Finite Group Databasebeta

← #45667174

Subgroups — #45667174

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×#22551548C211yesmaximal, characteristic
2#30656621C211yesmaximal, characteristic, complemented
3#30656621C211yesmaximal, characteristic, complemented
4#30656619C211yesmaximal, characteristic, complemented
5#30656619C211yesmaximal, characteristic, complemented
6#30656610C211yesmaximal, characteristic
7#30656609C211yesmaximal, characteristic
8#30656604C211yesmaximal, characteristic, complemented
9#30656601C211yesmaximal, characteristic, complemented
10#30656598C211yesmaximal, characteristic
11C4×#16861597C211yesmaximal, characteristic, complemented
12C22⋊C4×Dic257C211yesmaximal, characteristic
13#30656583C211yesmaximal, characteristic, complemented
14C2×#22551499C211yesmaximal, characteristic
15C4×C22⋊C4×C257C211yesmaximal, characteristic
16C23×Dic257C2211yescharacteristic
17C23×Dic257C2211yescharacteristic
18#22551548C418yescomplemented
19C2×#16861597C2211yescharacteristic
20C2×#16861597C2211yescharacteristic
21C2×#16861597C2211yescharacteristic
22C2×#16861597C2211yescharacteristic
23C2×#16861597C2211yescharacteristic
24C2×#16861597C2211yescharacteristic
25C2×C22⋊Dic257C2211yescharacteristic
26C2×C22⋊Dic257C2211yescharacteristic
27C2×C4×Dic257C2211yescharacteristic
28C2×C4×Dic257C2211yescharacteristic
29C2×C4×Dic257C2211yescharacteristic
30C2×C4×Dic257C2211yescharacteristic
31#22551499C2211yescharacteristic
32#22551499C2211yescharacteristic
33#22551499C2211yescharacteristic
34#22551499C2211yescharacteristic
35#22551499C2211yescharacteristic
36#22551499C2211yescharacteristic
37#22551499C2211yescharacteristic
38#22551499C2211yescharacteristic
39#22551499C2211yescharacteristic
40#22551499C2211yescharacteristic
41#22551499C2211yescharacteristic
42#22551499C2211yescharacteristic
43#22551499C2211yescharacteristic
44#22551499C2211yescharacteristic
45C23×C4×C257C2211yescharacteristic
46C2×C22⋊C4×C257C2211yescharacteristic
47C2×C22⋊C4×C257C2211yescharacteristic
48C2×C42×C257C2211yescharacteristic
49C2×C42×C257C2211yescharacteristic
50#95×C257C2211yescharacteristic
51#95×C257C2211yescharacteristic
52C22×Dic257—22no—
53C22×Dic257—22no—
54C22×Dic257—22no—
55C22×Dic257—22no—
56C22×Dic257—41no—
57C22×Dic257C2×C414yes—
58C22×Dic257C2311yescharacteristic
59C22×Dic257C2311yescharacteristic
60C22×Dic257C2311yescharacteristic
61C22×Dic257C2311yescharacteristic
62C22×Dic257C2311yescharacteristic
63C22×Dic257C2311yescharacteristic
64C22×Dic257C2311yescharacteristic
65C22×Dic257C2311yescharacteristic
66#16861597—22no—
67#16861597—22no—
68#16861597—41no—
69#16861597—41no—
70#16861597C2×C414yescomplemented
71#16861597C2×C414yescomplemented
72C22⋊Dic257—22no—
73C22⋊Dic257C2×C414yes—
74C4×Dic257—22no—
75C4×Dic257—41no—
76C4×Dic257C2×C414yescomplemented
77C4×Dic257C2×C414yescomplemented
78C24×C257C2311yescharacteristic
79C22×C4×C257—22no—
80C22×C4×C257—22no—
81C22×C4×C257—41no—
82C22×C4×C257C2311yescharacteristic
83C22×C4×C257C2311yescharacteristic
84C22×C4×C257C2311yescharacteristic
85C22×C4×C257C2311yescharacteristic
86C22×C4×C257C2311yescharacteristic
87C22×C4×C257C2311yescharacteristic
88C22⋊C4×C257—22no—
89C22⋊C4×C257C2×C414yes—
90C42×C257—22no—
91C42×C257—41no—
92C2×Dic257—28no—
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—21no—
111C2×Dic257—21no—
112C2×Dic257—44no—
113C2×Dic257C22×C412yes—
114C2×Dic257C22×C412yes—
115C2×Dic257C22×C412yes—
116C2×Dic257C22×C412yes—
117C2×Dic257#4112yes—
118C23×C257—21no—
119C23×C257—21no—
120C23×C257—21no—
121C23×C257—21no—
122C23×C257C22×C412yes—
123C23×C257#4112yes—
124C23×C257#4112yes—
125C23×C257C2411yescharacteristic
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×C257—44no—
142C2×C4×C257C22×C412yes—
143C2×C4×C257C22×C412yes—
144C2×C4×C257#4112yes—
145C22×C257—22no—
146C22×C257—22no—
147C22×C257—22no—
148C22×C257—22no—
149C22×C257—22no—
150C22×C257—22no—
151C22×C257#11714yes—
152C22×C257C23×C411yescharacteristic
153C22×C257C2×#4111yescharacteristic
154C22×C257C2×#4111yescharacteristic
155C22×C257C2×#4111yescharacteristic
156C22×C257C2×#4111yescharacteristic
157C22×C257H(5,𝔽2)11yescharacteristic
158C22×C2572−1+411yescharacteristic
159C4×C257—22no—
160C4×C257—22no—
161C4×C257—22no—
162C4×C257—41no—
163C4×C257—41no—
164C4×C257—41no—
165Dic257—22no—
166Dic257—22no—
167Dic257—22no—
168Dic257—22no—
169Dic257—22no—
170Dic257—41no—
171Dic257—41no—
172Dic257—41no—
173C2×C257—24no—
174C2×C257C2×#11711yescharacteristic
175C2×C257C4×#4111yescharacteristic
176C2×C257#52011yescharacteristic
177C2×C257#55111yescharacteristic
178C2×C257#55111yescharacteristic
179C2×C257#55211yescharacteristic
180C2×C257#55211yescharacteristic
181C257#234311yescharacteristic, complemented, Sylow
182#2343—2571nomaximal, Sylow
183C2×#117—2571no—
184#387—2571no—
185#387—2571no—
186#387—2571no—
187#387—2571no—
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⋊C4D25711yescharacteristic
196#376—2571no—
197C2×#95—2571no—
198C23×C4—2571no—
199C23×C4—2571no—
200C23×C4C2×D25711yescharacteristic
201#117—2578no—
202C2×C4⋊C4—2571no—
203C2×C4⋊C4—2571no—
204C2×C4⋊C4—2571no—
205C2×C4⋊C4—2571no—
206C2×C4⋊C4—2571no—
207C2×C4⋊C4—2571no—
208C2×C22⋊C4—2571no—
209C2×C22⋊C4—2571no—
210C2×C22⋊C4C2×D25711yescharacteristic
211C2×C22⋊C4C2×D25711yescharacteristic
212C2×C42—2571no—
213C2×C42—2571no—
214C2×C42—2571no—
215C2×C42—2571no—
216C2×C42C2×D25711yescharacteristic
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#95—2571no—
231#95—2571no—
232#95C2×D25711yescharacteristic
233#95C2×D25711yescharacteristic
234C24C22×D25711yescharacteristic
235C22×C4—2574no—
236C22×C4—2571no—
237C22×C4—2571no—
238C22×C4—2571no—
239C22×C4—2571no—
240C22×C4—2571no—
241C22×C4—2571no—
242C22×C4—2571no—
243C22×C4—2571no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—5142no—
247C22×C4—5142no—
248C22×C4—5142no—
249C22×C4—5142no—
250C22×C4—41no—
251C22×C4—10281no—
252C22×C4C22×D25711yescharacteristic
253C22×C4C22×D25711yescharacteristic
254C22×C4C22×D25711yescharacteristic
255C22×C4C22×D25711yescharacteristic
256C22×C4C22×D25711yescharacteristic
257C22×C4C22×D25711yescharacteristic
258C4⋊C4—2574no—
259C4⋊C4—2574no—
260C4⋊C4—5142no—
261C4⋊C4—5142no—
262C4⋊C4—10281no—
263C4⋊C4—10281no—
264C22⋊C4—2574no—
265C22⋊C4—22no—
266C22⋊C4—5142no—
267C22⋊C4C4×D25714yes—
268C42—2574no—
269C42—2574no—
270C42—22no—
271C42—5142no—
272C42—41no—
273C42—10281no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23C2×C4×D25712yes—
279C23#1686161812yes—
280C23#1686161812yes—
281C23C23×D25711yescharacteristic
282C2×C4—2572no—
283C2×C4—2572no—
284C2×C4—2572no—
285C2×C4—2572no—
286C2×C4—2572no—
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—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—5148no—
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—5141no—
322C2×C4—44no—
323C2×C4—10284no—
324C2×C4C2×C4×D25712yes—
325C2×C4C2×C4×D25712yes—
326C2×C4#1686161712yes—
327C22—22no—
328C22—22no—
329C22—22no—
330C22—22no—
331C22—22no—
332C22—22no—
333C22#2255154914yes—
334C22C22×C4×D25711yescharacteristic
335C22#41×D25711yescharacteristic
336C22C2×#1686161711yescharacteristic
337C22C2×#1686161811yescharacteristic
338C22C2×#1686161811yescharacteristic
339C22#2255163711yescharacteristic
340C22#2255163911yescharacteristic
341C4—22no—
342C4—22no—
343C4—22no—
344C4—5142no—
345C4—5142no—
346C4—5142no—
347C4—5142no—
348C4—5142no—
349C4—41no—
350C4—41no—
351C4—41no—
352C4—10281no—
353C4—10281no—
354C4—10281no—
355C2—24no—
356C2C2×#2255154911yescharacteristic
357C2C4×#1686161711yescharacteristic
358C2#3065721611yescharacteristic
359C2#3065741311yescharacteristic
360C2#3065741611yescharacteristic
361C2#3065742011yescharacteristic
362C2#3065742711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D257
commutatorC22×C257C23×C4
FittingC4×C22⋊C4×C257C2
socle, cosocleC23×C257C24
prime radicalC24×C257C23
hypercenterC4×C22⋊C4D257
hypocenterC257#2343
LCSG ▷ C22×C257 ▷ C257
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #30656621C2semidirect
3. #30656621C2semidirect
4. #30656619C2semidirect
5. #30656619C2semidirect
8. #30656604C2semidirect
9. #30656601C2semidirect
11. C4×#16861597C2semidirect
13. #30656583C2semidirect
18. #22551548C4semidirect
70. #16861597C2×C4semidirect
71. #16861597C2×C4semidirect
76. C4×Dic257C2×C4semidirect
77. C4×Dic257C2×C4semidirect
181. C257#2343semidirect

Maximal subgroups

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

subgroupindex
C2×#225515482
#306566212
#306566192
#306566102
#306566092
#306566042
#306566012
#306565982
C4×#168615972
C22⋊C4×Dic2572
#306565832
C2×#225514992
C4×C22⋊C4×C2572
#2343257