Finite Group Databasebeta

← #45665324

Subgroups — #45665324

Order 32896.

444 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 91 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
1C22×#16861606C211yesmaximal, characteristic, complemented
2C2×#22551572C211yesmaximal, characteristic, complemented
3C2×#22551563C211yesmaximal, characteristic
4C2×#22551517C211yesmaximal, characteristic, complemented
5C2×#22551504C211yesmaximal, characteristic, complemented
6#30656499C211yesmaximal, characteristic, complemented
7C2×#102×C257C211yesmaximal, characteristic, complemented
8C22×Dic514C2211yescharacteristic
9C22×C22⋊D257—21no—
10C22×C4×D257C2211yescharacteristic
11C2×#16861606—22no—
12C2×#16861606—22no—
13C2×#16861606—22no—
14C22×#14523339C2211yescharacteristic
15#22551572—22no—
16#22551572—22no—
17#22551563—22no—
18#22551563—22no—
19C2×C4⋊Dic257—21no—
20C2×C4⋊Dic257C2211yescharacteristic
21C2×C22⋊Dic257—21no—
22C2×#16861595—21no—
23#22551517—22no—
24#22551504—22no—
25C22×D4×C257C2211yescharacteristic
26C22×C8×C257C2211yescharacteristic
27C2×C4⋊C4×C257C2211yescharacteristic
28#102×C257—22no—
29C23×D257D411yescharacteristic
30C2×Dic514—21no—
31C2×Dic514—21no—
32C2×Dic514—21no—
33C2×Dic514—21no—
34C2×Dic514—21no—
35C2×Dic514D412yescomplemented
36C2×C22⋊D257—24no—
37C2×C22⋊D257—42no—
38C2×C22⋊D257—42no—
39C22×Dic257—21no—
40C22×Dic257—21no—
41C22×Dic257D411yescharacteristic
42C2×C4×D257—21no—
43C2×C4×D257—21no—
44C2×C4×D257—21no—
45C2×C4×D257—21no—
46C2×C4×D257—21no—
47C2×C4×D257—21no—
48#16861606—28no—
49C2×#14523339—21no—
50C2×#14523339—21no—
51C2×#14523339D412yescomplemented
52C4⋊Dic257—24no—
53C4⋊Dic257—21no—
54C4⋊Dic257D412yes—
55C22⋊Dic257—24no—
56#16861595—24no—
57C24×C257—21no—
58C2×D4×C257—21no—
59C2×D4×C257—21no—
60C2×D4×C257—21no—
61C2×D4×C257—21no—
62C2×D4×C257—21no—
63C2×D4×C257D412yes—
64C22×C4×C257—21no—
65C22×C4×C257C2311yescharacteristic
66C2×C8×C257—21no—
67C2×C8×C257—21no—
68C2×C8×C257—21no—
69C4⋊C4×C257—21no—
70C4⋊C4×C257D412yescomplemented
71C22×D257—21no—
72C22×D257—21no—
73C22×D257—21no—
74C22×D257—21no—
75C22×D257—21no—
76C22×D257—21no—
77C22×D257SD1612yescomplemented
78Dic514—22no—
79Dic514—22no—
80Dic514—41no—
81Dic514—41no—
82C22⋊D257—48no—
83C2×Dic257—22no—
84C2×Dic257—21no—
85C2×Dic257—22no—
86C2×Dic257—21no—
87C2×Dic257—21no—
88C2×Dic257—41no—
89C2×Dic257—41no—
90C2×Dic257—41no—
91C2×Dic257—41no—
92C4×D257—41no—
93C4×D257—41no—
94C4×D257—41no—
95C4×D257—41no—
96#14523339—22no—
97C23×C257—22no—
98C23×C257—41no—
99C23×C257—41no—
100C23×C257—41no—
101C23×C257—41no—
102C23×C257—41no—
103C23×C257—41no—
104C23×C257C2×D411yescharacteristic
105D4×C257—22no—
106D4×C257—22no—
107D4×C257—41no—
108D4×C257—41no—
109C2×C4×C257—22no—
110C2×C4×C257—41no—
111C2×C4×C257—41no—
112C2×C4×C257C2×D411yescharacteristic
113C2×C4×C257C2×D411yescharacteristic
114C2×C4×C257C2×D411yescharacteristic
115C2×C4×C257C2×D411yescharacteristic
116C2×C4×C257C2×D411yescharacteristic
117C2×C4×C257#4111yescharacteristic
118C8×C257—41no—
119C22×C257—42no—
120C22×C257—42no—
121C22×C257—42no—
122C22×C257—81no—
123C22×C257—81no—
124C22×C257—81no—
125C22×C257—81no—
126C22×C257#12011yescharacteristic
127C22×C257#12011yescharacteristic
128C22×C257#12111yescharacteristic
129C22×C257C2×SD1611yescharacteristic
130C22×C257#13511yescharacteristic
131C22×C257AGL(1,ℤ/8)11yescharacteristic
132C22×C257#13711yescharacteristic
133C2×D257—22no—
134C2×D257—22no—
135C2×D257—22no—
136C2×D257—41no—
137C2×D257—41no—
138C2×D257—41no—
139C2×D257—41no—
140C4×C257—42no—
141C4×C257#12011yescharacteristic
142C4×C257#12111yescharacteristic
143C4×C257#12111yescharacteristic
144C4×C257#12711yescharacteristic
145Dic257—41no—
146Dic257—42no—
147Dic257—42no—
148C2×C257—82no—
149C2×C257#39211yescharacteristic
150C2×C257#44811yescharacteristic
151C2×C257#44911yescharacteristic
152C2×C257#45011yescharacteristic
153C2×C257#45211yescharacteristic
154C2×C257#46511yescharacteristic
155C2×C257#46811yescharacteristic
156D257—42no—
157C257#200011yescharacteristic, complemented, Sylow
158#2000—2571nomaximal, Sylow
159C22×SD16—2571no—
160C2×C22⋊Q8—2571no—
161C2×#121—2571no—
162C2×#103—2571no—
163C2×#102D25711yescharacteristic, complemented
164C2×C22⋊C8—2571no—
165#340—2571no—
166C22×Q8—2571no—
167C22×D4—5141no—
168C22×D4C2×D25711yescharacteristic
169C23×C4—2571no—
170C2×SD16—5142no—
171C2×SD16—5142no—
172C2×SD16—5142no—
173C22×C8—2571no—
174C22×C8C2×D25711yescharacteristic
175C22⋊Q8—5142no—
176C22⋊Q8—5142no—
177#121—5142no—
178#121—5142no—
179C2×C4⋊C4—2571no—
180C2×C4⋊C4—5141no—
181C2×C4⋊C4C2×D25711yescharacteristic
182C2×C22⋊C4—5141no—
183C2×C22⋊C4—5141no—
184#103—5142no—
185#102—22no—
186C22⋊C8—5142no—
187C24—2571no—
188C24—21no—
189C2×Q8—2572no—
190C2×Q8—5141no—
191C2×Q8—5141no—
192C2×Q8—5141no—
193C2×Q8—5141no—
194C2×Q8—5141no—
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—5144no—
201C2×D4—10282no—
202C2×D4—10282no—
203C2×D4D102812yes—
204C22×C4—2571no—
205C22×C4—21no—
206C22×C4—5141no—
207C22×C4—5141no—
208C22×C4—5141no—
209C22×C4—5141no—
210C22×C4—5141no—
211C22×C4—5141no—
212C22×C4—5141no—
213C22×C4—5141no—
214C22×C4C22×D25711yescharacteristic
215SD16—5148no—
216C2×C8—2572no—
217C2×C8—21no—
218C2×C8—21no—
219C2×C8—21no—
220C2×C8—5141no—
221C2×C8—5141no—
222C4⋊C4—2572no—
223C4⋊C4—21no—
224C4⋊C4—5144no—
225C4⋊C4—5141no—
226C4⋊C4C22⋊D25712yescomplemented
227C22⋊C4—5144no—
228C22⋊C4—5144no—
229C23—2572no—
230C23—22no—
231C23—5141no—
232C23—5141no—
233C23—5141no—
234C23—5141no—
235C23—5141no—
236C23—5141no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23D4×D25711yescharacteristic
244Q8—5142no—
245Q8—5142no—
246Q8—10281no—
247Q8—10281no—
248D4—22no—
249D4—22no—
250D4—41no—
251D4—41no—
252D4—10288no—
253C2×C4—22no—
254C2×C4—5142no—
255C2×C4—5141no—
256C2×C4—5141no—
257C2×C4—5142no—
258C2×C4—5141no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—10281no—
262C2×C4—10281no—
263C2×C4—10281no—
264C2×C4—10281no—
265C2×C4—10281no—
266C2×C4—10281no—
267C2×C4—10281no—
268C2×C4—10281no—
269C2×C4D4×D25711yescharacteristic
270C2×C4D4×D25711yescharacteristic
271C2×C4D4×D25711yescharacteristic
272C2×C4C2×C22⋊D25711yescharacteristic
273C2×C4C2×D102811yescharacteristic
274C2×C4#1686161811yescharacteristic
275C8—5142no—
276C8—41no—
277C22—5142no—
278C22—5142no—
279C22—5142no—
280C22—42no—
281C22—42no—
282C22—42no—
283C22—10281no—
284C22—10281no—
285C22—10281no—
286C22—10281no—
287C22—81no—
288C22—81no—
289C22—81no—
290C22—81no—
291C22#2255156011yescharacteristic
292C22#2255156211yescharacteristic
293C22#2255156511yescharacteristic
294C22SD16×D25711yescharacteristic
295C22#2255161111yescharacteristic
296C22#2255161611yescharacteristic
297C22#2255162111yescharacteristic
298C4—42no—
299C4—10282no—
300C4—10281no—
301C4—10282no—
302C4#2255156211yescharacteristic
303C4#2255156511yescharacteristic
304C4#2255156611yescharacteristic
305C4#2255158811yescharacteristic
306C2—10282no—
307C2—82no—
308C2#3065664611yescharacteristic
309C2#3065687411yescharacteristic
310C2#3065687911yescharacteristic
311C2#3065688911yescharacteristic
312C2#3065690011yescharacteristic
313C2#3065696511yescharacteristic
314C2#3065698211yescharacteristic
subgroupG/H
centerC23D4×D257
commutator, cosocleC22×C4×C257C23
FrattiniC22×C4C22×D257
FittingC2×#102×C257C2
socleC23×C257C2×D4
prime radicalC2×#22551563C2
hypercenterC2×#102D257
hypocenterC257#2000
LCSG ▷ C22×C4×C257 ▷ C2×C257 ▷ C257
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. C22×#16861606C2semidirect
2. C2×#22551572C2semidirect
4. C2×#22551517C2semidirect
5. C2×#22551504C2semidirect
6. #30656499C2semidirect
7. C2×#102×C257C2semidirect
35. C2×Dic514D4semidirect
51. C2×#14523339D4semidirect
70. C4⋊C4×C257D4semidirect
77. C22×D257SD16semidirect
157. C257#2000semidirect
163. C2×#102D257semidirect
226. C4⋊C4C22⋊D257semidirect

Maximal subgroups

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

subgroupindex
C22×#168616062
C2×#225515722
C2×#225515632
C2×#225515172
C2×#225515042
#306564992
C2×#102×C2572
#2000257