal, characteristic
5#30656499C211yesmaximal, characteristic
6#30656491C211yesmaximal, characteristic
7#340×C257C211yesmaximal, characteristic
8C22×Dic514—21no—
9C22×Dic514C2211yescharacteristic
10C22×#14523339C2211yescharacteristic
11#22551591—22no—
12#22551591—22no—
13C2×C4⋊Dic257C2211yescharacteristic
14C2×#16861597—21no—
15C2×C4×Dic257C2211yescharacteristic
16#22551517—22no—
17#22551515—22no—
18#22551499—21no—
19#22551499—21no—
20C22×C8×C257C2211yescharacteristic
21C2×C4⋊C4×C257C2211yescharacteristic
22C2×C4⋊C4×C257C2211yescharacteristic
23C2×Dic514—24no—
24C2×Dic514—21no—
25C2×Dic514—21no—
26C2×Dic514—21no—
27C2×Dic514—21no—
28C2×Dic514—21no—
29C2×Dic514—42no—
30C2×Dic514—42no—
31C2×Dic514Q812yescomplemented
32C22×Dic257—21no—
33C22×Dic257—21no—
34C22×Dic257D411yescharacteristic
35C22×Dic257D411yescharacteristic
36C2×#14523339—21no—
37C2×#14523339—21no—
38C2×#14523339—21no—
39C4⋊Dic257—21no—
40C4⋊Dic257D412yes—
41#16861597—24no—
42C4×Dic257—21no—
43C4×Dic257—21no—
44C22×C4×C257—21no—
45C22×C4×C257—21no—
46C22×C4×C257C2311yescharacteristic
47C2×C8×C257—21no—
48C2×C8×C257—21no—
49C2×C8×C257—21no—
50C4⋊C4×C257—21no—
51C4⋊C4×C257—21no—
52C4⋊C4×C257—21no—
53C4⋊C4×C257D412yes—
54Dic514—22no—
55Dic514—22no—
56Dic514—48no—
57Dic514—41no—
58Dic514—41no—
59C2×Dic257—21no—
60C2×Dic257—21no—
61C2×Dic257—21no—
62C2×Dic257—21no—
63C2×Dic257—22no—
64C2×Dic257—22no—
65C2×Dic257—41no—
66C2×Dic257—41no—
67C2×Dic257—41no—
68C2×Dic257—41no—
69C2×Dic257SD1612yes—
70C2×Dic257Q1612yes—
71#14523339—41no—
72C23×C257C2×D411yescharacteristic
73C2×C4×C257—22no—
74C2×C4×C257—41no—
75C2×C4×C257—41no—
76C2×C4×C257—41no—
77C2×C4×C257—41no—
78C2×C4×C257—41no—
79C2×C4×C257C2×D411yescharacteristic
80C2×C4×C257C2×D411yescharacteristic
81C2×C4×C257C2×Q811yescharacteristic
82C2×C4×C257#4111yescharacteristic
83C2×C4×C257#4111yescharacteristic
84C2×C4×C257#4111yescharacteristic
85C8×C257—41no—
86C22×C257#12011yescharacteristic
87C22×C257C22⋊Q811yescharacteristic
88C22×C257#12411yescharacteristic
89C22×C257C2×SD1611yescharacteristic
90C22×C257C2×Q1611yescharacteristic
91C22×C257AGL(1,ℤ/8)11yescharacteristic
92C22×C257#13711yescharacteristic
93C4×C257—42no—
94C4×C257—81no—
95C4×C257C22⋊Q811yescharacteristic
96C4×C257C22⋊Q811yescharacteristic
97C4×C257#12411yescharacteristic
98C4×C257#12411yescharacteristic
99Dic257—22no—
100Dic257—22no—
101Dic257—42no—
102Dic257—42no—
103C2×C257#39311yescharacteristic
104C2×C257#45011yescharacteristic
105C2×C257#45111yescharacteristic
106C2×C257#47511yescharacteristic
107C2×C257#47711yescharacteristic
108C2×C257#48711yescharacteristic
109C2×C257#48911yescharacteristic
110C257#202411yescharacteristic, complemented, Sylow
111#2024—2571nomaximal, Sylow
112C2×C4⋊Q8—2571no—
113C2×#103—2571no—
114C2×#103—2571no—
115#391—2571no—
116#340—2571no—
117#340D25711yescharacteristic
118#336—2571no—
119C22×Q8—2571no—
120C22×Q8—5141no—
121C22×C8—2571no—
122C22×C8C2×D25711yescharacteristic
123C4⋊Q8—5142no—
124C4⋊Q8—5142no—
125C2×C4⋊C4—2571no—
126C2×C4⋊C4—5141no—
127C2×C4⋊C4C2×D25711yescharacteristic
128C2×C4⋊C4C2×D25711yescharacteristic
129C2×C42—2571no—
130#103—5142no—
131#103—5142no—
132#95—5141no—
133#95—5141no—
134C2×Q8—2572no—
135C2×Q8—5144no—
136C2×Q8—5141no—
137C2×Q8—5141no—
138C2×Q8—5141no—
139C2×Q8—5141no—
140C2×Q8—5141no—
141C2×Q8—10282no—
142C2×Q8—10282no—
143C22×C4—2571no—
144C22×C4—2571no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—5141no—
148C22×C4—5141no—
149C22×C4C22×D25711yescharacteristic
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—5141no—
154C2×C8—5141no—
155C2×C8—5141no—
156C4⋊C4—2572no—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—5144no—
161C4⋊C4—5141no—
162C4⋊C4D102812yes—
163C42—5141no—
164C42—5141no—
165C23D4×D25711yescharacteristic
166Q8—5142no—
167Q8—5142no—
168Q8—10288no—
169Q8—10281no—
170Q8—10281no—
171C2×C4—2572no—
172C2×C4—2572no—
173C2×C4—22no—
174C2×C4—5141no—
175C2×C4—5141no—
176C2×C4—5141no—
177C2×C4—5141no—
178C2×C4—5142no—
179C2×C4—5142no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—10281no—
186C2×C4—10281no—
187C2×C4—10281no—
188C2×C4—10281no—
189C2×C4D4×D25711yescharacteristic
190C2×C4C2×D102811yescharacteristic
191C2×C4Q8×D25711yescharacteristic
192C2×C4#1686161711yescharacteristic
193C2×C4#1686161811yescharacteristic
194C2×C4#1686161811yescharacteristic
195C8—41no—
196C8—10281no—
197C22#2255156211yescharacteristic
198C22#2255157011yescharacteristic
199C22#2255157811yescharacteristic
200C22SD16×D25711yescharacteristic
201C22Q16×D25711yescharacteristic
202C22#2255161611yescharacteristic
203C22#2255162111yescharacteristic
204C4—5142no—
205C4—5142no—
206C4—42no—
207C4—10282no—
208C4—10282no—
209C4—81no—
210C4#2255157011yescharacteristic
211C4#2255157211yescharacteristic
212C4#2255157811yescharacteristic
213C4#2255158011yescharacteristic
214C2#3065665011yescharacteristic
215C2#3065688911yescharacteristic
216C2#3065689411yescharacteristic
217C2#3065702111yescharacteristic
218C2#3065703111yescharacteristic
219C2#3065708611yescharacteristic
220C2#3065709511yescharacteristic
subgroupG/H
centerC23D4×D257
commutator, cosocleC22×C4×C257C23
FrattiniC22×C4C22×D257
Fitting#340×C257C2
socle, prime radicalC23×C257C2×D4
hypercenter#340D257
hypocenterC257#2024
LCSG ▷ C22×C4×C257 ▷ C2×C257 ▷ C257
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
31. C2×Dic514Q8semidirect
110. C257#2024semidirect

Maximal subgroups

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

subgroupindex
C2×#225515912
C2×#225515172
C2×#225515152
#306566422
#306564992
#306564912
#340×C2572
#2024257