Finite Group Databasebeta

← #45667138

Subgroups — #45667138

Order 32896.

1252 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×D257C211yesmaximal, characteristic
2C2×C4×C22⋊D257C211yesmaximal, characteristic, complemented
3C2×#22551552C211yesmaximal, characteristic, complemented
4C22×#16861595C211yesmaximal, characteristic, complemented
5C2×C22⋊C4×D257C211yesmaximal, characteristic, complemented
6#30656637C211yesmaximal, characteristic, complemented
7#30656637C211yesmaximal, characteristic, complemented
8#30656617C211yesmaximal, characteristic, complemented
9#30656615C211yesmaximal, characteristic, complemented
10#30656615C211yesmaximal, characteristic, complemented
11#30656596C211yesmaximal, characteristic, complemented
12#30656592C211yesmaximal, characteristic, complemented
13#30656592C211yesmaximal, characteristic, complemented
14#30656584C211yesmaximal, characteristic, complemented
15#386×C257C211yesmaximal, characteristic, complemented
16C24×D257C2211yescharacteristic
17C24×D257C2211yescharacteristic
18C22×D1028C2211yescharacteristic
19C22×C22⋊D257C2211yescharacteristic
20C22×C22⋊D257C2211yescharacteristic
21C2×D4×D257—24no—
22C2×D4×D257—24no—
23C2×D4×D257C418yescomplemented
24C23×Dic257C2211yescharacteristic, complemented
25C22×C4×D257C2211yescharacteristic
26C22×C4×D257C2211yescharacteristic, complemented
27C22×C4×D257C2211yescharacteristic, complemented
28C4×C22⋊D257—24no—
29#22551552—24no—
30C2×#16861597C2211yescharacteristic, complemented
31C2×#16861597C2211yescharacteristic, complemented
32C2×C22⋊Dic257C2211yescharacteristic, complemented
33C2×C22⋊Dic257C2211yescharacteristic, complemented
34C2×C22⋊Dic257C2211yescharacteristic, complemented
35C2×#16861595—24no—
36C2×#16861595C2211yescharacteristic
37C2×#16861595C2211yescharacteristic
38C2×#16861595C2211yescharacteristic
39C2×#16861595C2211yescharacteristic
40C2×#16861595C2211yescharacteristic, complemented
41C2×#16861595C2211yescharacteristic, complemented
42C2×#16861595C2211yescharacteristic, complemented
43C2×#16861595C2211yescharacteristic, complemented
44C2×#16861595C2211yescharacteristic, complemented
45C2×#16861595C2211yescharacteristic, complemented
46C22⋊C4×D257—24no—
47C2×C4×Dic257C2211yescharacteristic, complemented
48C2×C4×Dic257C2211yescharacteristic, complemented
49#22551499C2211yescharacteristic, complemented
50#22551499C2211yescharacteristic, complemented
51C22×D4×C257C2211yescharacteristic
52C23×C4×C257C2211yescharacteristic, complemented
53C2×C22⋊C4×C257C2211yescharacteristic, complemented
54C2×C22⋊C4×C257C2211yescharacteristic, complemented
55C2×C22⋊C4×C257C2211yescharacteristic, complemented
56#95×C257C2211yescharacteristic, complemented
57#95×C257C2211yescharacteristic, complemented
58C23×D257—22no—
59C23×D257—22no—
60C23×D257—22no—
61C23×D257—22no—
62C23×D257—22no—
63C23×D257—22no—
64C23×D257—22no—
65C23×D257—22no—
66C23×D257—41no—
67C23×D257—41no—
68C23×D257—41no—
69C23×D257—41no—
70C23×D257C2×C414yescomplemented
71C23×D257C2×C414yescomplemented
72C23×D257C2311yescharacteristic
73C23×D257C2311yescharacteristic
74C23×D257C2311yescharacteristic
75C2×D1028—22no—
76C2×D1028—22no—
77C2×D1028C2×C414yescomplemented
78C2×C22⋊D257—22no—
79C2×C22⋊D257—22no—
80C2×C22⋊D257C2×C414yes—
81C2×C22⋊D257C2×C414yescomplemented
82C2×C22⋊D257D414yes—
83C2×C22⋊D257D414yes—
84D4×D257—232no—
85C22×Dic257—22no—
86C22×Dic257—22no—
87C22×Dic257—41no—
88C22×Dic257C2311yescharacteristic
89C22×Dic257C2311yescharacteristic
90C22×Dic257C2311yescharacteristic
91C22×Dic257C2311yescharacteristic
92C22×Dic257C2311yescharacteristic
93C2×C4×D257—22no—
94C2×C4×D257—22no—
95C2×C4×D257—22no—
96C2×C4×D257—22no—
97C2×C4×D257—22no—
98C2×C4×D257—22no—
99C2×C4×D257—41no—
100C2×C4×D257—41no—
101C2×C4×D257C2×C414yescomplemented
102#16861597—22no—
103#16861597—22no—
104C22⋊Dic257—22no—
105C22⋊Dic257—22no—
106C22⋊Dic257—41no—
107#16861595—22no—
108#16861595—22no—
109#16861595—22no—
110#16861595—22no—
111#16861595—22no—
112#16861595—22no—
113#16861595—22no—
114#16861595—41no—
115#16861595—41no—
116#16861595D414yescomplemented
117C4×Dic257—22no—
118C4×Dic257—22no—
119C24×C257C2311yescharacteristic
120C24×C257C2311yescharacteristic
121C2×D4×C257—22no—
122C2×D4×C257—22no—
123C2×D4×C257C2×C414yescomplemented
124C22×C4×C257—22no—
125C22×C4×C257—22no—
126C22×C4×C257—41no—
127C22×C4×C257C2311yescharacteristic
128C22×C4×C257C2311yescharacteristic
129C22×C4×C257C2311yescharacteristic
130C22×C4×C257C2311yescharacteristic
131C22×C4×C257C2311yescharacteristic
132C22⋊C4×C257—22no—
133C22⋊C4×C257—22no—
134C22⋊C4×C257—41no—
135C22×D257—21no—
136C22×D257—21no—
137C22×D257—21no—
138C22×D257—21no—
139C22×D257—21no—
140C22×D257—21no—
141C22×D257—21no—
142C22×D257—21no—
143C22×D257—28no—
144C22×D257—21no—
145C22×D257—21no—
146C22×D257—21no—
147C22×D257—21no—
148C22×D257—21no—
149C22×D257—21no—
150C22×D257—28no—
151C22×D257—28no—
152C22×D257—28no—
153C22×D257—44no—
154C22×D257—44no—
155C22×D257—44no—
156C22×D257—44no—
157C22×D257—44no—
158C22×D257—44no—
159C22×D257—44no—
160C22×D257—44no—
161C22×D257—44no—
162C22×D257—44no—
163C22×D257C22×C412yes—
164C22×D257C22×C412yes—
165C22×D257C22×C412yes—
166C22×D257C2×D412yes—
167C22×D257C2×D412yes—
168C22×D257C2×D412yes—
169C22×D257#4112yes—
170D1028—28no—
171C22⋊D257—28no—
172C22⋊D257C22⋊C4116yescomplemented
173C2×Dic257—21no—
174C2×Dic257—21no—
175C2×Dic257—21no—
176C2×Dic257—21no—
177C2×Dic257—21no—
178C2×Dic257—21no—
179C2×Dic257—21no—
180C2×Dic257—21no—
181C2×Dic257—21no—
182C2×Dic257—21no—
183C2×Dic257—21no—
184C2×Dic257—44no—
185C2×Dic257C22×C412yes—
186C2×Dic257C2×D412yes—
187C2×Dic257C2×D412yes—
188C2×Dic257C2×D412yescomplemented
189C4×D257—28no—
190C4×D257—44no—
191C4×D257—44no—
192C23×C257—21no—
193C23×C257—21no—
194C23×C257—21no—
195C23×C257—21no—
196C23×C257—21no—
197C23×C257—21no—
198C23×C257—21no—
199C23×C257—21no—
200C23×C257—21no—
201C23×C257—21no—
202C23×C257C22×C412yes—
203C23×C257C22×C412yes—
204C23×C257C2×D412yes—
205C23×C257C2×D412yes—
206C23×C257C2411yescharacteristic
207D4×C257—28no—
208C2×C4×C257—21no—
209C2×C4×C257—21no—
210C2×C4×C257—21no—
211C2×C4×C257—21no—
212C2×C4×C257—21no—
213C2×C4×C257—21no—
214C2×C4×C257—21no—
215C2×C4×C257—21no—
216C2×C4×C257—21no—
217C2×C4×C257—21no—
218C2×C4×C257—21no—
219C2×C4×C257—21no—
220C2×C4×C257—21no—
221C2×C4×C257—44no—
222C2×C4×C257C22×C412yes—
223C2×C4×C257C2×D412yescomplemented
224C22×C257—22no—
225C22×C257—22no—
226C22×C257—22no—
227C22×C257—22no—
228C22×C257—22no—
229C22×C257—22no—
230C22×C257—22no—
231C22×C257—22no—
232C22×C257—22no—
233C22×C257—41no—
234C22×C257—41no—
235C22×C257—41no—
236C22×C257—41no—
237C22×C257C2×C22⋊C414yescomplemented
238C22×C257C23×C411yescharacteristic
239C22×C257C22×D411yescharacteristic
240C22×C257C22×D411yescharacteristic
241C22×C257C22×D411yescharacteristic
242C22×C257C2×#4111yescharacteristic
243C22×C257H(5,𝔽2)11yescharacteristic
244C22×C257H(5,𝔽2)11yescharacteristic
245C2×D257—22no—
246C2×D257—22no—
247C2×D257—22no—
248C2×D257—22no—
249C2×D257—22no—
250C2×D257—22no—
251C2×D257—22no—
252C2×D257—22no—
253C2×D257—22no—
254C2×D257—22no—
255C2×D257—22no—
256C2×D257—22no—
257C2×D257—22no—
258C2×D257—22no—
259C2×D257—22no—
260C2×D257—416no—
261C2×D257—416no—
262C2×D257—41no—
263C2×D257—41no—
264C2×D257—41no—
265C2×D257—41no—
266C2×D257C2×C22⋊C414yescomplemented
267C2×D257C4×D414yescomplemented
268C4×C257—22no—
269C4×C257—22no—
270C4×C257—41no—
271C4×C257—41no—
272C4×C257—41no—
273Dic257—22no—
274Dic257—41no—
275Dic257—41no—
276Dic257—41no—
277Dic257C2×C22⋊C414yescomplemented
278C2×C257—24no—
279C2×C257—42no—
280C2×C257C22×C22⋊C411yescharacteristic
281C2×C257C2×C4×D411yescharacteristic
282C2×C257#51811yescharacteristic
283C2×C257D4211yescharacteristic
284C2×C257D4211yescharacteristic
285C2×C257#54611yescharacteristic
286C2×C257#54611yescharacteristic
287D257—24no—
288D257—24no—
289D257—42no—
290C257D4×C22⋊C411yescharacteristic, complemented, Sylow
291D4×C22⋊C4—2571nomaximal, Sylow
292C23×D4—2571no—
293C2×C4×D4—2571no—
294C2×C4×D4—2571no—
295C22×C22⋊C4—2571no—
296C22×C22⋊C4—2571no—
297#390—2571no—
298#390—2571no—
299#386—2571no—
300#386—2571no—
301#386—2571no—
302#386D25711yescharacteristic, complemented
303#380—2571no—
304#379—2571no—
305#379—2571no—
306C4×C22⋊C4—2571no—
307C25—2571no—
308C25—2571no—
309C22×D4—2571no—
310C22×D4—2571no—
311C22×D4—2578no—
312C22×D4—2571no—
313C22×D4—5144no—
314C22×D4—5144no—
315C22×D4C2×D25711yescharacteristic
316C23×C4—2571no—
317C23×C4—2571no—
318C23×C4—2571no—
319C23×C4—2571no—
320C23×C4C2×D25711yescharacteristic, complemented
321C4×D4—5144no—
322C4×D4—5144no—
323C2×C4⋊C4—2571no—
324C2×C4⋊C4—2571no—
325C2×C22⋊C4—2571no—
326C2×C22⋊C4—2571no—
327C2×C22⋊C4—2571no—
328C2×C22⋊C4—2571no—
329C2×C22⋊C4—2571no—
330C2×C22⋊C4—2571no—
331C2×C22⋊C4—2571no—
332C2×C22⋊C4—2571no—
333C2×C22⋊C4—2571no—
334C2×C22⋊C4—2571no—
335C2×C22⋊C4—2571no—
336C2×C22⋊C4—2571no—
337C2×C22⋊C4—2571no—
338C2×C22⋊C4—5144no—
339C2×C22⋊C4—5144no—
340C2×C22⋊C4C2×D25711yescharacteristic, complemented
341C2×C22⋊C4C2×D25711yescharacteristic, complemented
342C2×C22⋊C4C2×D25711yescharacteristic, complemented
343C2×C42—2571no—
344C2×C42—2571no—
345#95—2571no—
346#95—2571no—
347#95C2×D25711yescharacteristic, complemented
348#95C2×D25711yescharacteristic, complemented
349C24—2571no—
350C24—2571no—
351C24—2571no—
352C24—2574no—
353C24—2574no—
354C24—5142no—
355C24—5142no—
356C24—5142no—
357C24—5142no—
358C24—5142no—
359C24—5142no—
360C24—5142no—
361C24—5142no—
362C24—10281no—
363C24—10281no—
364C24—10281no—
365C24—10281no—
366C24C22×D25711yescharacteristic
367C24C22×D25711yescharacteristic
368C2×D4—2574no—
369C2×D4—2574no—
370C2×D4—2574no—
371C2×D4—2574no—
372C2×D4—2574no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—5142no—
376C2×D4—5142no—
377C2×D4—5142no—
378C2×D4—5142no—
379C2×D4—51432no—
380C2×D4C4×D25714yescomplemented
381C22×C4—2571no—
382C22×C4—2571no—
383C22×C4—2574no—
384C22×C4—2571no—
385C22×C4—2571no—
386C22×C4—2571no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—5142no—
390C22×C4—5142no—
391C22×C4—5142no—
392C22×C4—5142no—
393C22×C4—5142no—
394C22×C4—5142no—
395C22×C4—5142no—
396C22×C4—5142no—
397C22×C4—41no—
398C22×C4—10281no—
399C22×C4—10281no—
400C22×C4—10281no—
401C22×C4C22×D25711yescharacteristic
402C22×C4C22×D25711yescharacteristic
403C22×C4C22×D25711yescharacteristic
404C22×C4C22×D25711yescharacteristic
405C22×C4C22×D25711yescharacteristic
406C4⋊C4—5142no—
407C4⋊C4—5142no—
408C22⋊C4—2574no—
409C22⋊C4—22no—
410C22⋊C4—22no—
411C22⋊C4—5142no—
412C22⋊C4—5142no—
413C22⋊C4—5142no—
414C22⋊C4—5142no—
415C22⋊C4—5142no—
416C22⋊C4—5142no—
417C22⋊C4—5142no—
418C22⋊C4—5142no—
419C22⋊C4—5142no—
420C22⋊C4—41no—
421C22⋊C4—10281no—
422C22⋊C4—10281no—
423C22⋊C4—10281no—
424C42—5142no—
425C42—5142no—
426C23—2572no—
427C23—2572no—
428C23—2572no—
429C23—2572no—
430C23—2572no—
431C23—2572no—
432C23—2572no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—5148no—
444C23—5148no—
445C23—5148no—
446C23—5141no—
447C23—5141no—
448C23—5141no—
449C23—5141no—
450C23—5141no—
451C23—5141no—
452C23—5141no—
453C23—5141no—
454C23—5141no—
455C23—5141no—
456C23—5141no—
457C23—5141no—
458C23—5141no—
459C23—5141no—
460C23—5148no—
461C23—10284no—
462C23—10284no—
463C23—10284no—
464C23—10284no—
465C23—10284no—
466C23—10284no—
467C23—10284no—
468C23—10284no—
469C23—10284no—
470C23—10284no—
471C23C2×C4×D25712yes—
472C23C2×C4×D25712yes—
473C23D4×D25712yes—
474C23D4×D25712yes—
475C23C23×D25711yescharacteristic
476D4—25716no—
477D4—28no—
478D4—5148no—
479D4—5148no—
480C2×C4—2572no—
481C2×C4—2572no—
482C2×C4—2572no—
483C2×C4—2572no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—5148no—
498C2×C4—5141no—
499C2×C4—5141no—
500C2×C4—5141no—
501C2×C4—5141no—
502C2×C4—5141no—
503C2×C4—5141no—
504C2×C4—5141no—
505C2×C4—5141no—
506C2×C4—5141no—
507C2×C4—5141no—
508C2×C4—5141no—
509C2×C4—44no—
510C2×C4—10284no—
511C2×C4—10284no—
512C2×C4—10284no—
513C2×C4C2×C4×D25712yes—
514C2×C4D4×D25712yescomplemented
515C22—2574no—
516C22—2574no—
517C22—22no—
518C22—22no—
519C22—22no—
520C22—22no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22—22no—
525C22—22no—
526C22—5142no—
527C22—5142no—
528C22—5142no—
529C22—5142no—
530C22—5142no—
531C22—5142no—
532C22—5142no—
533C22—5142no—
534C22—5142no—
535C22—5142no—
536C22—5142no—
537C22—5142no—
538C22—5142no—
539C22—5142no—
540C22—5142no—
541C22—41no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22—102816no—
546C22—102816no—
547C22—10281no—
548C22—10281no—
549C22—10281no—
550C22—10281no—
551C22C22⋊C4×D25714yescomplemented
552C22C22×C4×D25711yescharacteristic
553C22C2×D4×D25711yescharacteristic
554C22C2×D4×D25711yescharacteristic
555C22C2×D4×D25711yescharacteristic
556C22#41×D25711yescharacteristic
557C22#2255163711yescharacteristic
558C22#2255163811yescharacteristic
559C4—2574no—
560C4—22no—
561C4—22no—
562C4—5142no—
563C4—41no—
564C4—41no—
565C4—41no—
566C4—10281no—
567C4—10281no—
568C4—10281no—
569C2—24no—
570C2—5144no—
571C2—5144no—
572C2—42no—
573C2—10282no—
574C2C2×C22⋊C4×D25711yescharacteristic
575C2C4×D4×D25711yescharacteristic
576C2#3065720911yescharacteristic
577C2#3065736911yescharacteristic
578C2#3065737111yescharacteristic
579C2#3065737611yescharacteristic
580C2#3065737911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D257
commutatorC22×C257C23×C4
Fitting#386×C257C2
socle, cosocleC23×C257C24
prime radicalC22×D4×D257C2
hypercenter#386D257
hypocenterC257D4×C22⋊C4
LCSG ▷ C22×C257 ▷ C257
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C2×C4×C22⋊D257C2semidirect
3. C2×#22551552C2semidirect
4. C22×#16861595C2semidirect
5. C2×C22⋊C4×D257C2semidirect
6. #30656637C2semidirect
7. #30656637C2semidirect
8. #30656617C2semidirect
9. #30656615C2semidirect
10. #30656615C2semidirect
11. #30656596C2semidirect
12. #30656592C2semidirect
13. #30656592C2semidirect
14. #30656584C2semidirect
15. #386×C257C2semidirect
23. C2×D4×D257C4semidirect
24. C23×Dic257C22semidirect
26. C22×C4×D257C22semidirect
27. C22×C4×D257C22semidirect
30. C2×#16861597C22semidirect
31. C2×#16861597C22semidirect
32. C2×C22⋊Dic257C22semidirect
33. C2×C22⋊Dic257C22semidirect
34. C2×C22⋊Dic257C22semidirect
40. C2×#16861595C22semidirect
41. C2×#16861595C22semidirect
42. C2×#16861595C22semidirect
43. C2×#16861595C22semidirect
44. C2×#16861595C22semidirect
45. C2×#16861595C22semidirect
47. C2×C4×Dic257C22semidirect
48. C2×C4×Dic257C22semidirect
49. #22551499C22semidirect
50. #22551499C22semidirect
52. C23×C4×C257C22semidirect
53. C2×C22⋊C4×C257C22semidirect
54. C2×C22⋊C4×C257C22semidirect
55. C2×C22⋊C4×C257C22semidirect
56. #95×C257C22semidirect
57. #95×C257C22semidirect
70. C23×D257C2×C4semidirect
71. C23×D257C2×C4semidirect
77. C2×D1028C2×C4semidirect
81. C2×C22⋊D257C2×C4semidirect
101. C2×C4×D257C2×C4semidirect
116. #16861595D4semidirect
123. C2×D4×C257C2×C4semidirect
172. C22⋊D257C22⋊C4semidirect
188. C2×Dic257C2×D4semidirect
223. C2×C4×C257C2×D4semidirect
237. C22×C257C2×C22⋊C4semidirect
266. C2×D257C2×C22⋊C4semidirect
267. C2×D257C4×D4semidirect
277. Dic257C2×C22⋊C4semidirect
290. C257D4×C22⋊C4semidirect
302. #386D257semidirect
320. C23×C4C2×D257semidirect
340. C2×C22⋊C4C2×D257semidirect
341. C2×C22⋊C4C2×D257semidirect
342. C2×C22⋊C4C2×D257semidirect
347. #95C2×D257semidirect
348. #95C2×D257semidirect
380. C2×D4C4×D257semidirect
514. C2×C4D4×D257semidirect
551. C22C22⋊C4×D257semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D2572
C2×C4×C22⋊D2572
C2×#225515522
C22×#168615952
C2×C22⋊C4×D2572
#306566372
#306566172
#306566152
#306565962
#306565922
#306565842
#386×C2572
D4×C22⋊C4257