Finite Group Databasebeta

← #45665516

Subgroups — #45665516

Order 32896.

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

normal subgroups only

This group has 316 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×#22551571C211yesmaximal, characteristic
2C2×#22551523C211yesmaximal, characteristic, complemented
3C2×#22551515C211yesmaximal, characteristic, complemented
4C2×#22551515C211yesmaximal, characteristic, complemented
5C2×C22⋊#14523339C211yesmaximal, characteristic
6#30656499C211yesmaximal, characteristic, complemented
7#380×C257C211yesmaximal, characteristic
8C22×Dic514C2211yescharacteristic
9C22×#14523339C2211yescharacteristic
10C22×#14523339C2211yescharacteristic
11#22551571—22no—
12#22551571—22no—
13C2×C4⋊Dic257C2211yescharacteristic
14C2×#16861597—21no—
15C2×C22⋊Dic257—21no—
16#22551523—22no—
17#22551515—22no—
18#22551515—22no—
19C22⋊#14523339—22no—
20C23×C4×C257C2211yescharacteristic
21C2×C4⋊C4×C257C2211yescharacteristic
22C2×C4⋊C4×C257C2211yescharacteristic
23C2×C22⋊C4×C257—21no—
24#95×C257—21no—
25C2×Dic514—21no—
26C2×Dic514—21no—
27C2×Dic514—21no—
28C2×Dic514—21no—
29C2×Dic514—21no—
30C2×Dic514D412yes—
31C22×Dic257—21no—
32C22×Dic257—21no—
33C2×#14523339—21no—
34C2×#14523339—21no—
35C2×#14523339—21no—
36C2×#14523339—21no—
37C2×#14523339—21no—
38C2×#14523339D412yes—
39C4⋊Dic257—21no—
40C4⋊Dic257D412yes—
41#16861597—24no—
42C22⋊Dic257—24no—
43C24×C257D411yescharacteristic
44C22×C4×C257—21no—
45C22×C4×C257—21no—
46C22×C4×C257—21no—
47C22×C4×C257—21no—
48C22×C4×C257—21no—
49C22×C4×C257—21no—
50C22×C4×C257—21no—
51C22×C4×C257—21no—
52C22×C4×C257D411yescharacteristic
53C22×C4×C257C2311yescharacteristic
54C4⋊C4×C257—21no—
55C4⋊C4×C257—21no—
56C4⋊C4×C257—21no—
57C4⋊C4×C257—21no—
58C22⋊C4×C257—42no—
59Dic514—22no—
60Dic514—22no—
61Dic514—41no—
62Dic514—41no—
63C2×Dic257—22no—
64C2×Dic257—22no—
65C2×Dic257—41no—
66C2×Dic257—41no—
67C2×Dic257—41no—
68C2×Dic257—41no—
69#14523339—22no—
70#14523339—41no—
71C23×C257—21no—
72C23×C257—21no—
73C23×C257—21no—
74C23×C257—21no—
75C23×C257—21no—
76C23×C257—21no—
77C23×C257SD1612yes—
78C23×C257C2×D411yescharacteristic
79C2×C4×C257—21no—
80C2×C4×C257—21no—
81C2×C4×C257—21no—
82C2×C4×C257—41no—
83C2×C4×C257—41no—
84C2×C4×C257—41no—
85C2×C4×C257—41no—
86C2×C4×C257—41no—
87C2×C4×C257—41no—
88C2×C4×C257—41no—
89C2×C4×C257—41no—
90C2×C4×C257—41no—
91C2×C4×C257—41no—
92C2×C4×C257C2×D411yescharacteristic
93C2×C4×C257C2×D411yescharacteristic
94C2×C4×C257C2×D411yescharacteristic
95C2×C4×C257#4111yescharacteristic
96C2×C4×C257#4111yescharacteristic
97C2×C4×C257#4111yescharacteristic
98C22×C257—22no—
99C22×C257—22no—
100C22×C257—22no—
101C22×C257—41no—
102C22×C257—41no—
103C22×C257—41no—
104C22×C257—41no—
105C22×C257#12011yescharacteristic
106C22×C257#12111yescharacteristic
107C22×C257#12311yescharacteristic
108C22×C257C2×SD1611yescharacteristic
109C22×C257#13511yescharacteristic
110C22×C257#13711yescharacteristic
111C22×C257#13711yescharacteristic
112C4×C257—41no—
113C4×C257—81no—
114C4×C257—81no—
115C4×C257#12111yescharacteristic
116C4×C257#12111yescharacteristic
117C4×C257#12311yescharacteristic
118C4×C257#12411yescharacteristic
119Dic257—42no—
120Dic257—42no—
121C2×C257—42no—
122C2×C257#39411yescharacteristic
123C2×C257#44811yescharacteristic
124C2×C257#45211yescharacteristic
125C2×C257#46511yescharacteristic
126C2×C257#47011yescharacteristic
127C2×C257#48311yescharacteristic
128C2×C257#48511yescharacteristic
129C257#203511yescharacteristic, complemented, Sylow
130#2035—2571nomaximal, Sylow
131C2×C22⋊Q8—2571no—
132C2×#106—2571no—
133C2×#103—2571no—
134C2×#103—2571no—
135C2×C22⋊C8—2571no—
136#380D25711yescharacteristic
137#340—2571no—
138C22×Q8—2571no—
139C23×C4C2×D25711yescharacteristic
140C22×C8—2571no—
141C22×C8—2571no—
142C22⋊Q8—5142no—
143C22⋊Q8—5142no—
144C2×C4⋊C4—2571no—
145C2×C4⋊C4—5141no—
146C2×C4⋊C4C2×D25711yescharacteristic
147C2×C4⋊C4C2×D25711yescharacteristic
148C2×C22⋊C4—21no—
149C2×C22⋊C4—5141no—
150#106—5142no—
151#103—5142no—
152#103—5142no—
153C22⋊C8—5142no—
154#95—21no—
155C24C22⋊D25711yescharacteristic
156C2×Q8—2572no—
157C2×Q8—5141no—
158C2×Q8—5141no—
159C2×Q8—5141no—
160C2×Q8—5141no—
161C2×Q8—5141no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—5141no—
171C22×C4—5141no—
172C22×C4C22⋊D25711yescharacteristic
173C22×C4C22×D25711yescharacteristic
174C2×C8—2572no—
175C2×C8—5141no—
176C2×C8—5141no—
177C2×C8—5141no—
178C2×C8—5141no—
179C2×C8—5141no—
180C4⋊C4—2572no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—5144no—
186C4⋊C4—5141no—
187C22⋊C4—5144no—
188C22⋊C4—42no—
189C23—21no—
190C23—21no—
191C23—21no—
192C23—21no—
193C23—21no—
194C23—21no—
195C23#1686160612yes—
196C23C2×C22⋊D25711yescharacteristic
197Q8—5142no—
198Q8—5142no—
199Q8—10281no—
200Q8—10281no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—5142no—
205C2×C4—5142no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—10281no—
217C2×C4—10281no—
218C2×C4—10281no—
219C2×C4—10281no—
220C2×C4D4×D25711yescharacteristic
221C2×C4D4×D25711yescharacteristic
222C2×C4D4×D25711yescharacteristic
223C2×C4#1686161711yescharacteristic
224C2×C4#1686161811yescharacteristic
225C2×C4#1686161811yescharacteristic
226C8—5142no—
227C8—10281no—
228C22—22no—
229C22—22no—
230C22—22no—
231C22—41no—
232C22—41no—
233C22—41no—
234C22—41no—
235C22#2255156011yescharacteristic
236C22#2255156411yescharacteristic
237C22#2255157511yescharacteristic
238C22C2×#1686160611yescharacteristic
239C22#2255161211yescharacteristic
240C22#2255162211yescharacteristic
241C22#2255162311yescharacteristic
242C4—41no—
243C4—10282no—
244C4—10282no—
245C4—81no—
246C4—81no—
247C4#2255156511yescharacteristic
248C4#2255156511yescharacteristic
249C4#2255157411yescharacteristic
250C4#2255157811yescharacteristic
251C2—42no—
252C2#3065665211yescharacteristic
253C2#3065687511yescharacteristic
254C2#3065689711yescharacteristic
255C2#3065696411yescharacteristic
256C2#3065699311yescharacteristic
257C2#3065706711yescharacteristic
258C2#3065707811yescharacteristic
subgroupG/H
centerC23C2×C22⋊D257
commutator, cosocleC22×C4×C257C23
FrattiniC22×C4C22×D257
Fitting#380×C257C2
socleC23×C257C2×D4
prime radicalC24×C257D4
hypercenter#380D257
hypocenterC257#2035
LCSG ▷ C22×C4×C257 ▷ C2×C257 ▷ C257
UCSC1 ◁ C23 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
2. C2×#22551523C2semidirect
3. C2×#22551515C2semidirect
4. C2×#22551515C2semidirect
6. #30656499C2semidirect
129. C257#2035semidirect

Maximal subgroups

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

subgroupindex
C2×#225515712
C2×#225515232
C2×#225515152
C2×C22⋊#145233392
#306564992
#380×C2572
#2035257