Finite Group Databasebeta

← #481639

Subgroups — #481639

Order 512.

1139 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35010C211yesmaximal, characteristic
2C2×C4×#99C211yesmaximal, characteristic
3C2×#1466C211yesmaximal, characteristic
4#11789C214yesmaximal, complemented
5#11786C212yesmaximal, complemented
6#11785C212yesmaximal, complemented
7#11716C214yesmaximal, complemented
8C22×C22⋊Q8C411yescharacteristic, complemented
9C22×C4×D4C2211yescharacteristic
10C2×#432—21no—
11C22×#99—21no—
12#2855—21no—
13#2848—21no—
14#2627C412yescomplemented
15#2626C2212yes—
16#2579—21no—
17#2579—21no—
18#2564C412yescomplemented
19#2564C2212yes—
20#2487C411yescharacteristic, complemented
21#2487C2211yescharacteristic
22#2417—21no—
23#2417—21no—
24#2409C411yescharacteristic, complemented
25#2409C2211yescharacteristic
26#2315—21no—
27#2308—21no—
28C2×#380C411yescharacteristic, complemented
29C2×C4×C22⋊C4—21no—
30C4×#99—21no—
31C4×#99—21no—
32C4×#99C2212yes—
33C4×#99C2212yes—
34C2×#342C2212yes—
35#1466—21no—
36#1466—21no—
37#1466C2212yes—
38#1466C2212yes—
39C2×#323—21no—
40C2×#323C2212yes—
41#1352C2218yescomplemented
42#1351C2218yescomplemented
43C23×Q8C2×C411yescharacteristic, complemented
44C23×D4D411yescharacteristic
45C24×C4C2×C411yescharacteristic
46C24×C4D411yescharacteristic
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8—21no—
51C2×C22⋊Q8—41no—
52C2×C22⋊Q8—41no—
53C2×C22⋊Q8—41no—
54C2×C22⋊Q8—41no—
55C2×C22⋊Q8C2×C412yes—
56C2×C22⋊Q8C2×C412yescomplemented
57C2×C4×D4—41no—
58C2×C4×D4—41no—
59C2×C4×D4—41no—
60C2×C4×D4—41no—
61C2×C4×D4C2×C411yescharacteristic
62C2×C4×D4C2×C411yescharacteristic
63C2×C4×D4D411yescharacteristic
64C2×C4×D4D411yescharacteristic
65C2×C4×D4D411yescharacteristic
66C2×C4×D4D411yescharacteristic
67C2×C4×D4C2311yescharacteristic
68C2×C4×D4C2311yescharacteristic
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4C2×C411yescharacteristic
72C22×C4⋊C4C2×C411yescharacteristic, complemented
73C22×C22⋊C4—22no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4D411yescharacteristic
77C22×C22⋊C4C2311yescharacteristic
78C22×C42—41no—
79C22×C42D411yescharacteristic
80#432—22no—
81#432—22no—
82#432—41no—
83#432—41no—
84C2×C4⋊C8—41no—
85C2×#99—28no—
86C2×#99—22no—
87C2×#99—22no—
88C2×C22⋊C8—22no—
89C2×#97—41no—
90#400C2×C412yescomplemented
91#400C2×C412yescomplemented
92#399C2×C412yes—
93#399C2312yes—
94#398—22no—
95#398—22no—
96#396—22no—
97#396—22no—
98#395C2×C412yescomplemented
99#395C2×C412yescomplemented
100#393C2×C412yes—
101#393C2312yes—
102#391—21no—
103#391—21no—
104#389—21no—
105#389—21no—
106#387—22no—
107#387—22no—
108#386—22no—
109#386—21no—
110#386—21no—
111#386—22no—
112#386—21no—
113#386—21no—
114#385—22no—
115#385—22no—
116#385—22no—
117#385—22no—
118#385—21no—
119#385—21no—
120#385—21no—
121#385—21no—
122#381—22no—
123#380—21no—
124#380—21no—
125#380—21no—
126#380—21no—
127#380—22no—
128#380—22no—
129#380C2×C412yes—
130#380C2×C411yescharacteristic, complemented
131#380C2×C411yescharacteristic, complemented
132#379—22no—
133C4×C22⋊C4—22no—
134C4×C22⋊C4—22no—
135C4×C22⋊C4—41no—
136C4×C22⋊C4—41no—
137C2×#95—41no—
138C2×#95C2×C412yescomplemented
139#342—22no—
140#342C2314yes—
141#323—22no—
142#323—22no—
143#323—22no—
144#323C2314yes—
145C25C22⋊C411yescharacteristic
146C25C2×D411yescharacteristic
147C22×Q8—21no—
148C22×Q8—41no—
149C22×Q8—41no—
150C22×Q8—82no—
151C22×Q8C22×C412yes—
152C22×D4—24no—
153C22×D4—24no—
154C22×D4—41no—
155C22×D4—41no—
156C22×D4C22⋊C411yescharacteristic
157C22×D4C22⋊C411yescharacteristic
158C22×D4C2×D411yescharacteristic
159C22×D4C2×D411yescharacteristic
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—22no—
171C23×C4—42no—
172C23×C4—82no—
173C23×C4—82no—
174C23×C4C22⋊C411yescharacteristic
175C23×C4C22⋊C411yescharacteristic
176C23×C4C22⋊C411yescharacteristic
177C23×C4C22⋊C411yescharacteristic
178C23×C4C22⋊C411yescharacteristic
179C23×C4C22⋊C411yescharacteristic
180C23×C4C22⋊C411yescharacteristic
181C23×C4C22×C411yescharacteristic
182C23×C4C22×C411yescharacteristic
183C23×C4C22×C411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C22×C8—42no—
188C22⋊Q8—42no—
189C22⋊Q8—42no—
190C22⋊Q8—41no—
191C22⋊Q8—41no—
192C22⋊Q8—41no—
193C22⋊Q8—41no—
194C22⋊Q8—42no—
195C22⋊Q8—42no—
196C22⋊Q8—42no—
197C22⋊Q8—42no—
198C22⋊Q8—82no—
199C22⋊Q8—82no—
200C4×D4—41no—
201C4×D4—41no—
202C4×D4—41no—
203C4×D4—41no—
204C4×D4—41no—
205C4×D4—41no—
206C4×D4—41no—
207C4×D4—41no—
208C4×D4—41no—
209C4×D4—41no—
210C4×D4—41no—
211C4×D4—41no—
212C4×D4—41no—
213C4×D4—41no—
214C4×D4—41no—
215C4×D4—41no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—82no—
233C2×C4⋊C4C22⋊C411yescharacteristic
234C2×C4⋊C4C22⋊C411yescharacteristic
235C2×C4⋊C4C22×C412yes—
236C2×C4⋊C4C22×C411yescharacteristic
237C2×C4⋊C4C22×C411yescharacteristic
238C2×C22⋊C4—28no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—41no—
250C2×C22⋊C4—41no—
251C2×C22⋊C4—41no—
252C2×C22⋊C4—41no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4C22⋊C411yescharacteristic
259C2×C22⋊C4C22⋊C411yescharacteristic
260C2×C22⋊C4C22×C411yescharacteristic
261C2×C22⋊C4C2×D411yescharacteristic
262C2×C22⋊C4C2×D411yescharacteristic
263C2×C22⋊C4C2×D411yescharacteristic
264C2×C22⋊C4C2×D411yescharacteristic
265C2×C22⋊C4C2411yescharacteristic
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—42no—
271C2×C42—42no—
272C2×C42—84no—
273C2×C42C22⋊C411yescharacteristic
274C2×C42C22⋊C411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C4⋊C8—42no—
278C4⋊C8—42no—
279#99—28no—
280#99—28no—
281C22⋊C8—24no—
282C22⋊C8—42no—
283#97—42no—
284#97—42no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95—42no—
295#95—42no—
296#95C22×C412yes—
297#95C22×C412yes—
298C24—21no—
299C24—21no—
300C24—21no—
301C24—21no—
302C24—22no—
303C24—22no—
304C24—81no—
305C24—81no—
306C24—81no—
307C24—81no—
308C24#10414yes—
309C24C2×C22⋊C411yescharacteristic
310C24C2×C22⋊C411yescharacteristic
311C24C2×C22⋊C411yescharacteristic
312C24C2×C22⋊C411yescharacteristic
313C24#13714yes—
314C24C22×D411yescharacteristic
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8—41no—
319C2×Q8—41no—
320C2×Q8—82no—
321C2×Q8—84no—
322C2×Q8—84no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4—48no—
336C2×D4—48no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—42no—
368C22×C4—44no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—44no—
374C22×C4—48no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—44no—
378C22×C4—82no—
379C22×C4—82no—
380C22×C4—82no—
381C22×C4—81no—
382C22×C4—81no—
383C22×C4—81no—
384C22×C4—81no—
385C22×C4—82no—
386C22×C4—82no—
387C22×C4—82no—
388C22×C4—82no—
389C22×C4—82no—
390C22×C4—82no—
391C22×C4—81no—
392C22×C4—81no—
393C22×C4—81no—
394C22×C4—82no—
395C22×C4—82no—
396C22×C4—84no—
397C22×C4—84no—
398C22×C4#9911yescharacteristic
399C22×C4#9911yescharacteristic
400C22×C4#9911yescharacteristic
401C22×C4#9911yescharacteristic
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C2×C22⋊C411yescharacteristic
404C22×C4C2×C22⋊C411yescharacteristic
405C22×C4C2×C22⋊C411yescharacteristic
406C22×C4C2×C22⋊C411yescharacteristic
407C22×C4C2×C22⋊C411yescharacteristic
408C22×C4C2×C22⋊C411yescharacteristic
409C22×C4C2×C22⋊C411yescharacteristic
410C22×C4C23×C411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C2×C8—44no—
413C2×C8—84no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—42no—
419C4⋊C4—44no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—82no—
427C4⋊C4—82no—
428C4⋊C4—82no—
429C4⋊C4—82no—
430C4⋊C4—82no—
431C22⋊C4—28no—
432C22⋊C4—41no—
433C22⋊C4—41no—
434C22⋊C4—41no—
435C22⋊C4—41no—
436C22⋊C4—41no—
437C22⋊C4—41no—
438C22⋊C4—41no—
439C22⋊C4—41no—
440C22⋊C4—48no—
441C22⋊C4—44no—
442C22⋊C4—44no—
443C22⋊C4—42no—
444C22⋊C4—42no—
445C22⋊C4—42no—
446C22⋊C4—42no—
447C22⋊C4—44no—
448C22⋊C4—82no—
449C22⋊C4—82no—
450C22⋊C4—82no—
451C22⋊C4—82no—
452C42—21no—
453C42—21no—
454C42—21no—
455C42—21no—
456C42—21no—
457C42—21no—
458C42—21no—
459C42—21no—
460C42—84no—
461C42—84no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—24no—
469C23—21no—
470C23—21no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—42no—
481C23—42no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—81no—
487C23—81no—
488C23—81no—
489C23—81no—
490C23—81no—
491C23—81no—
492C23—84no—
493C23—84no—
494C23—84no—
495C23—84no—
496C23C2×#9911yescharacteristic
497C23C2×#9911yescharacteristic
498C23#41912yes—
499C23#41912yes—
500C23C2×#10412yes—
501C23C2×#10412yes—
502C23C2×#10412yes—
503C23C22×C22⋊C411yescharacteristic
504C23C2×#13712yes—
505Q8—82no—
506Q8—84no—
507Q8—82no—
508D4—44no—
509D4—44no—
510D4—44no—
511D4—44no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—48no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—81no—
562C2×C4—81no—
563C2×C4—81no—
564C2×C4—81no—
565C2×C4—81no—
566C2×C4—81no—
567C2×C4—81no—
568C2×C4—81no—
569C2×C4—81no—
570C2×C4—81no—
571C2×C4—81no—
572C2×C4—81no—
573C2×C4—84no—
574C2×C4—816no—
575C2×C4—84no—
576C2×C4—84no—
577C2×C4—84no—
578C2×C4—164no—
579C2×C4C2×#9911yescharacteristic
580C2×C4C2×#9911yescharacteristic
581C2×C4C2×#9911yescharacteristic
582C2×C4C2×#9911yescharacteristic
583C8—84no—
584C22—21no—
585C22—21no—
586C22—21no—
587C22—21no—
588C22—21no—
589C22—21no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22—48no—
601C22—42no—
602C22—42no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—81no—
612C22—81no—
613C22—84no—
614C22—84no—
615C22—84no—
616C22#150812yes—
617C22#150812yes—
618C22C22×#9911yescharacteristic
619C22C2×#41911yescharacteristic
620C22C22×#10411yescharacteristic
621C4—21no—
622C4—21no—
623C4—21no—
624C4—41no—
625C4—41no—
626C4—41no—
627C4—41no—
628C4—41no—
629C4—41no—
630C4—41no—
631C4—41no—
632C4—82no—
633C4—82no—
634C4—88no—
635C4—82no—
636C4—82no—
637C4#212112yes—
638C2—21no—
639C2—21no—
640C2—22no—
641C2—22no—
642C2—42no—
643C2—42no—
644C2—42no—
645C2—42no—
646C2C2×#150811yescharacteristic
647C2C2×#212111yescharacteristic
648C2#3335811yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11789C2semidirect
5. #11786C2semidirect
6. #11785C2semidirect
7. #11716C2semidirect
8. C22×C22⋊Q8C4semidirect
14. #2627C4semidirect
18. #2564C4semidirect
20. #2487C4semidirect
24. #2409C4semidirect
28. C2×#380C4semidirect
41. #1352C22semidirect
42. #1351C22semidirect
43. C23×Q8C2×C4semidirect
56. C2×C22⋊Q8C2×C4semidirect
72. C22×C4⋊C4C2×C4semidirect
90. #400C2×C4semidirect
91. #400C2×C4semidirect
98. #395C2×C4semidirect
99. #395C2×C4semidirect
130. #380C2×C4semidirect
131. #380C2×C4semidirect
138. C2×#95C2×C4semidirect

Maximal subgroups

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

subgroupindex
#350102
C2×C4×#992
C2×#14662
#117892
#117862
#117852
#117162