Finite Group Databasebeta

← #45667563

Subgroups — #45667563

Order 32896.

696 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×Dic257C211yesmaximal, characteristic, complemented
2C2×#22551552C211yesmaximal, characteristic, complemented
3C2×#22551552C211yesmaximal, characteristic, complemented
4C2×#22551548C211yesmaximal, characteristic, complemented
5#30656671C211yesmaximal, characteristic, complemented
6#30656665C211yesmaximal, characteristic, complemented
7#30656660C211yesmaximal, characteristic, complemented
8#30656659C211yesmaximal, characteristic, complemented
9#30656653C211yesmaximal, characteristic, complemented
10#30656652C211yesmaximal, characteristic, complemented
11#30656645C211yesmaximal, characteristic
12C4⋊C4×Dic257C211yesmaximal, characteristic, complemented
13C22⋊C4×Dic257C211yesmaximal, characteristic, complemented
14#30656584C211yesmaximal, characteristic, complemented
15#394×C257C211yesmaximal, characteristic, complemented
16C22×C22⋊D257C2211yescharacteristic
17C22×C22⋊D257C2211yescharacteristic
18C23×Dic257C2211yescharacteristic
19C23×Dic257C2211yescharacteristic
20C22×C4×D257C2211yescharacteristic
21D4×Dic257—24no—
22#22551552—24no—
23#22551552—24no—
24#22551548—24no—
25C2×C4⋊Dic257C2211yescharacteristic
26C2×C4⋊Dic257C2211yescharacteristic
27C2×#16861597C2211yescharacteristic
28C2×#16861597C2211yescharacteristic
29C2×#16861597C2211yescharacteristic
30C2×#16861597C2211yescharacteristic
31C2×C22⋊Dic257C2211yescharacteristic
32C2×C22⋊Dic257C2211yescharacteristic
33C2×C22⋊Dic257C2211yescharacteristic
34C2×C22⋊Dic257C2211yescharacteristic
35C2×#16861595C2211yescharacteristic
36C2×#16861595C2211yescharacteristic
37C2×#16861595C2211yescharacteristic
38C2×#16861595C2211yescharacteristic
39C2×C4×Dic257C2211yescharacteristic
40C2×C4×Dic257C2211yescharacteristic
41C2×C4×Dic257C2211yescharacteristic
42C2×C4×Dic257C2211yescharacteristic
43#22551499C2211yescharacteristic
44#22551499C2211yescharacteristic
45#22551499C2211yescharacteristic
46#22551499C2211yescharacteristic
47#22551499C2211yescharacteristic
48C22×D4×C257C2211yescharacteristic
49C2×C4⋊C4×C257C2211yescharacteristic
50C2×C22⋊C4×C257C2211yescharacteristic
51C2×C22⋊C4×C257C2211yescharacteristic
52C2×C22⋊C4×C257C2211yescharacteristic
53C2×C22⋊C4×C257C2211yescharacteristic
54#95×C257C2211yescharacteristic
55C23×D257C2311yescharacteristic
56C2×C22⋊D257—22no—
57C2×C22⋊D257—22no—
58C2×C22⋊D257—22no—
59C2×C22⋊D257—22no—
60C2×C22⋊D257—22no—
61C2×C22⋊D257—22no—
62C22×Dic257—22no—
63C22×Dic257—22no—
64C22×Dic257—22no—
65C22×Dic257—22no—
66C22×Dic257—22no—
67C22×Dic257D414yescomplemented
68C22×Dic257C2311yescharacteristic
69C22×Dic257C2311yescharacteristic
70C22×Dic257C2311yescharacteristic
71C22×Dic257C2311yescharacteristic
72C22×Dic257C2311yescharacteristic
73C22×Dic257C2311yescharacteristic
74C22×Dic257C2311yescharacteristic
75C2×C4×D257—22no—
76C2×C4×D257—22no—
77C2×C4×D257—22no—
78C4⋊Dic257—22no—
79C4⋊Dic257—22no—
80#16861597—22no—
81#16861597—22no—
82#16861597—22no—
83#16861597—22no—
84C22⋊Dic257—22no—
85C22⋊Dic257—22no—
86C22⋊Dic257—22no—
87C22⋊Dic257—22no—
88#16861595—22no—
89#16861595—22no—
90#16861595—22no—
91#16861595—22no—
92C4×Dic257—22no—
93C4×Dic257—22no—
94C4×Dic257—22no—
95C4×Dic257D414yescomplemented
96C24×C257C2311yescharacteristic
97C24×C257C2311yescharacteristic
98C2×D4×C257—22no—
99C2×D4×C257—22no—
100C2×D4×C257—22no—
101C22×C4×C257C2311yescharacteristic
102C22×C4×C257C2311yescharacteristic
103C22×C4×C257C2311yescharacteristic
104C22×C4×C257C2311yescharacteristic
105C22×C4×C257C2311yescharacteristic
106C4⋊C4×C257—22no—
107C22⋊C4×C257—22no—
108C22⋊C4×C257—22no—
109C22⋊C4×C257—22no—
110C22⋊C4×C257—22no—
111C22×D257—21no—
112C22×D257—21no—
113C22×D257—21no—
114C22×D257—21no—
115C22×D257—21no—
116C22×D257—21no—
117C22×D257#4112yes—
118C22⋊D257—28no—
119C22⋊D257—28no—
120C2×Dic257—21no—
121C2×Dic257—21no—
122C2×Dic257—21no—
123C2×Dic257—21no—
124C2×Dic257—21no—
125C2×Dic257—21no—
126C2×Dic257—28no—
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—44no—
137C2×Dic257C2×D412yes—
138C2×Dic257C2×D412yes—
139C2×Dic257C2×D412yes—
140C2×Dic257C2×D412yes—
141C2×Dic257#4112yes—
142C2×Dic257#4112yes—
143C4×D257—44no—
144C23×C257—21no—
145C23×C257—21no—
146C23×C257—21no—
147C23×C257—21no—
148C23×C257—21no—
149C23×C257—21no—
150C23×C257—21no—
151C23×C257—21no—
152C23×C257—21no—
153C23×C257—21no—
154C23×C257—21no—
155C23×C257—21no—
156C23×C257C2×D412yes—
157C23×C257#4112yes—
158C23×C257C2411yescharacteristic
159D4×C257—28no—
160C2×C4×C257—21no—
161C2×C4×C257—21no—
162C2×C4×C257—21no—
163C2×C4×C257—21no—
164C2×C4×C257—21no—
165C2×C4×C257—21no—
166C2×C4×C257—21no—
167C2×C4×C257—21no—
168C2×C4×C257—21no—
169C2×C4×C257—21no—
170C2×C4×C257—21no—
171C2×C4×C257—21no—
172C2×C4×C257—21no—
173C2×C4×C257C2×D412yes—
174C2×C4×C257#4112yes—
175C22×C257—22no—
176C22×C257—22no—
177C22×C257—22no—
178C22×C257—22no—
179C22×C257—22no—
180C22×C257—22no—
181C22×C257—41no—
182C22×C257—41no—
183C22×C257—41no—
184C22×C257—41no—
185C22×C257—41no—
186C22×C257—41no—
187C22×C257—41no—
188C22×C257—41no—
189C22×C257C22×D411yescharacteristic
190C22×C257C22×D411yescharacteristic
191C22×C257C2×#4111yescharacteristic
192C22×C257C2×#4111yescharacteristic
193C22×C257C2×#4111yescharacteristic
194C22×C257C2×#4111yescharacteristic
195C22×C257C2×#4111yescharacteristic
196C2×D257—22no—
197C2×D257—22no—
198C2×D257—22no—
199C2×D257—41no—
200C2×D257—41no—
201C2×D257—41no—
202C2×D257—41no—
203C4×C257—22no—
204C4×C257—22no—
205C4×C257—41no—
206C4×C257—41no—
207C4×C257—41no—
208Dic257—22no—
209Dic257—22no—
210Dic257—22no—
211Dic257—41no—
212Dic257—41no—
213Dic257—41no—
214Dic257#12114yes—
215C2×C257—42no—
216C2×C257—42no—
217C2×C257C2×#12111yescharacteristic
218C2×C257#52511yescharacteristic
219C2×C257#52511yescharacteristic
220C2×C257#52511yescharacteristic
221C2×C257#52911yescharacteristic
222C2×C257#52911yescharacteristic
223C2×C257#53211yescharacteristic
224D257—42no—
225C257#239211yescharacteristic, complemented, Sylow
226#2392—2571nomaximal, Sylow
227C2×C4×D4—2571no—
228C2×C4×D4—2571no—
229C2×C4×D4—2571no—
230C2×#117—2571no—
231#398—2571no—
232#397—2571no—
233#396—2571no—
234#396—2571no—
235#394—2571no—
236#394—2571no—
237#394D25711yescharacteristic, complemented
238#392—2571no—
239C4×C4⋊C4—2571no—
240C4×C22⋊C4—2571no—
241C4×C22⋊C4—2571no—
242C22×D4—2571no—
243C22×D4—2571no—
244C22×D4C2×D25711yescharacteristic
245C23×C4—2571no—
246C23×C4—2571no—
247C23×C4—2571no—
248C4×D4—5144no—
249C4×D4—5144no—
250C4×D4—5144no—
251#117—5144no—
252C2×C4⋊C4—2571no—
253C2×C4⋊C4—2571no—
254C2×C4⋊C4—2571no—
255C2×C4⋊C4—2571no—
256C2×C4⋊C4—2571no—
257C2×C4⋊C4—2571no—
258C2×C4⋊C4C2×D25711yescharacteristic
259C2×C22⋊C4—2571no—
260C2×C22⋊C4—2571no—
261C2×C22⋊C4—2571no—
262C2×C22⋊C4—2571no—
263C2×C22⋊C4—2571no—
264C2×C22⋊C4—2571no—
265C2×C22⋊C4—2571no—
266C2×C22⋊C4—2571no—
267C2×C22⋊C4C2×D25711yescharacteristic
268C2×C22⋊C4C2×D25711yescharacteristic
269C2×C22⋊C4C2×D25711yescharacteristic
270C2×C22⋊C4C2×D25711yescharacteristic
271C2×C42—2571no—
272C2×C42—2571no—
273C2×C42—2571no—
274C2×C42—2571no—
275#95—2571no—
276#95—2571no—
277#95—2571no—
278#95—2571no—
279#95—2571no—
280#95C2×D25711yescharacteristic
281C24—2571no—
282C24C22×D25711yescharacteristic
283C24C22×D25711yescharacteristic
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—5142no—
288C2×D4—5142no—
289C2×D4—5142no—
290C2×D4—5142no—
291C2×D4—5142no—
292C2×D4—5142no—
293C22×C4—2571no—
294C22×C4—2574no—
295C22×C4—2571no—
296C22×C4—2571no—
297C22×C4—2571no—
298C22×C4—2571no—
299C22×C4—2571no—
300C22×C4—2571no—
301C22×C4—5142no—
302C22×C4—5142no—
303C22×C4—5142no—
304C22×C4—5142no—
305C22×C4—5142no—
306C22×C4—5142no—
307C22×C4—5142no—
308C22×C4—5142no—
309C22×C4C22×D25711yescharacteristic
310C22×C4C22×D25711yescharacteristic
311C22×C4C22×D25711yescharacteristic
312C22×C4C22×D25711yescharacteristic
313C22×C4C22×D25711yescharacteristic
314C4⋊C4—22no—
315C4⋊C4—5142no—
316C4⋊C4—5142no—
317C4⋊C4—5142no—
318C4⋊C4—5142no—
319C4⋊C4—5142no—
320C4⋊C4—5142no—
321C22⋊C4—22no—
322C22⋊C4—22no—
323C22⋊C4—22no—
324C22⋊C4—22no—
325C22⋊C4—5142no—
326C22⋊C4—5142no—
327C22⋊C4—5142no—
328C22⋊C4—5142no—
329C22⋊C4—5142no—
330C22⋊C4—5142no—
331C22⋊C4—5142no—
332C22⋊C4—5142no—
333C42—2574no—
334C42—5142no—
335C42—5142no—
336C42—5142no—
337C23—2572no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—5141no—
351C23—5141no—
352C23—5141no—
353C23—5141no—
354C23—5141no—
355C23—5141no—
356C23D4×D25712yes—
357C23#1686161712yes—
358C23C23×D25711yescharacteristic
359D4—28no—
360D4—5148no—
361D4—5148no—
362C2×C4—2572no—
363C2×C4—2572no—
364C2×C4—2572no—
365C2×C4—2572no—
366C2×C4—2572no—
367C2×C4—2572no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—5148no—
382C2×C4—5141no—
383C2×C4—5141no—
384C2×C4—5141no—
385C2×C4—5141no—
386C2×C4—5141no—
387C2×C4—5141no—
388C2×C4—5141no—
389C2×C4—5141no—
390C2×C4—5141no—
391C2×C4—5141no—
392C2×C4—5141no—
393C2×C4—5141no—
394C2×C4—5141no—
395C2×C4—5141no—
396C2×C4—5141no—
397C2×C4—10284no—
398C2×C4—10284no—
399C2×C4D4×D25712yes—
400C2×C4#1686161712yes—
401C22—22no—
402C22—22no—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—5142no—
408C22—5142no—
409C22—5142no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—10281no—
419C22—10281no—
420C22—10281no—
421C22—10281no—
422C22C2×D4×D25711yescharacteristic
423C22C2×D4×D25711yescharacteristic
424C22#41×D25711yescharacteristic
425C22#41×D25711yescharacteristic
426C22#41×D25711yescharacteristic
427C22C2×#1686161711yescharacteristic
428C22C2×#1686161711yescharacteristic
429C4—2574no—
430C4—22no—
431C4—22no—
432C4—5142no—
433C4—5142no—
434C4—5142no—
435C4—41no—
436C4—41no—
437C4—41no—
438C4—10281no—
439C4—10281no—
440C4—10281no—
441C2—42no—
442C2—42no—
443C2—10282no—
444C2#121×D25711yescharacteristic
445C2#3065724311yescharacteristic
446C2#3065724511yescharacteristic
447C2#3065724711yescharacteristic
448C2#3065726411yescharacteristic
449C2#3065726611yescharacteristic
450C2#3065728211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D257
commutator, socle, cosocleC23×C257C24
Fitting#394×C257C2
prime radical#30656645C2
hypercenter#394D257
hypocenterC257#2392
LCSG ▷ C23×C257 ▷ C257
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×D4×Dic257C2semidirect
2. C2×#22551552C2semidirect
3. C2×#22551552C2semidirect
4. C2×#22551548C2semidirect
5. #30656671C2semidirect
6. #30656665C2semidirect
7. #30656660C2semidirect
8. #30656659C2semidirect
9. #30656653C2semidirect
10. #30656652C2semidirect
12. C4⋊C4×Dic257C2semidirect
13. C22⋊C4×Dic257C2semidirect
14. #30656584C2semidirect
15. #394×C257C2semidirect
67. C22×Dic257D4semidirect
95. C4×Dic257D4semidirect
225. C257#2392semidirect
237. #394D257semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic2572
C2×#225515522
C2×#225515482
#306566712
#306566652
#306566602
#306566592
#306566532
#306566522
#306566452
C4⋊C4×Dic2572
C22⋊C4×Dic2572
#306565842
#394×C2572
#2392257