Finite Group Databasebeta

← #205792

Subgroups — #205792

Order 512.

1183 conjugacy classes of subgroups.

The subgroup lattice has 669 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3187C211yesmaximal, characteristic
2#21939C211yesmaximal, characteristic, complemented
3#21825C211yesmaximal, characteristic, complemented
4#21793C211yesmaximal, characteristic
5#20527C211yesmaximal, characteristic, complemented
6#17578C211yesmaximal, characteristic, complemented
7#17577C211yesmaximal, characteristic
8#9786C214yesmaximal, complemented
9#9781C214yesmaximal, complemented
10C22×C22⋊Q8C2211yescharacteristic
11C22×#121C2211yescharacteristic
12#3187—22no—
13#3187—22no—
14#3187—22no—
15#3187—22no—
16C2×#481C412yescomplemented
17C2×#483C412yescomplemented
18C2×#465C2212yes—
19C2×#465C2212yes—
20C2×#448C412yescomplemented
21C2×#450C412yescomplemented
22C2×#407C2211yescharacteristic
23C22×C22⋊C8C2211yescharacteristic
24#2280C2211yescharacteristic
25#2280C2211yescharacteristic
26#2280C2211yescharacteristic
27#1996—22no—
28#1996C2214yescomplemented
29#1964—22no—
30#1964C2214yes—
31#1854—22no—
32#1854—22no—
33#1819—22no—
34#1808C2212yescomplemented
35#1807C2212yescomplemented
36#1781—22no—
37#1753—22no—
38#1752C2212yescomplemented
39C8×C22⋊C4C2212yescomplemented
40#1750—22no—
41#1520C2214yescomplemented
42#1519C2214yes—
43#1511—22no—
44#1510—22no—
45#1508—22no—
46#1507—22no—
47C23×Q8C2×C411yescharacteristic
48C23×D4—21no—
49C23×D4C2×C411yescharacteristic
50C24×C4C2311yescharacteristic
51C22×SD16C2×C412yes—
52C22×SD16C2×C412yes—
53C22×M16—21no—
54C23×C8C2311yescharacteristic
55C23×C8C2311yescharacteristic
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8—21no—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8—21no—
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8C2×C412yes—
67C2×C22⋊Q8C2312yes—
68C2×#121—21no—
69C2×#121—21no—
70C2×#121—21no—
71C2×#121—21no—
72C2×#121—21no—
73C2×#121—21no—
74C2×#121—21no—
75C2×#121—21no—
76C2×#121—21no—
77C2×#121—21no—
78C2×#121C2×C412yes—
79C2×#121C2312yes—
80C2×#117—21no—
81C2×#117—21no—
82C2×#117—21no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4C2×C411yescharacteristic
85C22×C4⋊C4C2×C411yescharacteristic
86C22×C22⋊C4—21no—
87C22×C22⋊C4—21no—
88#483—24no—
89#483—24no—
90#481—24no—
91#481—24no—
92#465—24no—
93#465—24no—
94#465—24no—
95#465—24no—
96#450—24no—
97#450—24no—
98#448—24no—
99#448—24no—
100C2×#107—22no—
101C2×#106—22no—
102C2×#106C2×C412yes—
103C2×#106C2×C412yes—
104C2×#103C2×C412yes—
105C2×#103C2×C412yes—
106C2×#102C2×C412yes—
107C2×#102C2×C412yes—
108C2×#99—28no—
109#407—21no—
110#407—21no—
111#407D412yes—
112#407D412yes—
113C2×C22⋊C8—21no—
114C2×C22⋊C8C2×C412yes—
115C2×C22⋊C8C2×C412yes—
116C2×C22⋊C8D412yes—
117C2×C22⋊C8D412yes—
118C2×C22⋊C8D412yes—
119C2×C22⋊C8D412yes—
120C2×C22⋊C8C2312yes—
121C2×#97—22no—
122C2×C4×C8—22no—
123#381—21no—
124#381—21no—
125#381—21no—
126#380—21no—
127#380—21no—
128#380—21no—
129#380D412yes—
130#380C2312yes—
131#380C2312yes—
132#379—21no—
133#379—21no—
134#379D412yes—
135C4×C22⋊C4—21no—
136C4×C22⋊C4—21no—
137C4×C22⋊C4—21no—
138C4×C22⋊C4—21no—
139C4×C22⋊C4—21no—
140C4×C22⋊C4C2312yes—
141#342—28no—
142#340—22no—
143#340—22no—
144#336—22no—
145#336—22no—
146#328—28no—
147#327—28no—
148C25—21no—
149C25C2×D411yescharacteristic
150C22×Q8—21no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×Q8—21no—
155C22×Q8—81no—
156C22×Q8—81no—
157C22×Q8C22×C412yes—
158C22×D4—22no—
159C22×D4—22no—
160C22×D4—21no—
161C22×D4—21no—
162C22×D4—21no—
163C22×D4—21no—
164C22×D4—21no—
165C22×D4—41no—
166C22×D4—41no—
167C22×D4—41no—
168C22×D4—41no—
169C22×D4—81no—
170C22×D4—81no—
171C22×D4—84no—
172C22×D4C22×C412yes—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—81no—
177C23×C4—81no—
178C23×C4C22×C411yescharacteristic
179C23×C4C22×C411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4#4111yescharacteristic
188C23×C4#4111yescharacteristic
189C23×C4#4111yescharacteristic
190C23×C4#4111yescharacteristic
191C23×C4C2411yescharacteristic
192C2×SD16—24no—
193C2×SD16—24no—
194C2×SD16—82no—
195C2×SD16—82no—
196C2×M16—22no—
197C2×M16—22no—
198C2×M16—81no—
199C2×M16—81no—
200C22×C8—22no—
201C22×C8—81no—
202C22×C8—81no—
203C22×C8C22×C412yes—
204C22×C8C22×C412yes—
205C22×C8C2×D412yes—
206C22×C8C2×D412yes—
207C22×C8C2×D412yes—
208C22×C8C2×D412yes—
209C22⋊Q8—22no—
210C22⋊Q8—22no—
211C22⋊Q8—22no—
212C22⋊Q8—22no—
213C22⋊Q8—22no—
214C22⋊Q8—22no—
215C22⋊Q8—22no—
216C22⋊Q8—22no—
217C22⋊Q8—41no—
218C22⋊Q8—41no—
219C22⋊Q8—41no—
220C22⋊Q8—41no—
221C22⋊Q8—41no—
222C22⋊Q8—41no—
223C22⋊Q8—41no—
224C22⋊Q8—41no—
225#121—22no—
226#121—22no—
227#121—22no—
228#121—22no—
229#121—22no—
230#121—22no—
231#121—22no—
232#121—22no—
233#121—41no—
234#121—41no—
235#121—41no—
236#121—41no—
237#121—41no—
238#121—41no—
239#121—41no—
240#121—41no—
241#117—81no—
242#117—81no—
243#117—81no—
244#117—81no—
245#117—81no—
246#117—81no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—41no—
263C2×C4⋊C4—41no—
264C2×C4⋊C4—41no—
265C2×C4⋊C4—41no—
266C2×C4⋊C4C22×C412yes—
267C2×C4⋊C4C22×C412yes—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—41no—
277C2×C22⋊C4—41no—
278C2×C22⋊C4—41no—
279C2×C22⋊C4—41no—
280C2×C22⋊C4—41no—
281C2×C22⋊C4—41no—
282C2×C22⋊C4—41no—
283C2×C22⋊C4—41no—
284C2×C22⋊C4—41no—
285C2×C22⋊C4—41no—
286C2×C22⋊C4—41no—
287C2×C22⋊C4—41no—
288C2×C22⋊C4C2×D414yes—
289C2×C42—22no—
290C2×C42—41no—
291C2×C42—41no—
292#107—82no—
293#106—24no—
294#106—24no—
295#106—82no—
296#103—24no—
297#103—24no—
298#102—24no—
299#102—24no—
300#99—88no—
301C22⋊C8—22no—
302C22⋊C8—22no—
303C22⋊C8—22no—
304C22⋊C8C22⋊C414yes—
305C22⋊C8C22⋊C414yes—
306C22⋊C8C22⋊C414yes—
307C22⋊C8C22⋊C414yes—
308C22⋊C8C2×D414yes—
309#97—82no—
310C4×C8—82no—
311#95—22no—
312#95—22no—
313#95—22no—
314#95—22no—
315#95—41no—
316#95—41no—
317C24—24no—
318C24—22no—
319C24—41no—
320C24—41no—
321C24—82no—
322C24—81no—
323C24—81no—
324C24—82no—
325C24—82no—
326C24C4×D411yescharacteristic
327C24C4×D411yescharacteristic
328C24#12111yescharacteristic
329C24#12111yescharacteristic
330C24#12711yescharacteristic
331C24#12711yescharacteristic
332C24C22×D411yescharacteristic
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8—42no—
336C2×Q8—42no—
337C2×Q8—41no—
338C2×Q8—41no—
339C2×Q8—81no—
340C2×Q8—81no—
341C2×Q8—81no—
342C2×Q8—81no—
343C2×Q8—82no—
344C2×Q8—82no—
345C2×Q8—82no—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—42no—
349C2×D4—42no—
350C2×D4—42no—
351C2×D4—42no—
352C2×D4—48no—
353C2×D4—42no—
354C2×D4—42no—
355C2×D4—41no—
356C2×D4—41no—
357C2×D4—81no—
358C2×D4—81no—
359C2×D4—81no—
360C2×D4—81no—
361C2×D4—84no—
362C2×D4—82no—
363C2×D4—82no—
364C2×D4—82no—
365C2×D4—84no—
366C2×D4—84no—
367C2×D4—84no—
368C2×D4—84no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—81no—
396C22×C4—81no—
397C22×C4—81no—
398C22×C4—81no—
399C22×C4—81no—
400C22×C4—81no—
401C22×C4—81no—
402C22×C4—82no—
403C22×C4—82no—
404C22×C4—82no—
405C22×C4C2×C22⋊C411yescharacteristic
406C22×C4C2×C22⋊C411yescharacteristic
407C22×C4C2×C22⋊C411yescharacteristic
408C22×C4C2×C22⋊C411yescharacteristic
409C22×C4C4×D411yescharacteristic
410C22×C4C4×D411yescharacteristic
411C22×C4C4×D411yescharacteristic
412C22×C4C4×D411yescharacteristic
413C22×C4C4×D411yescharacteristic
414C22×C4C4×D411yescharacteristic
415C22×C4#12111yescharacteristic
416C22×C4#12111yescharacteristic
417C22×C4#12111yescharacteristic
418C22×C4#12111yescharacteristic
419C22×C4#12111yescharacteristic
420C22×C4#12111yescharacteristic
421C22×C4#12411yescharacteristic
422C22×C4#12411yescharacteristic
423C22×C4#12411yescharacteristic
424C22×C4#12411yescharacteristic
425C22×C4#12711yescharacteristic
426C22×C4#12711yescharacteristic
427C22×C4C23×C411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C2×#4111yescharacteristic
432C22×C4C2×#4111yescharacteristic
433SD16—84no—
434SD16—84no—
435M16—82no—
436M16—82no—
437C2×C8—24no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8—42no—
441C2×C8—42no—
442C2×C8—42no—
443C2×C8—42no—
444C2×C8—42no—
445C2×C8—42no—
446C2×C8C2×C22⋊C414yes—
447C2×C8C2×C22⋊C414yes—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—41no—
459C4⋊C4—41no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—82no—
465C22⋊C4—28no—
466C22⋊C4—42no—
467C22⋊C4—42no—
468C22⋊C4—42no—
469C22⋊C4—42no—
470C22⋊C4—42no—
471C22⋊C4—42no—
472C22⋊C4—42no—
473C22⋊C4—42no—
474C22⋊C4—44no—
475C22⋊C4—44no—
476C22⋊C4—82no—
477C22⋊C4—82no—
478C22⋊C4—82no—
479C22⋊C4—82no—
480C22⋊C4—82no—
481C22⋊C4—82no—
482C42—42no—
483C42—42no—
484C42—82no—
485C42—82no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—42no—
507C23—44no—
508C23—82no—
509C23—82no—
510C23—82no—
511C23—82no—
512C23—82no—
513C23—82no—
514C23—82no—
515C23—82no—
516C23—88no—
517C23—168no—
518C23#39011yescharacteristic
519C23#39011yescharacteristic
520C23#39011yescharacteristic
521C23#39011yescharacteristic
522C23C2×C4×D411yescharacteristic
523C23C2×#12111yescharacteristic
524C23C2×#12711yescharacteristic
525C23#57714yes—
526Q8—42no—
527Q8—42no—
528Q8—82no—
529Q8—82no—
530Q8—82no—
531D4—42no—
532D4—42no—
533D4—82no—
534D4—82no—
535D4—82no—
536D4—84no—
537D4—84no—
538D4—84no—
539D4—84no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—82no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4—82no—
586C2×C4—82no—
587C2×C4—82no—
588C2×C4—82no—
589C2×C4—82no—
590C2×C4—82no—
591C2×C4—82no—
592C2×C4—82no—
593C2×C4—82no—
594C2×C4—82no—
595C2×C4—82no—
596C2×C4#39011yescharacteristic
597C2×C4#39011yescharacteristic
598C2×C4#39011yescharacteristic
599C2×C4#39011yescharacteristic
600C2×C4C22×C22⋊C411yescharacteristic
601C2×C4C2×C4×D411yescharacteristic
602C2×C4C2×#12111yescharacteristic
603C2×C4C2×#12411yescharacteristic
604C8—24no—
605C8—24no—
606C8—44no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—21no—
618C22—21no—
619C22—21no—
620C22—21no—
621C22—21no—
622C22—21no—
623C22—21no—
624C22—21no—
625C22—44no—
626C22—44no—
627C22—44no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22—82no—
635C22—164no—
636C22—164no—
637C22—164no—
638C22C2×#39011yescharacteristic
639C22#294912yes—
640C22#305712yes—
641C22#315412yes—
642C4—21no—
643C4—21no—
644C4—21no—
645C4—21no—
646C4—21no—
647C4—21no—
648C4—21no—
649C4—84no—
650C4—82no—
651C4—82no—
652C4—82no—
653C4—162no—
654C4—162no—
655C4#288311yescharacteristic
656C4#288311yescharacteristic
657C2—22no—
658C2—22no—
659C2—22no—
660C2—41no—
661C2—41no—
662C2—41no—
663C2—41no—
664C2—162no—
665C2#2121311yescharacteristic
666C2#2123111yescharacteristic
667C2#2124411yescharacteristic
subgroupG/H
center, socleC22C2×#390
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21939C2semidirect
3. #21825C2semidirect
5. #20527C2semidirect
6. #17578C2semidirect
8. #9786C2semidirect
9. #9781C2semidirect
16. C2×#481C4semidirect
17. C2×#483C4semidirect
20. C2×#448C4semidirect
21. C2×#450C4semidirect
28. #1996C22semidirect
34. #1808C22semidirect
35. #1807C22semidirect
38. #1752C22semidirect
39. C8×C22⋊C4C22semidirect
41. #1520C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31872
#219392
#218252
#217932
#205272
#175782
#175772
#97862
#97812