Finite Group Databasebeta

← #45667297

Subgroups — #45667297

Order 32896.

928 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22551560C211yesmaximal, characteristic
2C2×C4×C22⋊D257C211yesmaximal, characteristic, complemented
3C22×#16861595C211yesmaximal, characteristic, complemented
4C2×C22⋊C4×D257C211yesmaximal, characteristic, complemented
5#30656637C211yesmaximal, characteristic, complemented
6#30656628C211yesmaximal, characteristic, complemented
7#30656627C211yesmaximal, characteristic, complemented
8#30656617C211yesmaximal, characteristic, complemented
9#30656616C211yesmaximal, characteristic, complemented
10#30656613C211yesmaximal, characteristic, complemented
11#30656610C211yesmaximal, characteristic, complemented
12#30656608C211yesmaximal, characteristic, complemented
13C22⋊C4×Dic257C211yesmaximal, characteristic, complemented
14C4×#16861595C211yesmaximal, characteristic, complemented
15C2×C4×D4×C257C211yesmaximal, characteristic, complemented
16C24×D257C2211yescharacteristic
17C22×C22⋊D257C2211yescharacteristic
18C22×C22⋊D257C2211yescharacteristic
19C23×Dic257C2211yescharacteristic
20C22×C4×D257C2211yescharacteristic
21C22×C4×D257C2211yescharacteristic
22#22551560C418yescomplemented
23C4×C22⋊D257—24no—
24C2×#16861597C2211yescharacteristic, complemented
25C2×#16861597C2211yescharacteristic, complemented
26C2×C22⋊Dic257C2211yescharacteristic
27C2×C22⋊Dic257C2211yescharacteristic
28C2×C22⋊Dic257C2211yescharacteristic, complemented
29C2×C22⋊Dic257C2211yescharacteristic, complemented
30C2×#16861595—24no—
31C2×#16861595C2211yescharacteristic
32C2×#16861595C2211yescharacteristic
33C2×#16861595C2211yescharacteristic
34C2×#16861595C2211yescharacteristic
35C2×#16861595C2211yescharacteristic
36C2×#16861595C2211yescharacteristic
37C2×#16861595C2211yescharacteristic
38C2×#16861595C2211yescharacteristic
39C22⋊C4×D257—24no—
40C2×C4×Dic257C2211yescharacteristic, complemented
41C2×C4×Dic257C2211yescharacteristic, complemented
42#22551499C2211yescharacteristic, complemented
43#22551499C2211yescharacteristic, complemented
44#22551499C2211yescharacteristic, complemented
45#22551499C2211yescharacteristic, complemented
46#22551499C2211yescharacteristic, complemented
47#22551499C2211yescharacteristic, complemented
48C22×D4×C257C2211yescharacteristic
49C23×C4×C257C2211yescharacteristic
50C23×C4×C257C2211yescharacteristic, complemented
51C4×D4×C257—24no—
52C2×C4⋊C4×C257C2211yescharacteristic, complemented
53C2×C22⋊C4×C257C2211yescharacteristic
54C2×C22⋊C4×C257C2211yescharacteristic, complemented
55C2×C42×C257C2211yescharacteristic, complemented
56C23×D257—22no—
57C23×D257—22no—
58C23×D257—22no—
59C23×D257—22no—
60C23×D257—41no—
61C23×D257—41no—
62C23×D257C2×C414yescomplemented
63C23×D257C2311yescharacteristic
64C23×D257C2311yescharacteristic
65C2×C22⋊D257—22no—
66C2×C22⋊D257—22no—
67C2×C22⋊D257—41no—
68C2×C22⋊D257—41no—
69C2×C22⋊D257C2×C414yescomplemented
70C2×C22⋊D257C2×C414yescomplemented
71C22×Dic257—22no—
72C22×Dic257—22no—
73C22×Dic257—41no—
74C22×Dic257C2311yescharacteristic
75C22×Dic257C2311yescharacteristic
76C22×Dic257C2311yescharacteristic
77C22×Dic257C2311yescharacteristic
78C22×Dic257C2311yescharacteristic
79C22×Dic257C2311yescharacteristic
80C2×C4×D257—22no—
81C2×C4×D257—22no—
82C2×C4×D257—22no—
83C2×C4×D257—22no—
84C2×C4×D257—41no—
85C2×C4×D257—41no—
86#16861597—22no—
87#16861597—41no—
88C22⋊Dic257—22no—
89C22⋊Dic257—22no—
90C22⋊Dic257—41no—
91C22⋊Dic257C2×C414yescomplemented
92#16861595—22no—
93#16861595—22no—
94#16861595—22no—
95#16861595—22no—
96#16861595—22no—
97#16861595C2×C414yescomplemented
98#16861595C2×C414yescomplemented
99#16861595D414yescomplemented
100C4×Dic257—22no—
101C4×Dic257—41no—
102C24×C257C2311yescharacteristic
103C24×C257C2311yescharacteristic
104C2×D4×C257—22no—
105C2×D4×C257—22no—
106C2×D4×C257C2×C414yescomplemented
107C22×C4×C257—22no—
108C22×C4×C257—22no—
109C22×C4×C257—22no—
110C22×C4×C257—22no—
111C22×C4×C257—41no—
112C22×C4×C257—41no—
113C22×C4×C257C2311yescharacteristic
114C22×C4×C257C2311yescharacteristic
115C22×C4×C257C2311yescharacteristic
116C22×C4×C257C2311yescharacteristic
117C22×C4×C257C2311yescharacteristic
118C4⋊C4×C257—22no—
119C22⋊C4×C257—22no—
120C22⋊C4×C257D414yescomplemented
121C42×C257—22no—
122C22×D257—21no—
123C22×D257—21no—
124C22×D257—21no—
125C22×D257—21no—
126C22×D257—28no—
127C22×D257—21no—
128C22×D257—21no—
129C22×D257—21no—
130C22×D257—21no—
131C22×D257—21no—
132C22×D257—21no—
133C22×D257—28no—
134C22×D257—44no—
135C22×D257—44no—
136C22×D257—44no—
137C22×D257—44no—
138C22×D257—44no—
139C22×D257C22×C412yes—
140C22×D257C22×C412yes—
141C22×D257C2×D412yes—
142C22×D257#4112yes—
143C22⋊D257—28no—
144C22⋊D257—44no—
145C2×Dic257—21no—
146C2×Dic257—21no—
147C2×Dic257—21no—
148C2×Dic257—21no—
149C2×Dic257—21no—
150C2×Dic257—21no—
151C2×Dic257—21no—
152C2×Dic257—21no—
153C2×Dic257—21no—
154C2×Dic257—21no—
155C2×Dic257—21no—
156C2×Dic257—21no—
157C2×Dic257—21no—
158C2×Dic257—21no—
159C2×Dic257—21no—
160C2×Dic257—44no—
161C2×Dic257C22×C412yes—
162C2×Dic257C22×C412yes—
163C2×Dic257C2×D412yes—
164C4×D257—44no—
165C4×D257—44no—
166C23×C257—21no—
167C23×C257—21no—
168C23×C257—21no—
169C23×C257—21no—
170C23×C257—21no—
171C23×C257—21no—
172C23×C257—21no—
173C23×C257—21no—
174C23×C257—21no—
175C23×C257—21no—
176C23×C257C22×C412yes—
177C23×C257C22×C412yes—
178C23×C257C2×D412yes—
179C23×C257#4112yes—
180C23×C257C2411yescharacteristic
181D4×C257—28no—
182C2×C4×C257—21no—
183C2×C4×C257—21no—
184C2×C4×C257—21no—
185C2×C4×C257—21no—
186C2×C4×C257—21no—
187C2×C4×C257—21no—
188C2×C4×C257—21no—
189C2×C4×C257—21no—
190C2×C4×C257—21no—
191C2×C4×C257—21no—
192C2×C4×C257—21no—
193C2×C4×C257—44no—
194C2×C4×C257—44no—
195C2×C4×C257C22×C412yes—
196C2×C4×C257C2×D412yes—
197C2×C4×C257C2×D412yes—
198C2×C4×C257C2×D412yes—
199C22×C257—22no—
200C22×C257—22no—
201C22×C257—22no—
202C22×C257—22no—
203C22×C257—22no—
204C22×C257—22no—
205C22×C257—22no—
206C22×C257—22no—
207C22×C257—22no—
208C22×C257—41no—
209C22×C257—41no—
210C22×C257—41no—
211C22×C257—41no—
212C22×C257C4×D414yescomplemented
213C22×C257C23×C411yescharacteristic
214C22×C257C22×D411yescharacteristic
215C22×C257C22×D411yescharacteristic
216C22×C257C2×#4111yescharacteristic
217C22×C257C2×#4111yescharacteristic
218C22×C257H(5,𝔽2)11yescharacteristic
219C22×C257H(5,𝔽2)11yescharacteristic
220C2×D257—22no—
221C2×D257—22no—
222C2×D257—22no—
223C2×D257—22no—
224C2×D257—22no—
225C2×D257—22no—
226C2×D257—22no—
227C2×D257—22no—
228C2×D257—22no—
229C2×D257—416no—
230C2×D257—41no—
231C2×D257—41no—
232C2×D257—41no—
233C2×D257—41no—
234C2×D257C4×D414yescomplemented
235C4×C257—22no—
236C4×C257—22no—
237C4×C257—22no—
238C4×C257—22no—
239C4×C257—41no—
240Dic257—22no—
241Dic257—22no—
242Dic257—22no—
243Dic257—41no—
244Dic257—41no—
245Dic257—41no—
246C2×C257—24no—
247C2×C257—42no—
248C2×C257C2×C4×D411yescharacteristic
249C2×C257C2×C4×D411yescharacteristic
250C2×C257#51811yescharacteristic
251C2×C257D4211yescharacteristic
252C2×C257#54611yescharacteristic
253C2×C257#54611yescharacteristic
254C2×C257#55111yescharacteristic
255D257—24no—
256D257—42no—
257C257#235811yescharacteristic, complemented, Sylow
258#2358—2571nomaximal, Sylow
259C2×#120—2571no—
260C2×C4×D4—2571no—
261C2×C4×D4D25711yescharacteristic, complemented
262C22×C22⋊C4—2571no—
263C22×C22⋊C4—2571no—
264#390—2571no—
265#388—2571no—
266#388—2571no—
267#386—2571no—
268#386—2571no—
269#386—2571no—
270#385—2571no—
271#385—2571no—
272C4×C22⋊C4—2571no—
273C4×C22⋊C4—2571no—
274C25—2571no—
275C22×D4—2571no—
276C22×D4—2571no—
277C22×D4C2×D25711yescharacteristic
278C23×C4—2571no—
279C23×C4—2571no—
280C23×C4—2571no—
281C23×C4C2×D25711yescharacteristic
282C23×C4C2×D25711yescharacteristic, complemented
283#120—2578no—
284C4×D4—24no—
285C4×D4—5144no—
286C2×C4⋊C4—2571no—
287C2×C4⋊C4—2571no—
288C2×C4⋊C4C2×D25711yescharacteristic, complemented
289C2×C22⋊C4—2571no—
290C2×C22⋊C4—2571no—
291C2×C22⋊C4—2571no—
292C2×C22⋊C4—2571no—
293C2×C22⋊C4—2571no—
294C2×C22⋊C4—2571no—
295C2×C22⋊C4—2571no—
296C2×C22⋊C4—2571no—
297C2×C22⋊C4—2571no—
298C2×C22⋊C4—2571no—
299C2×C22⋊C4—2571no—
300C2×C22⋊C4—2571no—
301C2×C22⋊C4—5144no—
302C2×C22⋊C4—5144no—
303C2×C22⋊C4C2×D25711yescharacteristic
304C2×C22⋊C4C2×D25711yescharacteristic, complemented
305C2×C42—2571no—
306C2×C42—2571no—
307C2×C42C2×D25711yescharacteristic, complemented
308#95—2571no—
309#95—2571no—
310#95—2571no—
311#95—2571no—
312#95—2571no—
313#95—2571no—
314C24—2571no—
315C24—2571no—
316C24—2574no—
317C24—5142no—
318C24—5142no—
319C24—5142no—
320C24—5142no—
321C24—10281no—
322C24—10281no—
323C24C22×D25711yescharacteristic
324C24C22×D25711yescharacteristic
325C2×D4—2574no—
326C2×D4—2574no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—5142no—
330C2×D4—5142no—
331C2×D4—10281no—
332C2×D4—10281no—
333C2×D4C4×D25714yescomplemented
334C22×C4—2571no—
335C22×C4—2571no—
336C22×C4—2571no—
337C22×C4—2571no—
338C22×C4—2571no—
339C22×C4—2571no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—5142no—
345C22×C4—5142no—
346C22×C4—5142no—
347C22×C4—5142no—
348C22×C4—5142no—
349C22×C4—5142no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—10281no—
353C22×C4—10281no—
354C22×C4—10281no—
355C22×C4C22×D25711yescharacteristic
356C22×C4C22×D25711yescharacteristic
357C22×C4C22×D25711yescharacteristic
358C22×C4C22×D25711yescharacteristic
359C22×C4C22×D25711yescharacteristic
360C4⋊C4—22no—
361C4⋊C4—5142no—
362C4⋊C4—10281no—
363C22⋊C4—2574no—
364C22⋊C4—2574no—
365C22⋊C4—2574no—
366C22⋊C4—2574no—
367C22⋊C4—22no—
368C22⋊C4—5142no—
369C22⋊C4—5142no—
370C22⋊C4—5142no—
371C22⋊C4—5142no—
372C22⋊C4—5142no—
373C22⋊C4—5142no—
374C22⋊C4—5142no—
375C22⋊C4—10281no—
376C22⋊C4C22⋊D25714yescomplemented
377C42—22no—
378C42—5142no—
379C42—10281no—
380C23—2572no—
381C23—2572no—
382C23—2572no—
383C23—2572no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—5148no—
395C23—5141no—
396C23—5141no—
397C23—5141no—
398C23—5141no—
399C23—5148no—
400C23—5141no—
401C23—5141no—
402C23—5141no—
403C23—5141no—
404C23—5141no—
405C23—5141no—
406C23—10284no—
407C23—10284no—
408C23—10284no—
409C23—10284no—
410C23—10284no—
411C23C2×C4×D25712yes—
412C23C2×C4×D25712yes—
413C23C2×C22⋊D25712yes—
414C23#1686161812yes—
415C23C23×D25711yescharacteristic
416D4—28no—
417D4—5148no—
418D4—10284no—
419C2×C4—2572no—
420C2×C4—2572no—
421C2×C4—2572no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—5141no—
434C2×C4—5141no—
435C2×C4—5141no—
436C2×C4—5141no—
437C2×C4—5141no—
438C2×C4—5141no—
439C2×C4—5141no—
440C2×C4—5141no—
441C2×C4—5141no—
442C2×C4—5141no—
443C2×C4—5141no—
444C2×C4—5141no—
445C2×C4—5141no—
446C2×C4—5141no—
447C2×C4—5141no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—10284no—
451C2×C4—10284no—
452C2×C4—10284no—
453C2×C4C2×C4×D25712yes—
454C2×C4D4×D25712yes—
455C2×C4C2×C22⋊D25712yes—
456C2×C4C2×C22⋊D25712yes—
457C22—2574no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—22no—
464C22—22no—
465C22—22no—
466C22—22no—
467C22—5142no—
468C22—5142no—
469C22—5142no—
470C22—5142no—
471C22—5142no—
472C22—5142no—
473C22—5142no—
474C22—5142no—
475C22—5142no—
476C22—41no—
477C22—41no—
478C22—41no—
479C22—41no—
480C22—102816no—
481C22—10281no—
482C22—10281no—
483C22—10281no—
484C22—10281no—
485C22C4×C22⋊D25714yescomplemented
486C22C22×C4×D25711yescharacteristic
487C22C2×D4×D25711yescharacteristic
488C22C22×C22⋊D25711yescharacteristic
489C22#41×D25711yescharacteristic
490C22C2×#1686161811yescharacteristic
491C22#2255163711yescharacteristic
492C22#2255163811yescharacteristic
493C4—22no—
494C4—22no—
495C4—22no—
496C4—22no—
497C4—5142no—
498C4—5142no—
499C4—5142no—
500C4—41no—
501C4—10281no—
502C4—10281no—
503C4—10281no—
504C2—24no—
505C2—5144no—
506C2—42no—
507C2—10282no—
508C2C4×D4×D25711yescharacteristic
509C2C2×C4×C22⋊D25711yescharacteristic
510C2#3065720911yescharacteristic
511C2D4×C22⋊D25711yescharacteristic
512C2#3065737511yescharacteristic
513C2#3065738311yescharacteristic
514C2#3065741611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D257
commutatorC22×C257C23×C4
FittingC2×C4×D4×C257C2
socle, cosocleC23×C257C24
prime radicalC2×#22551560C2
hypercenterC2×C4×D4D257
hypocenterC257#2358
LCSG ▷ C22×C257 ▷ C257
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
2. C2×C4×C22⋊D257C2semidirect
3. C22×#16861595C2semidirect
4. C2×C22⋊C4×D257C2semidirect
5. #30656637C2semidirect
6. #30656628C2semidirect
7. #30656627C2semidirect
8. #30656617C2semidirect
9. #30656616C2semidirect
10. #30656613C2semidirect
11. #30656610C2semidirect
12. #30656608C2semidirect
13. C22⋊C4×Dic257C2semidirect
14. C4×#16861595C2semidirect
15. C2×C4×D4×C257C2semidirect
22. #22551560C4semidirect
24. C2×#16861597C22semidirect
25. C2×#16861597C22semidirect
28. C2×C22⋊Dic257C22semidirect
29. C2×C22⋊Dic257C22semidirect
40. C2×C4×Dic257C22semidirect
41. C2×C4×Dic257C22semidirect
42. #22551499C22semidirect
43. #22551499C22semidirect
44. #22551499C22semidirect
45. #22551499C22semidirect
46. #22551499C22semidirect
47. #22551499C22semidirect
50. C23×C4×C257C22semidirect
52. C2×C4⋊C4×C257C22semidirect
54. C2×C22⋊C4×C257C22semidirect
55. C2×C42×C257C22semidirect
62. C23×D257C2×C4semidirect
69. C2×C22⋊D257C2×C4semidirect
70. C2×C22⋊D257C2×C4semidirect
91. C22⋊Dic257C2×C4semidirect
97. #16861595C2×C4semidirect
98. #16861595C2×C4semidirect
99. #16861595D4semidirect
106. C2×D4×C257C2×C4semidirect
120. C22⋊C4×C257D4semidirect
212. C22×C257C4×D4semidirect
234. C2×D257C4×D4semidirect
257. C257#2358semidirect
261. C2×C4×D4D257semidirect
282. C23×C4C2×D257semidirect
288. C2×C4⋊C4C2×D257semidirect
304. C2×C22⋊C4C2×D257semidirect
307. C2×C42C2×D257semidirect
333. C2×D4C4×D257semidirect
376. C22⋊C4C22⋊D257semidirect
485. C22C4×C22⋊D257semidirect

Maximal subgroups

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

subgroupindex
C2×#225515602
C2×C4×C22⋊D2572
C22×#168615952
C2×C22⋊C4×D2572
#306566372
#306566282
#306566272
#306566172
#306566162
#306566132
#306566102
#306566082
C22⋊C4×Dic2572
C4×#168615952
C2×C4×D4×C2572
#2358257