Finite Group Databasebeta

← #45667447

Subgroups — #45667447

Order 32896.

496 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22551585C211yesmaximal, characteristic
2#30656632C211yesmaximal, characteristic, complemented
3#30656631C211yesmaximal, characteristic
4#30656625C211yesmaximal, characteristic
5#30656624C211yesmaximal, characteristic, complemented
6#30656624C211yesmaximal, characteristic, complemented
7#30656622C211yesmaximal, characteristic, complemented
8#30656612C211yesmaximal, characteristic
9C4⋊C4×Dic257C211yesmaximal, characteristic, complemented
10C4×#16861597C211yesmaximal, characteristic, complemented
11C4×C4⋊Dic257C211yesmaximal, characteristic, complemented
12C22⋊C4×Dic257C211yesmaximal, characteristic
13C22⋊C4×Dic257C211yesmaximal, characteristic
14#30656582C211yesmaximal, characteristic, complemented
15#388×C257C211yesmaximal, characteristic
16C23×Dic257C2211yescharacteristic
17#22551585C418yescomplemented
18C2×C4⋊Dic257C2211yescharacteristic
19C2×C4⋊Dic257C2211yescharacteristic
20C2×C4⋊Dic257C2211yescharacteristic
21C2×#16861597C2211yescharacteristic
22C2×#16861597C2211yescharacteristic
23C2×#16861597C2211yescharacteristic
24C2×#16861597C2211yescharacteristic
25C2×C22⋊Dic257C2211yescharacteristic
26C2×C22⋊Dic257C2211yescharacteristic
27C2×C22⋊Dic257C2211yescharacteristic
28C2×C4×Dic257C2211yescharacteristic
29C2×C4×Dic257C2211yescharacteristic
30C2×C4×Dic257C2211yescharacteristic
31C2×C4×Dic257C2211yescharacteristic
32C2×C4×Dic257C2211yescharacteristic
33C2×C4×Dic257C2211yescharacteristic
34C2×C4×Dic257C2211yescharacteristic
35#22551499C2211yescharacteristic
36#22551499C2211yescharacteristic
37#22551499C2211yescharacteristic
38#22551499C2211yescharacteristic
39#22551499C2211yescharacteristic
40#22551499C2211yescharacteristic
41#22551499C2211yescharacteristic
42#22551499C2211yescharacteristic
43#22551499C2211yescharacteristic
44#22551499C2211yescharacteristic
45C2×C4⋊C4×C257C2211yescharacteristic
46C2×C22⋊C4×C257C2211yescharacteristic
47C2×C22⋊C4×C257C2211yescharacteristic
48C2×C22⋊C4×C257C2211yescharacteristic
49C2×C42×C257C2211yescharacteristic
50#95×C257C2211yescharacteristic
51#95×C257C2211yescharacteristic
52C22×Dic257—22no—
53C22×Dic257—22no—
54C22×Dic257—41no—
55C22×Dic257C2311yescharacteristic
56C22×Dic257C2311yescharacteristic
57C22×Dic257C2311yescharacteristic
58C22×Dic257C2311yescharacteristic
59C22×Dic257C2311yescharacteristic
60C22×Dic257C2311yescharacteristic
61C22×Dic257C2311yescharacteristic
62C22×Dic257C2311yescharacteristic
63C4⋊Dic257—22no—
64C4⋊Dic257—22no—
65C4⋊Dic257C2×C414yescomplemented
66#16861597—22no—
67#16861597—41no—
68#16861597C2×C414yescomplemented
69#16861597C2×C414yescomplemented
70C22⋊Dic257—41no—
71C22⋊Dic257C2×C414yes—
72C22⋊Dic257C2×C414yes—
73C4×Dic257—22no—
74C4×Dic257—22no—
75C4×Dic257—22no—
76C4×Dic257—41no—
77C4×Dic257—41no—
78C4×Dic257—41no—
79C4×Dic257C2×C414yescomplemented
80C24×C257C2311yescharacteristic
81C22×C4×C257C2311yescharacteristic
82C22×C4×C257C2311yescharacteristic
83C22×C4×C257C2311yescharacteristic
84C22×C4×C257C2311yescharacteristic
85C22×C4×C257C2311yescharacteristic
86C22×C4×C257C2311yescharacteristic
87C4⋊C4×C257—22no—
88C22⋊C4×C257—22no—
89C22⋊C4×C257—22no—
90C22⋊C4×C257C2×C414yes—
91C42×C257—22no—
92C2×Dic257—21no—
93C2×Dic257—21no—
94C2×Dic257—21no—
95C2×Dic257—21no—
96C2×Dic257—21no—
97C2×Dic257—21no—
98C2×Dic257—21no—
99C2×Dic257—21no—
100C2×Dic257—21no—
101C2×Dic257—21no—
102C2×Dic257—21no—
103C2×Dic257—21no—
104C2×Dic257—21no—
105C2×Dic257—21no—
106C2×Dic257—21no—
107C2×Dic257—21no—
108C2×Dic257—21no—
109C2×Dic257—21no—
110C2×Dic257—44no—
111C2×Dic257C22×C412yes—
112C2×Dic257C22×C412yes—
113C2×Dic257C22×C412yes—
114C2×Dic257C22×C412yes—
115C2×Dic257#4112yes—
116C2×Dic257#4112yes—
117C23×C257—21no—
118C23×C257—21no—
119C23×C257—21no—
120C23×C257—21no—
121C23×C257—21no—
122C23×C257—21no—
123C23×C257C22×C412yes—
124C23×C257C2411yescharacteristic
125C2×C4×C257—21no—
126C2×C4×C257—21no—
127C2×C4×C257—21no—
128C2×C4×C257—21no—
129C2×C4×C257—21no—
130C2×C4×C257—21no—
131C2×C4×C257—21no—
132C2×C4×C257—21no—
133C2×C4×C257—21no—
134C2×C4×C257—21no—
135C2×C4×C257—21no—
136C2×C4×C257—21no—
137C2×C4×C257—21no—
138C2×C4×C257—21no—
139C2×C4×C257C22×C412yes—
140C2×C4×C257C22×C412yes—
141C2×C4×C257#4112yes—
142C2×C4×C257#4112yes—
143C22×C257—22no—
144C22×C257—22no—
145C22×C257—22no—
146C22×C257—41no—
147C22×C257—41no—
148C22×C257—41no—
149C22×C257—41no—
150C22×C257C23×C411yescharacteristic
151C22×C257C2×#4111yescharacteristic
152C22×C257C2×#4111yescharacteristic
153C22×C257C2×#4111yescharacteristic
154C22×C257C2×#4111yescharacteristic
155C22×C257H(5,𝔽2)11yescharacteristic
156C22×C2572−1+411yescharacteristic
157C4×C257—22no—
158C4×C257—22no—
159C4×C257—22no—
160C4×C257—22no—
161C4×C257—41no—
162C4×C257—41no—
163Dic257—22no—
164Dic257—22no—
165Dic257—22no—
166Dic257—22no—
167Dic257—22no—
168Dic257—22no—
169Dic257—41no—
170Dic257—41no—
171C2×C257—42no—
172C2×C257C4×#4111yescharacteristic
173C2×C257C4×#4111yescharacteristic
174C2×C257#52011yescharacteristic
175C2×C257#55111yescharacteristic
176C2×C257#55211yescharacteristic
177C2×C257#55311yescharacteristic
178C2×C257#55611yescharacteristic
179C257#237311yescharacteristic, complemented, Sylow
180#2373—2571nomaximal, Sylow
181C2×#126—2571no—
182#389—2571no—
183#388—2571no—
184#388—2571no—
185#388D25711yescharacteristic
186#387—2571no—
187#387—2571no—
188#387—2571no—
189#385—2571no—
190C4×C4⋊C4—2571no—
191C4×C4⋊C4—2571no—
192C4×C4⋊C4—2571no—
193C4×C22⋊C4—2571no—
194C4×C22⋊C4—2571no—
195#376—2571no—
196C23×C4—2571no—
197#126—2578no—
198C2×C4⋊C4—2571no—
199C2×C4⋊C4—2571no—
200C2×C4⋊C4—2571no—
201C2×C4⋊C4—2571no—
202C2×C4⋊C4—2571no—
203C2×C4⋊C4—2571no—
204C2×C4⋊C4—2571no—
205C2×C4⋊C4C2×D25711yescharacteristic
206C2×C22⋊C4—2571no—
207C2×C22⋊C4—2571no—
208C2×C22⋊C4—2571no—
209C2×C22⋊C4C2×D25711yescharacteristic
210C2×C22⋊C4C2×D25711yescharacteristic
211C2×C22⋊C4C2×D25711yescharacteristic
212C2×C42—2571no—
213C2×C42—2571no—
214C2×C42—2571no—
215C2×C42—2571no—
216C2×C42—2571no—
217C2×C42—2571no—
218C2×C42—2571no—
219C2×C42C2×D25711yescharacteristic
220#95—2571no—
221#95—2571no—
222#95—2571no—
223#95—2571no—
224#95—2571no—
225#95—2571no—
226#95—2571no—
227#95—2571no—
228#95—2571no—
229#95—2571no—
230#95C2×D25711yescharacteristic
231#95C2×D25711yescharacteristic
232C24C22×D25711yescharacteristic
233C22×C4—2571no—
234C22×C4—2571no—
235C22×C4—2571no—
236C22×C4—2571no—
237C22×C4—2571no—
238C22×C4—2571no—
239C22×C4—2571no—
240C22×C4—2571no—
241C22×C4—5142no—
242C22×C4—5142no—
243C22×C4—10281no—
244C22×C4C22×D25711yescharacteristic
245C22×C4C22×D25711yescharacteristic
246C22×C4C22×D25711yescharacteristic
247C22×C4C22×D25711yescharacteristic
248C22×C4C22×D25711yescharacteristic
249C22×C4C22×D25711yescharacteristic
250C4⋊C4—2574no—
251C4⋊C4—2574no—
252C4⋊C4—2574no—
253C4⋊C4—22no—
254C4⋊C4—5142no—
255C4⋊C4—5142no—
256C4⋊C4—5142no—
257C4⋊C4—10281no—
258C22⋊C4—2574no—
259C22⋊C4—2574no—
260C22⋊C4—22no—
261C22⋊C4—22no—
262C22⋊C4—10281no—
263C22⋊C4C4×D25714yes—
264C42—2574no—
265C42—22no—
266C42—5142no—
267C42—5142no—
268C42—5142no—
269C42—10281no—
270C42—10281no—
271C42—10281no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23C2×C4×D25712yes—
279C23C23×D25711yescharacteristic
280C2×C4—2572no—
281C2×C4—2572no—
282C2×C4—2572no—
283C2×C4—2572no—
284C2×C4—2572no—
285C2×C4—2572no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—5141no—
301C2×C4—5141no—
302C2×C4—5141no—
303C2×C4—5141no—
304C2×C4—5141no—
305C2×C4—5141no—
306C2×C4—5141no—
307C2×C4—5141no—
308C2×C4—5141no—
309C2×C4—5141no—
310C2×C4—5141no—
311C2×C4—5141no—
312C2×C4—5141no—
313C2×C4—5141no—
314C2×C4—5141no—
315C2×C4—5141no—
316C2×C4—5141no—
317C2×C4—5141no—
318C2×C4—10284no—
319C2×C4C2×C4×D25712yes—
320C2×C4C2×C4×D25712yes—
321C2×C4#1686161712yes—
322C2×C4#1686161712yes—
323C22—22no—
324C22—22no—
325C22—22no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22C22×C4×D25711yescharacteristic
331C22#41×D25711yescharacteristic
332C22#41×D25711yescharacteristic
333C22C2×#1686161711yescharacteristic
334C22C2×#1686161711yescharacteristic
335C22#2255163811yescharacteristic
336C22#2255163911yescharacteristic
337C4—22no—
338C4—22no—
339C4—22no—
340C4—22no—
341C4—5142no—
342C4—5142no—
343C4—5142no—
344C4—5142no—
345C4—5142no—
346C4—5142no—
347C4—41no—
348C4—41no—
349C4—10281no—
350C4—10281no—
351C2—42no—
352C2C4×#1686161711yescharacteristic
353C2C4×#1686161711yescharacteristic
354C2#3065721711yescharacteristic
355C2#3065741211yescharacteristic
356C2#3065742411yescharacteristic
357C2#3065743911yescharacteristic
358C2#3065745611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D257
commutatorC22×C257C23×C4
Fitting#388×C257C2
socle, cosocleC23×C257C24
prime radicalC24×C257C23
hypercenter#388D257
hypocenterC257#2373
LCSG ▷ C22×C257 ▷ C257
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. #30656632C2semidirect
5. #30656624C2semidirect
6. #30656624C2semidirect
7. #30656622C2semidirect
9. C4⋊C4×Dic257C2semidirect
10. C4×#16861597C2semidirect
11. C4×C4⋊Dic257C2semidirect
14. #30656582C2semidirect
17. #22551585C4semidirect
65. C4⋊Dic257C2×C4semidirect
68. #16861597C2×C4semidirect
69. #16861597C2×C4semidirect
79. C4×Dic257C2×C4semidirect
179. C257#2373semidirect

Maximal subgroups

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

subgroupindex
C2×#225515852
#306566322
#306566312
#306566252
#306566242
#306566222
#306566122
C4⋊C4×Dic2572
C4×#168615972
C4×C4⋊Dic2572
C22⋊C4×Dic2572
#306565822
#388×C2572
#2373257