Finite Group Databasebeta

← #45666952

Subgroups — #45666952

Order 32896.

568 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22551554C211yesmaximal, characteristic
2#30656628C211yesmaximal, characteristic, complemented
3#30656626C211yesmaximal, characteristic, complemented
4#30656623C211yesmaximal, characteristic, complemented
5#30656615C211yesmaximal, characteristic
6#30656615C211yesmaximal, characteristic
7#30656611C211yesmaximal, characteristic, complemented
8#30656608C211yesmaximal, characteristic, complemented
9#30656604C211yesmaximal, characteristic, complemented
10#30656601C211yesmaximal, characteristic, complemented
11#30656599C211yesmaximal, characteristic, complemented
12#30656594C211yesmaximal, characteristic, complemented
13C4×#16861595C211yesmaximal, characteristic, complemented
14#95×D257C211yesmaximal, characteristic, complemented
15#387×C257C211yesmaximal, characteristic, complemented
16C22×D1028C2211yescharacteristic
17C22×C4×D257C2211yescharacteristic
18C22×C4×D257C2211yescharacteristic
19C22×C4×D257C2211yescharacteristic
20C22×C4×D257C2211yescharacteristic
21#22551554C418yescomplemented
22C2×C4⋊Dic257C2211yescharacteristic
23C2×#16861597C2211yescharacteristic
24C2×#16861597C2211yescharacteristic
25C2×#16861595C2211yescharacteristic
26C2×#16861595C2211yescharacteristic
27C2×#16861595C2211yescharacteristic
28C2×#16861595C2211yescharacteristic
29C2×#16861595C2211yescharacteristic
30C2×#16861595C2211yescharacteristic
31C2×#16861595C2211yescharacteristic
32C2×#16861595C2211yescharacteristic
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×C4⋊C4×C257C2211yescharacteristic
47C2×C42×C257C2211yescharacteristic
48#95×C257C2211yescharacteristic
49#95×C257C2211yescharacteristic
50#95×C257C2211yescharacteristic
51#95×C257C2211yescharacteristic
52C23×D257C2311yescharacteristic
53C23×D257C2311yescharacteristic
54C2×D1028—22no—
55C2×D1028—22no—
56C2×D1028C2×C414yes—
57C22×Dic257C2311yescharacteristic
58C22×Dic257C2311yescharacteristic
59C22×Dic257C2311yescharacteristic
60C22×Dic257C2311yescharacteristic
61C22×Dic257C2311yescharacteristic
62C22×Dic257C2311yescharacteristic
63C2×C4×D257—22no—
64C2×C4×D257—22no—
65C2×C4×D257—41no—
66C2×C4×D257—41no—
67C2×C4×D257—41no—
68C2×C4×D257—41no—
69C2×C4×D257—41no—
70C2×C4×D257—41no—
71C2×C4×D257—41no—
72C2×C4×D257—41no—
73C2×C4×D257C2×C414yes—
74C2×C4×D257C2×C414yescomplemented
75C4⋊Dic257—41no—
76#16861597—41no—
77#16861597—41no—
78#16861595—22no—
79#16861595—22no—
80#16861595—41no—
81#16861595—41no—
82#16861595—41no—
83#16861595—41no—
84#16861595C2×C414yes—
85#16861595C2×C414yescomplemented
86C4×Dic257—41no—
87C4×Dic257C2×C414yescomplemented
88C22×C4×C257C2311yescharacteristic
89C22×C4×C257C2311yescharacteristic
90C22×C4×C257C2311yescharacteristic
91C22×C4×C257C2311yescharacteristic
92C22×C4×C257C2311yescharacteristic
93C22×C4×C257C2311yescharacteristic
94C22×C4×C257C2311yescharacteristic
95C4⋊C4×C257—41no—
96C4⋊C4×C257C2×C414yescomplemented
97C42×C257—41no—
98C22×D257—21no—
99C22×D257—21no—
100C22×D257—21no—
101C22×D257—21no—
102C22×D257—21no—
103C22×D257—21no—
104C22×D257—21no—
105C22×D257—21no—
106C22×D257—21no—
107C22×D257—21no—
108C22×D257C22×C412yes—
109C22×D257C22×C412yes—
110C22×D257#4112yes—
111C22×D257#4112yes—
112D1028—28no—
113C2×Dic257—21no—
114C2×Dic257—21no—
115C2×Dic257—21no—
116C2×Dic257—21no—
117C2×Dic257—21no—
118C2×Dic257—21no—
119C2×Dic257—21no—
120C2×Dic257—21no—
121C2×Dic257—21no—
122C2×Dic257—21no—
123C2×Dic257—21no—
124C2×Dic257—21no—
125C2×Dic257—21no—
126C2×Dic257—21no—
127C2×Dic257—21no—
128C2×Dic257—21no—
129C2×Dic257C22×C412yes—
130C2×Dic257C22×C412yes—
131C4×D257—44no—
132C4×D257—44no—
133C4×D257—44no—
134C4×D257—44no—
135C23×C257C2411yescharacteristic
136C2×C4×C257—21no—
137C2×C4×C257—21no—
138C2×C4×C257—21no—
139C2×C4×C257—21no—
140C2×C4×C257—21no—
141C2×C4×C257—21no—
142C2×C4×C257—21no—
143C2×C4×C257—21no—
144C2×C4×C257—21no—
145C2×C4×C257—21no—
146C2×C4×C257—21no—
147C2×C4×C257—21no—
148C2×C4×C257—21no—
149C2×C4×C257—21no—
150C2×C4×C257—21no—
151C2×C4×C257—21no—
152C2×C4×C257—21no—
153C2×C4×C257—21no—
154C2×C4×C257C22×C412yes—
155C2×C4×C257C22×C412yes—
156C2×C4×C257C22×C412yes—
157C22×C257C23×C411yescharacteristic
158C22×C257C2×#4111yescharacteristic
159C22×C257C2×#4111yescharacteristic
160C22×C257H(5,𝔽2)11yescharacteristic
161C22×C257H(5,𝔽2)11yescharacteristic
162C22×C257H(5,𝔽2)11yescharacteristic
163C22×C2572−1+411yescharacteristic
164C2×D257—22no—
165C2×D257—22no—
166C2×D257—22no—
167C2×D257—22no—
168C2×D257—22no—
169C2×D257—22no—
170C2×D257—22no—
171C2×D257—22no—
172C2×D257—22no—
173C2×D257—41no—
174C2×D257—41no—
175C2×D257—41no—
176C2×D257—41no—
177C2×D257#11714yes—
178C4×C257—22no—
179C4×C257—22no—
180C4×C257—22no—
181C4×C257—41no—
182C4×C257—41no—
183C4×C257—41no—
184C4×C257—41no—
185Dic257—22no—
186Dic257—22no—
187Dic257—41no—
188Dic257—41no—
189Dic257—41no—
190Dic257—41no—
191C2×C257C2×#11711yescharacteristic
192C2×C257#51811yescharacteristic
193C2×C257#52011yescharacteristic
194C2×C257#53811yescharacteristic
195C2×C257#53811yescharacteristic
196C2×C257#53911yescharacteristic
197C2×C257#53911yescharacteristic
198D257—24no—
199D257—42no—
200C257#231811yescharacteristic, complemented, Sylow
201#2318—2571nomaximal, Sylow
202C2×C4×D4—2571no—
203#388—2571no—
204#388—2571no—
205#387—2571no—
206#387D25711yescharacteristic, complemented
207#386—2571no—
208#386—2571no—
209#385—2571no—
210#385—2571no—
211#383—2571no—
212#382—2571no—
213#381—2571no—
214#380—2571no—
215C4×C22⋊C4—2571no—
216C2×#95—2571no—
217C22×D4—2571no—
218C23×C4—2571no—
219C23×C4—2571no—
220C23×C4—2571no—
221C23×C4—2571no—
222C4×D4—2578no—
223C2×C4⋊C4—2571no—
224C2×C4⋊C4—2571no—
225C2×C4⋊C4—2571no—
226C2×C4⋊C4C2×D25711yescharacteristic
227C2×C4⋊C4C2×D25711yescharacteristic
228C2×C22⋊C4—2571no—
229C2×C22⋊C4—2571no—
230C2×C22⋊C4—2571no—
231C2×C22⋊C4—2571no—
232C2×C22⋊C4—2571no—
233C2×C22⋊C4—2571no—
234C2×C22⋊C4—2571no—
235C2×C22⋊C4—2571no—
236C2×C42—2571no—
237C2×C42—2571no—
238C2×C42C2×D25711yescharacteristic
239#95—2571no—
240#95—2571no—
241#95—2571no—
242#95—2571no—
243#95—2571no—
244#95—2571no—
245#95—2571no—
246#95—2571no—
247#95—2571no—
248#95—2571no—
249#95C2×D25711yescharacteristic
250#95C2×D25711yescharacteristic
251#95C2×D25711yescharacteristic
252#95C2×D25711yescharacteristic
253C24—2571no—
254C24—2571no—
255C2×D4—2574no—
256C2×D4—5142no—
257C2×D4—5142no—
258C22×C4—2571no—
259C22×C4—2571no—
260C22×C4—2574no—
261C22×C4—2574no—
262C22×C4—2571no—
263C22×C4—2571no—
264C22×C4—2571no—
265C22×C4—2571no—
266C22×C4—5142no—
267C22×C4—5142no—
268C22×C4—10281no—
269C22×C4—10281no—
270C22×C4—10281no—
271C22×C4—10281no—
272C22×C4—10281no—
273C22×C4—10281no—
274C22×C4—10281no—
275C22×C4—10281no—
276C22×C4C22×D25711yescharacteristic
277C22×C4C22×D25711yescharacteristic
278C22×C4C22×D25711yescharacteristic
279C22×C4C22×D25711yescharacteristic
280C22×C4C22×D25711yescharacteristic
281C22×C4C22×D25711yescharacteristic
282C22×C4C22×D25711yescharacteristic
283C4⋊C4—41no—
284C4⋊C4—10281no—
285C4⋊C4—10281no—
286C4⋊C4—10281no—
287C4⋊C4C4×D25714yescomplemented
288C22⋊C4—2574no—
289C22⋊C4—2574no—
290C22⋊C4—5142no—
291C22⋊C4—5142no—
292C22⋊C4—10281no—
293C22⋊C4—10281no—
294C22⋊C4—10281no—
295C22⋊C4—10281no—
296C42—2574no—
297C42—41no—
298C42—10281no—
299C23—2572no—
300C23—2572no—
301C23—2572no—
302C23—2572no—
303C23—5141no—
304C23—5141no—
305C23—5141no—
306C23—5141no—
307C23—5141no—
308C23—5141no—
309C23—5141no—
310C23—5141no—
311C23—5141no—
312C23—5141no—
313C23C23×D25711yescharacteristic
314D4—5148no—
315C2×C4—2572no—
316C2×C4—2572no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—5141no—
336C2×C4—5141no—
337C2×C4—5141no—
338C2×C4—5141no—
339C2×C4—5141no—
340C2×C4—5141no—
341C2×C4—5141no—
342C2×C4—5141no—
343C2×C4—5141no—
344C2×C4—5141no—
345C2×C4—5141no—
346C2×C4—5141no—
347C2×C4—5141no—
348C2×C4—5141no—
349C2×C4—5141no—
350C2×C4—5141no—
351C2×C4—10284no—
352C2×C4—10284no—
353C2×C4—10284no—
354C2×C4—10284no—
355C2×C4C2×C4×D25712yes—
356C2×C4C2×C4×D25712yes—
357C2×C4C2×C4×D25712yes—
358C22—2574no—
359C22—5142no—
360C22—5142no—
361C22—5142no—
362C22—5142no—
363C22—5142no—
364C22—5142no—
365C22—5142no—
366C22—5142no—
367C22—5142no—
368C22—10281no—
369C22—10281no—
370C22—10281no—
371C22—10281no—
372C22C22×C4×D25711yescharacteristic
373C22#41×D25711yescharacteristic
374C22#41×D25711yescharacteristic
375C22#2255163711yescharacteristic
376C22#2255163811yescharacteristic
377C22#2255163811yescharacteristic
378C22#2255164011yescharacteristic
379C4—22no—
380C4—22no—
381C4—22no—
382C4—5142no—
383C4—5142no—
384C4—41no—
385C4—41no—
386C4—41no—
387C4—41no—
388C4—10281no—
389C4—10281no—
390C4—10281no—
391C4—10281no—
392C2—5144no—
393C2—10282no—
394C2#117×D25711yescharacteristic
395C2#3065720911yescharacteristic
396C2#3065721911yescharacteristic
397C2#3065732311yescharacteristic
398C2#3065732611yescharacteristic
399C2#3065733311yescharacteristic
400C2#3065733511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D257
commutatorC22×C257C23×C4
Fitting#387×C257C2
socle, cosocleC23×C257C24
prime radicalC22×D1028C22
hypercenter#387D257
hypocenterC257#2318
LCSG ▷ C22×C257 ▷ C257
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
2. #30656628C2semidirect
3. #30656626C2semidirect
4. #30656623C2semidirect
7. #30656611C2semidirect
8. #30656608C2semidirect
9. #30656604C2semidirect
10. #30656601C2semidirect
11. #30656599C2semidirect
12. #30656594C2semidirect
13. C4×#16861595C2semidirect
14. #95×D257C2semidirect
15. #387×C257C2semidirect
21. #22551554C4semidirect
74. C2×C4×D257C2×C4semidirect
85. #16861595C2×C4semidirect
87. C4×Dic257C2×C4semidirect
96. C4⋊C4×C257C2×C4semidirect
200. C257#2318semidirect
206. #387D257semidirect
287. C4⋊C4C4×D257semidirect

Maximal subgroups

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

subgroupindex
C2×#225515542
#306566282
#306566262
#306566232
#306566152
#306566112
#306566082
#306566042
#306566012
#306565992
#306565942
C4×#168615952
#95×D2572
#387×C2572
#2318257