Finite Group Databasebeta

← #45667476

Subgroups — #45667476

Order 32896.

576 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22551565C211yesmaximal, characteristic
2#30656638C211yesmaximal, characteristic, complemented
3#30656637C211yesmaximal, characteristic, complemented
4#30656630C211yesmaximal, characteristic, complemented
5#30656627C211yesmaximal, characteristic, complemented
6#30656619C211yesmaximal, characteristic, complemented
7#30656618C211yesmaximal, characteristic, complemented
8#30656616C211yesmaximal, characteristic, complemented
9#30656616C211yesmaximal, characteristic, complemented
10#30656615C211yesmaximal, characteristic, complemented
11#30656613C211yesmaximal, characteristic, complemented
12#30656611C211yesmaximal, characteristic, complemented
13#30656599C211yesmaximal, characteristic, complemented
14#30656594C211yesmaximal, characteristic, complemented
15#385×C257C211yesmaximal, characteristic, complemented
16C22×D1028C2211yescharacteristic
17C22×C22⋊D257C2211yescharacteristic
18C22×C22⋊D257C2211yescharacteristic
19C22×C4×D257C2211yescharacteristic
20C22×C4×D257C2211yescharacteristic
21#22551565C418yescomplemented
22C2×C4⋊Dic257C2211yescharacteristic
23C2×#16861597C2211yescharacteristic
24C2×#16861597C2211yescharacteristic
25C2×C22⋊Dic257C2211yescharacteristic
26C2×C22⋊Dic257C2211yescharacteristic
27C2×#16861595C2211yescharacteristic
28C2×#16861595C2211yescharacteristic
29C2×#16861595C2211yescharacteristic
30C2×#16861595C2211yescharacteristic
31C2×#16861595C2211yescharacteristic
32C2×#16861595C2211yescharacteristic
33C2×#16861595C2211yescharacteristic
34C2×#16861595C2211yescharacteristic
35C2×C4×Dic257C2211yescharacteristic
36C2×C4×Dic257C2211yescharacteristic
37#22551499C2211yescharacteristic
38#22551499C2211yescharacteristic
39#22551499C2211yescharacteristic
40#22551499C2211yescharacteristic
41#22551499C2211yescharacteristic
42#22551499C2211yescharacteristic
43#22551499C2211yescharacteristic
44#22551499C2211yescharacteristic
45C23×C4×C257C2211yescharacteristic
46C2×C4⋊C4×C257C2211yescharacteristic
47C2×C4⋊C4×C257C2211yescharacteristic
48C2×C22⋊C4×C257C2211yescharacteristic
49C2×C22⋊C4×C257C2211yescharacteristic
50#95×C257C2211yescharacteristic
51#95×C257C2211yescharacteristic
52C23×D257C2311yescharacteristic
53C23×D257C2311yescharacteristic
54C2×D1028—41no—
55C2×D1028—41no—
56C2×D1028C2×C414yescomplemented
57C2×C22⋊D257—41no—
58C2×C22⋊D257—41no—
59C2×C22⋊D257—41no—
60C2×C22⋊D257—41no—
61C2×C22⋊D257C2×C414yes—
62C2×C22⋊D257C2×C414yescomplemented
63C22×Dic257C2311yescharacteristic
64C22×Dic257C2311yescharacteristic
65C22×Dic257C2311yescharacteristic
66C22×Dic257C2311yescharacteristic
67C22×Dic257C2311yescharacteristic
68C22×Dic257C2311yescharacteristic
69C2×C4×D257—41no—
70C2×C4×D257—41no—
71C2×C4×D257—41no—
72C2×C4×D257—41no—
73C2×C4×D257—41no—
74C2×C4×D257C2×C414yescomplemented
75C4⋊Dic257—41no—
76#16861597—41no—
77#16861597C2×C414yescomplemented
78C22⋊Dic257—41no—
79C22⋊Dic257—41no—
80#16861595—41no—
81#16861595—41no—
82#16861595—41no—
83#16861595—41no—
84#16861595—41no—
85#16861595—41no—
86#16861595—41no—
87#16861595C2×C414yescomplemented
88C4×Dic257—41no—
89C4×Dic257—41no—
90C24×C257C2311yescharacteristic
91C22×C4×C257—41no—
92C22×C4×C257—41no—
93C22×C4×C257—41no—
94C22×C4×C257C2311yescharacteristic
95C22×C4×C257C2311yescharacteristic
96C22×C4×C257C2311yescharacteristic
97C22×C4×C257C2311yescharacteristic
98C22×C4×C257C2311yescharacteristic
99C22×C4×C257C2311yescharacteristic
100C4⋊C4×C257—41no—
101C4⋊C4×C257—41no—
102C22⋊C4×C257—41no—
103C22⋊C4×C257C2×C414yescomplemented
104C22×D257—21no—
105C22×D257—21no—
106C22×D257—21no—
107C22×D257—21no—
108C22×D257—21no—
109C22×D257—21no—
110C22×D257—21no—
111C22×D257—21no—
112C22×D257—21no—
113C22×D257—21no—
114C22×D257—21no—
115C22×D257—21no—
116C22×D257C22×C412yes—
117C22×D257C22×C412yes—
118D1028—44no—
119C22⋊D257—44no—
120C22⋊D257—44no—
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×Dic257—21no—
130C2×Dic257—21no—
131C2×Dic257—21no—
132C2×Dic257—21no—
133C2×Dic257—21no—
134C2×Dic257—21no—
135C2×Dic257—21no—
136C2×Dic257—21no—
137C2×Dic257C22×C412yes—
138C2×Dic257C22×C412yes—
139C4×D257—44no—
140C4×D257—44no—
141C23×C257—21no—
142C23×C257—21no—
143C23×C257—21no—
144C23×C257—21no—
145C23×C257—21no—
146C23×C257—21no—
147C23×C257C22×C412yes—
148C23×C257C2411yescharacteristic
149C2×C4×C257—21no—
150C2×C4×C257—21no—
151C2×C4×C257—21no—
152C2×C4×C257—21no—
153C2×C4×C257—21no—
154C2×C4×C257—21no—
155C2×C4×C257—21no—
156C2×C4×C257—21no—
157C2×C4×C257—21no—
158C2×C4×C257—21no—
159C2×C4×C257—21no—
160C2×C4×C257—21no—
161C2×C4×C257—21no—
162C2×C4×C257—21no—
163C2×C4×C257—21no—
164C2×C4×C257—21no—
165C2×C4×C257—44no—
166C2×C4×C257C22×C412yes—
167C2×C4×C257C22×C412yes—
168C22×C257—22no—
169C22×C257—22no—
170C22×C257—22no—
171C22×C257—41no—
172C22×C257—41no—
173C22×C257—41no—
174C22×C257—41no—
175C22×C257C23×C411yescharacteristic
176C22×C257H(5,𝔽2)11yescharacteristic
177C22×C257H(5,𝔽2)11yescharacteristic
178C22×C257H(5,𝔽2)11yescharacteristic
179C22×C257H(5,𝔽2)11yescharacteristic
180C22×C257H(5,𝔽2)11yescharacteristic
181C22×C2572−1+411yescharacteristic
182C2×D257—22no—
183C2×D257—22no—
184C2×D257—22no—
185C2×D257—22no—
186C2×D257—22no—
187C2×D257—22no—
188C2×D257—41no—
189C2×D257—41no—
190C2×D257—41no—
191C2×D257—41no—
192C2×D257—41no—
193C2×D257—41no—
194C2×D257—41no—
195C2×D257—41no—
196C4×C257—22no—
197C4×C257—22no—
198C4×C257—41no—
199C4×C257—41no—
200C4×C257—41no—
201C4×C257—41no—
202Dic257—22no—
203Dic257—22no—
204Dic257—41no—
205Dic257—41no—
206Dic257—41no—
207Dic257—41no—
208C2×C257—42no—
209C2×C257#51811yescharacteristic
210C2×C257#51811yescharacteristic
211C2×C257#52011yescharacteristic
212C2×C257#56011yescharacteristic
213C2×C257#56011yescharacteristic
214C2×C257#56211yescharacteristic
215C2×C257#56211yescharacteristic
216D257—42no—
217D257—42no—
218C257#237711yescharacteristic, complemented, Sylow
219#2377—2571nomaximal, Sylow
220C2×#121—2571no—
221#390—2571no—
222#390—2571no—
223#388—2571no—
224#388—2571no—
225#387—2571no—
226#387—2571no—
227#386—2571no—
228#386—2571no—
229#386—2571no—
230#386—2571no—
231#385—2571no—
232#385D25711yescharacteristic, complemented
233#381—2571no—
234#380—2571no—
235C22×D4—2571no—
236C22×D4—2571no—
237C22×D4—2571no—
238C23×C4—2571no—
239C23×C4—2571no—
240C23×C4C2×D25711yescharacteristic
241#121—2578no—
242C2×C4⋊C4—2571no—
243C2×C4⋊C4—2571no—
244C2×C4⋊C4—2571no—
245C2×C4⋊C4C2×D25711yescharacteristic
246C2×C4⋊C4C2×D25711yescharacteristic
247C2×C22⋊C4—2571no—
248C2×C22⋊C4—2571no—
249C2×C22⋊C4—2571no—
250C2×C22⋊C4—2571no—
251C2×C22⋊C4—2571no—
252C2×C22⋊C4—2571no—
253C2×C22⋊C4—2571no—
254C2×C22⋊C4—2571no—
255C2×C22⋊C4—2571no—
256C2×C22⋊C4—2571no—
257C2×C22⋊C4C2×D25711yescharacteristic
258C2×C22⋊C4C2×D25711yescharacteristic
259C2×C42—2571no—
260C2×C42—2571no—
261#95—2571no—
262#95—2571no—
263#95—2571no—
264#95—2571no—
265#95—2571no—
266#95—2571no—
267#95—2571no—
268#95—2571no—
269#95C2×D25711yescharacteristic
270#95C2×D25711yescharacteristic
271C24—2571no—
272C24—2571no—
273C24C22×D25711yescharacteristic
274C2×D4—2574no—
275C2×D4—2574no—
276C2×D4—2574no—
277C2×D4—10281no—
278C2×D4—10281no—
279C2×D4—10281no—
280C2×D4—10281no—
281C2×D4—10281no—
282C2×D4—10281no—
283C22×C4—2571no—
284C22×C4—2571no—
285C22×C4—2574no—
286C22×C4—2571no—
287C22×C4—2571no—
288C22×C4—2571no—
289C22×C4—2571no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—10281no—
294C22×C4—10281no—
295C22×C4—10281no—
296C22×C4—10281no—
297C22×C4—10281no—
298C22×C4C22×D25711yescharacteristic
299C22×C4C22×D25711yescharacteristic
300C22×C4C22×D25711yescharacteristic
301C22×C4C22×D25711yescharacteristic
302C22×C4C22×D25711yescharacteristic
303C22×C4C22×D25711yescharacteristic
304C4⋊C4—2574no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—10281no—
308C4⋊C4—10281no—
309C22⋊C4—2574no—
310C22⋊C4—41no—
311C22⋊C4—10281no—
312C22⋊C4—10281no—
313C22⋊C4—10281no—
314C22⋊C4—10281no—
315C22⋊C4—10281no—
316C22⋊C4—10281no—
317C22⋊C4—10281no—
318C22⋊C4—10281no—
319C22⋊C4—10281no—
320C22⋊C4C4×D25714yescomplemented
321C42—10281no—
322C42—10281no—
323C23—2572no—
324C23—2572no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—5141no—
332C23—5141no—
333C23—5141no—
334C23—5141no—
335C23—5141no—
336C23—5141no—
337C23—5141no—
338C23—5141no—
339C23—5141no—
340C23—5141no—
341C23—5141no—
342C23—5141no—
343C23C2×C4×D25712yes—
344C23C23×D25711yescharacteristic
345D4—10284no—
346D4—10284no—
347D4—10284no—
348C2×C4—2572no—
349C2×C4—2572no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—5141no—
367C2×C4—5141no—
368C2×C4—5141no—
369C2×C4—5141no—
370C2×C4—5141no—
371C2×C4—5141no—
372C2×C4—5141no—
373C2×C4—5141no—
374C2×C4—5141no—
375C2×C4—5141no—
376C2×C4—5141no—
377C2×C4—5141no—
378C2×C4—5141no—
379C2×C4—5141no—
380C2×C4—5141no—
381C2×C4—5141no—
382C2×C4—44no—
383C2×C4—10284no—
384C2×C4—10284no—
385C2×C4C2×C4×D25712yes—
386C2×C4C2×C4×D25712yes—
387C22—22no—
388C22—22no—
389C22—22no—
390C22—5142no—
391C22—5142no—
392C22—5142no—
393C22—5142no—
394C22—5142no—
395C22—5142no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—10281no—
401C22—10281no—
402C22—10281no—
403C22—10281no—
404C22—10281no—
405C22—10281no—
406C22—10281no—
407C22—10281no—
408C22C22×C4×D25711yescharacteristic
409C22#2255163711yescharacteristic
410C22#2255163711yescharacteristic
411C22#2255163711yescharacteristic
412C22#2255163811yescharacteristic
413C22#2255163811yescharacteristic
414C22#2255164011yescharacteristic
415C4—22no—
416C4—22no—
417C4—5142no—
418C4—5142no—
419C4—41no—
420C4—41no—
421C4—41no—
422C4—41no—
423C4—10281no—
424C4—10281no—
425C4—10281no—
426C4—10281no—
427C2—42no—
428C2—10282no—
429C2—10282no—
430C2#3065720911yescharacteristic
431C2#3065720911yescharacteristic
432C2#3065721811yescharacteristic
433C2#3065747611yescharacteristic
434C2#3065747711yescharacteristic
435C2#3065748311yescharacteristic
436C2#3065748311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D257
commutatorC22×C257C23×C4
Fitting#385×C257C2
socle, cosocleC23×C257C24
prime radicalC2×#22551565C2
hypercenter#385D257
hypocenterC257#2377
LCSG ▷ C22×C257 ▷ C257
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. #30656638C2semidirect
3. #30656637C2semidirect
4. #30656630C2semidirect
5. #30656627C2semidirect
6. #30656619C2semidirect
7. #30656618C2semidirect
8. #30656616C2semidirect
9. #30656616C2semidirect
10. #30656615C2semidirect
11. #30656613C2semidirect
12. #30656611C2semidirect
13. #30656599C2semidirect
14. #30656594C2semidirect
15. #385×C257C2semidirect
21. #22551565C4semidirect
56. C2×D1028C2×C4semidirect
62. C2×C22⋊D257C2×C4semidirect
74. C2×C4×D257C2×C4semidirect
77. #16861597C2×C4semidirect
87. #16861595C2×C4semidirect
103. C22⋊C4×C257C2×C4semidirect
218. C257#2377semidirect
232. #385D257semidirect
320. C22⋊C4C4×D257semidirect

Maximal subgroups

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

subgroupindex
C2×#225515652
#306566382
#306566372
#306566302
#306566272
#306566192
#306566182
#306566162
#306566152
#306566132
#306566112
#306565992
#306565942
#385×C2572
#2377257