Finite Group Databasebeta

← #45665518

Subgroups — #45665518

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×#22551572C211yesmaximal, characteristic
2C2×#22551517C211yesmaximal, characteristic, complemented
3C2×#22551515C211yesmaximal, characteristic, complemented
4C2×#22551504C211yesmaximal, characteristic
5#30656594C211yesmaximal, characteristic
6#30656499C211yesmaximal, characteristic, complemented
7C2×#106×C257C211yesmaximal, characteristic, complemented
8C22×Dic514C2211yescharacteristic
9C22×C4×D257C2211yescharacteristic
10C22×#14523339C2211yescharacteristic
11#22551572—22no—
12#22551572—22no—
13C2×C4⋊Dic257—21no—
14C2×C4⋊Dic257C2211yescharacteristic
15C2×#16861595—21no—
16C2×#16861595—21no—
17#22551517—22no—
18#22551515—22no—
19#22551504—22no—
20#22551499—21no—
21C22×C8×C257C2211yescharacteristic
22C2×C4⋊C4×C257C2211yescharacteristic
23C2×C4⋊C4×C257C2211yescharacteristic
24#106×C257—22no—
25C23×D257D411yescharacteristic
26C2×Dic514—21no—
27C2×Dic514—21no—
28C2×Dic514—21no—
29C2×Dic514—21no—
30C2×Dic514—21no—
31C2×Dic514D412yes—
32C22×Dic257—21no—
33C22×Dic257—21no—
34C22×Dic257D411yescharacteristic
35C2×C4×D257—21no—
36C2×C4×D257—21no—
37C2×C4×D257—21no—
38C2×C4×D257—21no—
39C2×C4×D257—21no—
40C2×C4×D257—21no—
41C2×#14523339—21no—
42C2×#14523339—21no—
43C2×#14523339—21no—
44C4⋊Dic257—24no—
45C4⋊Dic257—21no—
46C4⋊Dic257—21no—
47#16861595—24no—
48#16861595—42no—
49C22×C4×C257—21no—
50C22×C4×C257—21no—
51C22×C4×C257C2311yescharacteristic
52C2×C8×C257—21no—
53C2×C8×C257—21no—
54C2×C8×C257D412yes—
55C4⋊C4×C257—21no—
56C4⋊C4×C257—21no—
57C4⋊C4×C257—21no—
58C4⋊C4×C257D412yes—
59C22×D257—21no—
60C22×D257—21no—
61C22×D257—21no—
62C22×D257—21no—
63C22×D257—21no—
64C22×D257—21no—
65C22×D257SD1612yes—
66Dic514—22no—
67Dic514—22no—
68Dic514—41no—
69Dic514—41no—
70C2×Dic257—22no—
71C2×Dic257—21no—
72C2×Dic257—21no—
73C2×Dic257—21no—
74C2×Dic257—41no—
75C2×Dic257—41no—
76C2×Dic257—41no—
77C2×Dic257—41no—
78C2×Dic257—41no—
79C4×D257—41no—
80C4×D257—41no—
81C4×D257—41no—
82C4×D257—41no—
83#14523339—41no—
84C23×C257C2×D411yescharacteristic
85C2×C4×C257—22no—
86C2×C4×C257—41no—
87C2×C4×C257—41no—
88C2×C4×C257—41no—
89C2×C4×C257—41no—
90C2×C4×C257—41no—
91C2×C4×C257C2×D411yescharacteristic
92C2×C4×C257C2×D411yescharacteristic
93C2×C4×C257C2×D411yescharacteristic
94C2×C4×C257#4111yescharacteristic
95C2×C4×C257#4111yescharacteristic
96C2×C4×C257#4111yescharacteristic
97C8×C257—22no—
98C22×C257#12011yescharacteristic
99C22×C257#12111yescharacteristic
100C22×C257#12311yescharacteristic
101C22×C257C2×SD1611yescharacteristic
102C22×C257#13511yescharacteristic
103C22×C257#13711yescharacteristic
104C22×C257#13711yescharacteristic
105C2×D257—22no—
106C2×D257—22no—
107C2×D257—22no—
108C2×D257—41no—
109C2×D257—41no—
110C2×D257—41no—
111C2×D257—41no—
112C4×C257—42no—
113C4×C257—81no—
114C4×C257#12111yescharacteristic
115C4×C257#12111yescharacteristic
116C4×C257#12311yescharacteristic
117C4×C257#12411yescharacteristic
118Dic257—41no—
119Dic257—42no—
120Dic257—81no—
121C2×C257#39411yescharacteristic
122C2×C257#44811yescharacteristic
123C2×C257#45211yescharacteristic
124C2×C257#46511yescharacteristic
125C2×C257#47011yescharacteristic
126C2×C257#48311yescharacteristic
127C2×C257#48511yescharacteristic
128D257—42no—
129C257#203511yescharacteristic, complemented, Sylow
130#2035—2571nomaximal, Sylow
131C2×C22⋊Q8—2571no—
132C2×#106D25711yescharacteristic, complemented
133C2×#103—2571no—
134C2×#103—2571no—
135C2×C22⋊C8—2571no—
136#380—2571no—
137#340—2571no—
138C22×Q8—2571no—
139C23×C4—2571no—
140C22×C8—2571no—
141C22×C8C2×D25711yescharacteristic
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—5141no—
149C2×C22⋊C4—5141no—
150#106—22no—
151#103—5142no—
152#103—5142no—
153C22⋊C8—5142no—
154#95—5141no—
155C24—2571no—
156C2×Q8—2572no—
157C2×Q8—5141no—
158C2×Q8—5141no—
159C2×Q8—5141no—
160C2×Q8—5141no—
161C2×Q8—5141no—
162C22×C4—2571no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—5141no—
166C22×C4—5141no—
167C22×C4—5141no—
168C22×C4—5141no—
169C22×C4—5141no—
170C22×C4—5141no—
171C22×C4—5141no—
172C22×C4—5141no—
173C22×C4C22×D25711yescharacteristic
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—5141no—
177C2×C8—5141no—
178C2×C8—5141no—
179C2×C8D102812yes—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—5144no—
184C4⋊C4—5141no—
185C4⋊C4—5141no—
186C4⋊C4C22⋊D25712yes—
187C22⋊C4—5144no—
188C22⋊C4—10282no—
189C23—2572no—
190C23—5141no—
191C23—5141no—
192C23—5141no—
193C23—5141no—
194C23—5141no—
195C23—5141no—
196C23D4×D25711yescharacteristic
197Q8—5142no—
198Q8—5142no—
199Q8—10281no—
200Q8—10281no—
201C2×C4—22no—
202C2×C4—5142no—
203C2×C4—5141no—
204C2×C4—5141no—
205C2×C4—5141no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—10281no—
212C2×C4—10281no—
213C2×C4—10281no—
214C2×C4—10281no—
215C2×C4—10281no—
216C2×C4—10281no—
217C2×C4—10281no—
218C2×C4—10281no—
219C2×C4—10281no—
220C2×C4D4×D25711yescharacteristic
221C2×C4C2×C22⋊D25711yescharacteristic
222C2×C4C2×D102811yescharacteristic
223C2×C4#1686161811yescharacteristic
224C2×C4#1686161911yescharacteristic
225C2×C4#1686161911yescharacteristic
226C8—22no—
227C8—10281no—
228C22—5142no—
229C22—5142no—
230C22—5142no—
231C22—10281no—
232C22—10281no—
233C22—10281no—
234C22—10281no—
235C22#2255156011yescharacteristic
236C22#2255156711yescharacteristic
237C22#2255157711yescharacteristic
238C22SD16×D25711yescharacteristic
239C22#2255161311yescharacteristic
240C22#2255162111yescharacteristic
241C22#2255162111yescharacteristic
242C4—42no—
243C4—10281no—
244C4—10282no—
245C4—81no—
246C4—20561no—
247C4#2255156611yescharacteristic
248C4#2255156711yescharacteristic
249C4#2255157711yescharacteristic
250C4#2255158111yescharacteristic
251C2—10282no—
252C2#3065665411yescharacteristic
253C2#3065687411yescharacteristic
254C2#3065689611yescharacteristic
255C2#3065696711yescharacteristic
256C2#3065699611yescharacteristic
257C2#3065706911yescharacteristic
258C2#3065708111yescharacteristic
subgroupG/H
centerC23D4×D257
commutator, cosocleC22×C4×C257C23
FrattiniC22×C4C22×D257
FittingC2×#106×C257C2
socleC23×C257C2×D4
prime radicalC23×D257D4
hypercenterC2×#106D257
hypocenterC257#2035
LCSG ▷ C22×C4×C257 ▷ C2×C257 ▷ C257
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#106

Complements

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

NH ≅ G/Ntype
2. C2×#22551517C2semidirect
3. C2×#22551515C2semidirect
6. #30656499C2semidirect
7. C2×#106×C257C2semidirect
129. C257#2035semidirect
132. C2×#106D257semidirect

Maximal subgroups

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

subgroupindex
C2×#225515722
C2×#225515172
C2×#225515152
C2×#225515042
#306565942
#306564992
C2×#106×C2572
#2035257