Finite Group Databasebeta

← #45667175

Subgroups — #45667175

Order 32896.

532 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22551550C211yesmaximal, characteristic
2#30656623C211yesmaximal, characteristic, complemented
3#30656622C211yesmaximal, characteristic, complemented
4#30656620C211yesmaximal, characteristic, complemented
5#30656611C211yesmaximal, characteristic
6#30656608C211yesmaximal, characteristic
7#30656604C211yesmaximal, characteristic, complemented
8#30656601C211yesmaximal, characteristic, complemented
9#30656599C211yesmaximal, characteristic
10C4⋊C4×Dic257C211yesmaximal, characteristic, complemented
11C4×#16861595C211yesmaximal, characteristic
12#30656584C211yesmaximal, characteristic
13#30656583C211yesmaximal, characteristic, complemented
14#95×D257C211yesmaximal, characteristic
15#387×C257C211yesmaximal, characteristic, complemented
16C22×C4×D257C2211yescharacteristic
17C22×C4×D257C2211yescharacteristic
18C22×C4×D257C2211yescharacteristic
19#22551550C418yescomplemented
20C2×C4⋊Dic257C2211yescharacteristic
21C2×C4⋊Dic257C2211yescharacteristic
22C2×#16861597C2211yescharacteristic
23C2×#16861597C2211yescharacteristic
24C2×#16861595C2211yescharacteristic
25C2×#16861595C2211yescharacteristic
26C2×#16861595C2211yescharacteristic
27C2×#16861595C2211yescharacteristic
28C2×C4×Dic257C2211yescharacteristic
29C2×C4×Dic257C2211yescharacteristic
30C2×C4×Dic257C2211yescharacteristic
31C2×C4×Dic257C2211yescharacteristic
32C2×C4×Dic257C2211yescharacteristic
33#22551499C2211yescharacteristic
34#22551499C2211yescharacteristic
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×C4⋊C4×C257C2211yescharacteristic
47C2×C42×C257C2211yescharacteristic
48#95×C257C2211yescharacteristic
49#95×C257C2211yescharacteristic
50#95×C257C2211yescharacteristic
51#95×C257C2211yescharacteristic
52C23×D257C2311yescharacteristic
53C22×Dic257C2311yescharacteristic
54C22×Dic257C2311yescharacteristic
55C22×Dic257C2311yescharacteristic
56C22×Dic257C2311yescharacteristic
57C22×Dic257C2311yescharacteristic
58C22×Dic257C2311yescharacteristic
59C22×Dic257C2311yescharacteristic
60C2×C4×D257—22no—
61C2×C4×D257—22no—
62C2×C4×D257—22no—
63C2×C4×D257—22no—
64C2×C4×D257—22no—
65C2×C4×D257—22no—
66C2×C4×D257—41no—
67C2×C4×D257—41no—
68C2×C4×D257C2×C414yes—
69C4⋊Dic257—22no—
70C4⋊Dic257C2×C414yescomplemented
71#16861597—41no—
72#16861597—41no—
73#16861595—22no—
74#16861595—22no—
75#16861595C2×C414yes—
76#16861595C2×C414yes—
77C4×Dic257—22no—
78C4×Dic257—41no—
79C4×Dic257—41no—
80C4×Dic257C2×C414yescomplemented
81C4×Dic257C2×C414yescomplemented
82C22×C4×C257C2311yescharacteristic
83C22×C4×C257C2311yescharacteristic
84C22×C4×C257C2311yescharacteristic
85C22×C4×C257C2311yescharacteristic
86C22×C4×C257C2311yescharacteristic
87C22×C4×C257C2311yescharacteristic
88C22×C4×C257C2311yescharacteristic
89C4⋊C4×C257—22no—
90C4⋊C4×C257C2×C414yescomplemented
91C42×C257—22no—
92C22×D257—21no—
93C22×D257—21no—
94C22×D257—21no—
95C22×D257—21no—
96C22×D257C22×C412yes—
97C22×D257#4112yes—
98C22×D257#4112yes—
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—21no—
111C2×Dic257—21no—
112C2×Dic257—21no—
113C2×Dic257—21no—
114C2×Dic257—21no—
115C2×Dic257—21no—
116C2×Dic257C22×C412yes—
117C2×Dic257C22×C412yes—
118C2×Dic257C22×C412yes—
119C2×Dic257#4112yes—
120C4×D257—28no—
121C4×D257—44no—
122C4×D257—44no—
123C23×C257C2411yescharacteristic
124C2×C4×C257—21no—
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×C257—21no—
140C2×C4×C257—21no—
141C2×C4×C257C22×C412yes—
142C2×C4×C257C22×C412yes—
143C2×C4×C257C22×C412yes—
144C2×C4×C257#4112yes—
145C22×C257C23×C411yescharacteristic
146C22×C257C2×#4111yescharacteristic
147C22×C257C2×#4111yescharacteristic
148C22×C257C2×#4111yescharacteristic
149C22×C257C2×#4111yescharacteristic
150C22×C257H(5,𝔽2)11yescharacteristic
151C22×C2572−1+411yescharacteristic
152C2×D257—22no—
153C2×D257—22no—
154C2×D257—22no—
155C2×D257—22no—
156C2×D257—22no—
157C2×D257—22no—
158C2×D257#11714yes—
159C4×C257—22no—
160C4×C257—22no—
161C4×C257—22no—
162C4×C257—22no—
163C4×C257—41no—
164C4×C257—41no—
165C4×C257—41no—
166Dic257—22no—
167Dic257—22no—
168Dic257—22no—
169Dic257—22no—
170Dic257—41no—
171Dic257—41no—
172Dic257—41no—
173C2×C257C2×#11711yescharacteristic
174C2×C257C4×#4111yescharacteristic
175C2×C257#52011yescharacteristic
176C2×C257#55111yescharacteristic
177C2×C257#55111yescharacteristic
178C2×C257#55211yescharacteristic
179C2×C257#55211yescharacteristic
180D257—24no—
181C257#234311yescharacteristic, complemented, Sylow
182#2343—2571nomaximal, Sylow
183C2×#117—2571no—
184#387—2571no—
185#387—2571no—
186#387—2571no—
187#387D25711yescharacteristic, complemented
188#385—2571no—
189#385—2571no—
190#383—2571no—
191#382—2571no—
192#381—2571no—
193C4×C4⋊C4—2571no—
194C4×C22⋊C4—2571no—
195C4×C22⋊C4—2571no—
196#376—2571no—
197C2×#95—2571no—
198C23×C4—2571no—
199C23×C4—2571no—
200C23×C4—2571no—
201#117—2578no—
202C2×C4⋊C4—2571no—
203C2×C4⋊C4—2571no—
204C2×C4⋊C4—2571no—
205C2×C4⋊C4—2571no—
206C2×C4⋊C4C2×D25711yescharacteristic
207C2×C4⋊C4C2×D25711yescharacteristic
208C2×C22⋊C4—2571no—
209C2×C22⋊C4—2571no—
210C2×C22⋊C4—2571no—
211C2×C22⋊C4—2571no—
212C2×C42—2571no—
213C2×C42—2571no—
214C2×C42—2571no—
215C2×C42—2571no—
216C2×C42—2571no—
217C2×C42C2×D25711yescharacteristic
218#95—2571no—
219#95—2571no—
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
232#95C2×D25711yescharacteristic
233#95C2×D25711yescharacteristic
234C24—2571no—
235C22×C4—2571no—
236C22×C4—2571no—
237C22×C4—2571no—
238C22×C4—2571no—
239C22×C4—2574no—
240C22×C4—2571no—
241C22×C4—2571no—
242C22×C4—2571no—
243C22×C4—5142no—
244C22×C4—5142no—
245C22×C4—5142no—
246C22×C4—5142no—
247C22×C4—5142no—
248C22×C4—5142no—
249C22×C4—10281no—
250C22×C4—10281no—
251C22×C4C22×D25711yescharacteristic
252C22×C4C22×D25711yescharacteristic
253C22×C4C22×D25711yescharacteristic
254C22×C4C22×D25711yescharacteristic
255C22×C4C22×D25711yescharacteristic
256C22×C4C22×D25711yescharacteristic
257C22×C4C22×D25711yescharacteristic
258C4⋊C4—2574no—
259C4⋊C4—22no—
260C4⋊C4—5142no—
261C4⋊C4—10281no—
262C4⋊C4—10281no—
263C4⋊C4C4×D25714yescomplemented
264C22⋊C4—2574no—
265C22⋊C4—2574no—
266C22⋊C4—5142no—
267C22⋊C4—5142no—
268C42—2574no—
269C42—2574no—
270C42—22no—
271C42—5142no—
272C42—10281no—
273C42—10281no—
274C23—2572no—
275C23—2572no—
276C23—2572no—
277C23—5141no—
278C23—5141no—
279C23—5141no—
280C23—5141no—
281C23C23×D25711yescharacteristic
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—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—5148no—
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—5141no—
319C2×C4—5141no—
320C2×C4—5141no—
321C2×C4—10284no—
322C2×C4—10284no—
323C2×C4C2×C4×D25712yes—
324C2×C4C2×C4×D25712yes—
325C2×C4C2×C4×D25712yes—
326C2×C4#1686161712yes—
327C22—2574no—
328C22—5142no—
329C22—5142no—
330C22—5142no—
331C22—5142no—
332C22—5142no—
333C22—5142no—
334C22C22×C4×D25711yescharacteristic
335C22#41×D25711yescharacteristic
336C22#41×D25711yescharacteristic
337C22#41×D25711yescharacteristic
338C22C2×#1686161711yescharacteristic
339C22#2255163811yescharacteristic
340C22#2255164011yescharacteristic
341C4—22no—
342C4—22no—
343C4—22no—
344C4—22no—
345C4—5142no—
346C4—5142no—
347C4—5142no—
348C4—5142no—
349C4—41no—
350C4—41no—
351C4—41no—
352C4—10281no—
353C4—10281no—
354C4—10281no—
355C2—5144no—
356C2#117×D25711yescharacteristic
357C2C4×#1686161711yescharacteristic
358C2#3065721911yescharacteristic
359C2#3065741011yescharacteristic
360C2#3065741211yescharacteristic
361C2#3065741911yescharacteristic
362C2#3065742111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D257
commutatorC22×C257C23×C4
Fitting#387×C257C2
socle, cosocleC23×C257C24
prime radicalC23×D257C23
hypercenter#387D257
hypocenterC257#2343
LCSG ▷ C22×C257 ▷ C257
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
2. #30656623C2semidirect
3. #30656622C2semidirect
4. #30656620C2semidirect
7. #30656604C2semidirect
8. #30656601C2semidirect
10. C4⋊C4×Dic257C2semidirect
13. #30656583C2semidirect
15. #387×C257C2semidirect
19. #22551550C4semidirect
70. C4⋊Dic257C2×C4semidirect
80. C4×Dic257C2×C4semidirect
81. C4×Dic257C2×C4semidirect
90. C4⋊C4×C257C2×C4semidirect
181. C257#2343semidirect
187. #387D257semidirect
263. C4⋊C4C4×D257semidirect

Maximal subgroups

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

subgroupindex
C2×#225515502
#306566232
#306566222
#306566202
#306566112
#306566082
#306566042
#306566012
#306565992
C4⋊C4×Dic2572
C4×#168615952
#306565842
#306565832
#95×D2572
#387×C2572
#2343257