Finite Group Databasebeta

← #495596

Subgroups — #495596

Order 512.

1174 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35084C211yesmaximal, characteristic
2#20726C211yesmaximal, characteristic
3C2×C4×#99C211yesmaximal, characteristic
4#13296C212yesmaximal, complemented
5#13220C212yesmaximal, complemented
6#13084C214yesmaximal
7#12621C214yesmaximal, complemented
8C22×C22⋊Q8C411yescharacteristic, complemented
9C22×C4×D4C2211yescharacteristic
10#2920—21no—
11C22×#99C2212yes—
12#2855—21no—
13#2848—21no—
14#2638—21no—
15#2638—21no—
16#2583—21no—
17#2561—21no—
18#2486C411yescharacteristic, complemented
19#2486C2211yescharacteristic
20#2483C412yescomplemented
21#2415—21no—
22#2415—21no—
23#2409C411yescharacteristic, complemented
24#2409C2211yescharacteristic
25#2405C2212yes—
26#2322C412yescomplemented
27#2322C2212yes—
28#2280C411yescharacteristic, complemented
29C2×C4×C22⋊C4—21no—
30C2×#352C2218yes—
31#1821—21no—
32#1800—21no—
33#1800C2212yes—
34#1783C2212yes—
35C4×#100—21no—
36C4×#100C2212yes—
37C4×#99—21no—
38C4×#99—21no—
39C4×#99C2212yes—
40C4×#99C2212yes—
41C2×#342—21no—
42#1390C2218yes—
43C23×Q8C2×C411yescharacteristic, complemented
44C23×D4C2311yescharacteristic
45C24×C4C2×C411yescharacteristic
46C24×C4D411yescharacteristic
47C2×#123C2×C412yes—
48C2×#123C2312yes—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8—21no—
51C2×C22⋊Q8—21no—
52C2×C22⋊Q8—21no—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8—41no—
56C2×C22⋊Q8—41no—
57C2×C22⋊Q8—41no—
58C2×C22⋊Q8—41no—
59C2×C4×D4—41no—
60C2×C4×D4—41no—
61C2×C4×D4—41no—
62C2×C4×D4—41no—
63C2×C4×D4C2×C411yescharacteristic
64C2×C4×D4C2×C411yescharacteristic
65C2×C4×D4D411yescharacteristic
66C2×C4×D4D411yescharacteristic
67C2×C4×D4D411yescharacteristic
68C2×C4×D4D411yescharacteristic
69C2×C4×D4C2311yescharacteristic
70C2×C4×D4C2311yescharacteristic
71C2×#117—21no—
72C22×C4⋊C4C2×C412yescomplemented
73C22×C4⋊C4C2×C411yescharacteristic, complemented
74C22×C4⋊C4D411yescharacteristic
75C22×C22⋊C4—22no—
76C22×C22⋊C4C2×C412yes—
77C22×C22⋊C4C2×C411yescharacteristic
78C22×C22⋊C4D411yescharacteristic
79C22×C42—41no—
80C22×C42D411yescharacteristic
81#432—41no—
82#432—41no—
83#431—41no—
84#431—41no—
85#423—22no—
86C2×#100—22no—
87C2×#100C2314yes—
88C2×#99—22no—
89C2×#99—28no—
90C2×#99C2314yes—
91#407—22no—
92C4×M16—22no—
93#398—21no—
94#398—21no—
95#397—21no—
96#397—22no—
97#397—22no—
98#397—21no—
99#395—21no—
100#395—21no—
101#394—22no—
102#394—22no—
103#393—21no—
104#393—21no—
105#391—21no—
106#391—21no—
107#389—22no—
108#389—22no—
109#389—21no—
110#389—21no—
111#388—22no—
112#388—22no—
113#387—22no—
114#387—22no—
115#386C2×C412yes—
116#386C2312yes—
117#385—22no—
118#385—22no—
119#385—22no—
120#385—22no—
121#385C2×C412yescomplemented
122#385C2×C412yescomplemented
123#381—22no—
124#381—21no—
125#381C2×C412yescomplemented
126#381C2×C412yescomplemented
127#380—21no—
128#380—21no—
129#380—21no—
130#380C2×C411yescharacteristic, complemented
131#380C2×C411yescharacteristic, complemented
132#379—22no—
133#379C2×C412yes—
134C4×C22⋊C4—21no—
135C4×C22⋊C4—22no—
136C4×C22⋊C4—22no—
137C4×C22⋊C4—41no—
138C4×C22⋊C4—41no—
139C4×C22⋊C4C2×C412yescomplemented
140C2×#95—41no—
141#352—216no—
142#343—22no—
143#343—22no—
144#342—22no—
145#342—22no—
146C25C22×C411yescharacteristic
147C25C2×D411yescharacteristic
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8—41no—
151C22×Q8—41no—
152C22×Q8—82no—
153C22×D4—24no—
154C22×D4—41no—
155C22×D4—41no—
156C22×D4—42no—
157C22×D4C22×C411yescharacteristic
158C22×D4C2×D411yescharacteristic
159C22×D4C2×D411yescharacteristic
160C22×D4C2411yescharacteristic
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—42no—
170C23×C4—82no—
171C23×C4—82no—
172C23×C4C22⋊C411yescharacteristic
173C23×C4C22⋊C411yescharacteristic
174C23×C4C22⋊C411yescharacteristic
175C23×C4C22⋊C411yescharacteristic
176C23×C4C22⋊C411yescharacteristic
177C23×C4C22⋊C411yescharacteristic
178C23×C4C22⋊C411yescharacteristic
179C23×C4C22⋊C411yescharacteristic
180C23×C4C22×C412yes—
181C23×C4C22×C411yescharacteristic
182C23×C4C22×C411yescharacteristic
183C23×C4C22×C412yes—
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C2×M16—24no—
188#123—28no—
189#123—44no—
190C22⋊Q8—42no—
191C22⋊Q8—42no—
192C22⋊Q8—42no—
193C22⋊Q8—42no—
194C22⋊Q8—42no—
195C22⋊Q8—42no—
196C22⋊Q8—41no—
197C22⋊Q8—41no—
198C22⋊Q8—41no—
199C22⋊Q8—41no—
200C22⋊Q8—82no—
201C22⋊Q8—82no—
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—
216C4×D4—41no—
217C4×D4—41no—
218#117—82no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4C22⋊C411yescharacteristic
236C2×C4⋊C4C22⋊C411yescharacteristic
237C2×C4⋊C4C2×D411yescharacteristic
238C2×C4⋊C4C2×D411yescharacteristic
239C2×C22⋊C4—28no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
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—44no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4C22⋊C411yescharacteristic
258C2×C22⋊C4C22⋊C411yescharacteristic
259C2×C22⋊C4C22⋊C411yescharacteristic
260C2×C22⋊C4C22⋊C411yescharacteristic
261C2×C22⋊C4C22×C412yes—
262C2×C22⋊C4C22×C412yes—
263C2×C22⋊C4C22×C411yescharacteristic
264C2×C22⋊C4C22×C411yescharacteristic
265C2×C22⋊C4C2×D411yescharacteristic
266C2×C22⋊C4C2×D411yescharacteristic
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—22no—
272C2×C42—42no—
273C2×C42—42no—
274C2×C42—84no—
275C2×C42C22⋊C411yescharacteristic
276C2×C42C22⋊C411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C4⋊C8—42no—
280C4⋊C8—42no—
281#100—28no—
282#100—44no—
283#99—28no—
284#99—24no—
285#99#4118yes—
286C22⋊C8—42no—
287C22⋊C8—42no—
288#97—42no—
289C4×C8—42no—
290#95—21no—
291#95—21no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—22no—
299#95—22no—
300#95—22no—
301#95—42no—
302#95—42no—
303C24—21no—
304C24—22no—
305C24—21no—
306C24—22no—
307C24—21no—
308C24—21no—
309C24—22no—
310C24—41no—
311C24—42no—
312C24—42no—
313C24—81no—
314C24—81no—
315C24C2×C22⋊C411yescharacteristic
316C24C2×C22⋊C411yescharacteristic
317C24C2×C22⋊C411yescharacteristic
318C24C23×C411yescharacteristic
319C24C22×D411yescharacteristic
320C2×Q8—42no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—41no—
324C2×Q8—41no—
325C2×Q8—82no—
326C2×Q8—84no—
327C2×Q8—84no—
328C2×D4—22no—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4—41no—
336C2×D4—41no—
337C2×D4—41no—
338C2×D4—41no—
339C2×D4—48no—
340C2×D4—48no—
341C2×D4C2×#4114yes—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—22no—
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—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—42no—
373C22×C4—41no—
374C22×C4—44no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—44no—
381C22×C4—48no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—82no—
386C22×C4—82no—
387C22×C4—82no—
388C22×C4—81no—
389C22×C4—81no—
390C22×C4—81no—
391C22×C4—82no—
392C22×C4—82no—
393C22×C4—82no—
394C22×C4—82no—
395C22×C4—81no—
396C22×C4—81no—
397C22×C4—81no—
398C22×C4—82no—
399C22×C4—84no—
400C22×C4—84no—
401C22×C4#9911yescharacteristic
402C22×C4#9911yescharacteristic
403C22×C4#9911yescharacteristic
404C22×C4#9911yescharacteristic
405C22×C4C2×C22⋊C411yescharacteristic
406C22×C4C2×C22⋊C411yescharacteristic
407C22×C4C2×C22⋊C411yescharacteristic
408C22×C4C2×C22⋊C411yescharacteristic
409C22×C4C2×C22⋊C411yescharacteristic
410C22×C4C2×C22⋊C411yescharacteristic
411C22×C4C2×C22⋊C411yescharacteristic
412C22×C4C2×C22⋊C411yescharacteristic
413C22×C4C2×C22⋊C411yescharacteristic
414C22×C4C22×D411yescharacteristic
415M16—44no—
416M16—44no—
417C2×C8—44no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—44no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—44no—
427C4⋊C4—44no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—82no—
433C4⋊C4—82no—
434C22⋊C4—24no—
435C22⋊C4—28no—
436C22⋊C4—41no—
437C22⋊C4—41no—
438C22⋊C4—41no—
439C22⋊C4—41no—
440C22⋊C4—41no—
441C22⋊C4—41no—
442C22⋊C4—41no—
443C22⋊C4—41no—
444C22⋊C4—44no—
445C22⋊C4—48no—
446C22⋊C4—44no—
447C22⋊C4—44no—
448C22⋊C4—44no—
449C22⋊C4—42no—
450C22⋊C4—42no—
451C22⋊C4—44no—
452C22⋊C4—42no—
453C22⋊C4—82no—
454C22⋊C4—82no—
455C22⋊C4—82no—
456C42—21no—
457C42—21no—
458C42—21no—
459C42—21no—
460C42—21no—
461C42—21no—
462C42—21no—
463C42—21no—
464C42—82no—
465C42—84no—
466C42—84no—
467C23—24no—
468C23—24no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—22no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—41no—
482C23—42no—
483C23—42no—
484C23—41no—
485C23—41no—
486C23—42no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—42no—
492C23—41no—
493C23—41no—
494C23—42no—
495C23—42no—
496C23—81no—
497C23—81no—
498C23—81no—
499C23—81no—
500C23—84no—
501C23—84no—
502C23—84no—
503C23—84no—
504C23C2×#9911yescharacteristic
505C23C2×#9911yescharacteristic
506C23#41012yes—
507C23C22×C22⋊C411yescharacteristic
508C23C4×#4112yes—
509C23#52512yes—
510Q8—82no—
511Q8—84no—
512Q8—82no—
513D4—44no—
514D4—44no—
515D4—44no—
516D4—44no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
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—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—48no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
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—81no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4—84no—
578C2×C4—816no—
579C2×C4—84no—
580C2×C4—84no—
581C2×C4—84no—
582C2×C4C2×#9911yescharacteristic
583C2×C4C2×#9911yescharacteristic
584C2×C4C2×#9911yescharacteristic
585C2×C4C2×#9911yescharacteristic
586C2×C4#53212yes—
587C8—84no—
588C22—21no—
589C22—21no—
590C22—21no—
591C22—21no—
592C22—21no—
593C22—21no—
594C22—22no—
595C22—21no—
596C22—22no—
597C22—22no—
598C22—42no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22—48no—
604C22—42no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—81no—
615C22—81no—
616C22—84no—
617C22—81no—
618C22—81no—
619C22—84no—
620C22—84no—
621C22#212312yes—
622C22#229711yescharacteristic
623C22C22×#9911yescharacteristic
624C22C2×#41011yescharacteristic
625C4—21no—
626C4—21no—
627C4—21no—
628C4—22no—
629C4—41no—
630C4—41no—
631C4—41no—
632C4—41no—
633C4—41no—
634C4—41no—
635C4—41no—
636C4—82no—
637C4—82no—
638C4—88no—
639C4—82no—
640C4—82no—
641C4#212112yes—
642C2—21no—
643C2—21no—
644C2—22no—
645C2—42no—
646C2—41no—
647C2—42no—
648C2—42no—
649C2—81no—
650C2#2142011yescharacteristic
651C2C2×#212111yescharacteristic
652C2C2×#212311yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC24C23×C4
Frattini, cosocleC22×D4C24
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #13296C2semidirect
5. #13220C2semidirect
7. #12621C2semidirect
8. C22×C22⋊Q8C4semidirect
18. #2486C4semidirect
20. #2483C4semidirect
23. #2409C4semidirect
26. #2322C4semidirect
28. #2280C4semidirect
43. C23×Q8C2×C4semidirect
72. C22×C4⋊C4C2×C4semidirect
73. C22×C4⋊C4C2×C4semidirect
121. #385C2×C4semidirect
122. #385C2×C4semidirect
125. #381C2×C4semidirect
126. #381C2×C4semidirect
130. #380C2×C4semidirect
131. #380C2×C4semidirect
139. C4×C22⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#350842
#207262
C2×C4×#992
#132962
#132202
#130842
#126212