Finite Group Databasebeta

← #364855

Subgroups — #364855

Order 512.

1382 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×M16C211yesmaximal, characteristic
2C2×#2582C211yesmaximal, characteristic
3#21303C211yesmaximal, characteristic
4C2×#1801C211yesmaximal, characteristic
5C2×C4×#101C211yesmaximal, characteristic
6#11204C212yesmaximal
7#11045C212yesmaximal
8#10963C212yesmaximal
9#10913C212yesmaximal
10#10857C211yesmaximal, characteristic
11#10857C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14Q8×M16—24no—
15Q8×M16—24no—
16C2×#446—22no—
17C2×#446—21no—
18C2×C8×Q8—21no—
19C2×#423C2212yes—
20C2×#423C2211yescharacteristic
21C2×#423C2211yescharacteristic
22C22×#101C2211yescharacteristic
23C22×#101C2212yes—
24C22×#101C2211yescharacteristic
25C2×C4×M16C2212yes—
26C2×C4×M16C2211yescharacteristic
27C2×C4×M16C2211yescharacteristic
28#2582—21no—
29#2582—21no—
30#2582—22no—
31#2582—21no—
32#2582—21no—
33#2582—22no—
34C2×#398C414yes—
35C2×#391C412yes—
36C2×#391C2212yes—
37C2×#389C412yes—
38C2×#389C2212yes—
39C2×#384C2211yescharacteristic
40#1985—22no—
41#1984—21no—
42#1984—21no—
43#1801—24no—
44C4×#101—24no—
45C2×#343C2212yes—
46C2×#343C2212yes—
47C2×#343C2212yes—
48C2×#343C2211yescharacteristic
49C2×#343C2211yescharacteristic
50C2×#343C2212yes—
51C2×#343C2212yes—
52#1728C2212yes—
53#1728C2212yes—
54C23×Q8C2×C411yescharacteristic
55C23×Q8C2311yescharacteristic
56C22×M16C2312yes—
57C22×M16C2312yes—
58C22×M16C2311yescharacteristic
59C22×M16C2311yescharacteristic
60C22×M16C2311yescharacteristic
61C22×M16C2311yescharacteristic
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8—22no—
65C2×C4×Q8—21no—
66C2×C4×Q8—21no—
67C2×C4×Q8—21no—
68C2×C4×Q8—21no—
69C2×C4×Q8—22no—
70C2×C4×Q8—22no—
71C2×C4×Q8—22no—
72C2×C4×Q8—22no—
73C2×C4×Q8—22no—
74C2×C4×Q8D411yescharacteristic
75C2×C4×Q8D412yes—
76C2×C4×Q8D411yescharacteristic
77C2×C4×Q8D411yescharacteristic
78C2×C4×Q8D411yescharacteristic
79C2×C4×Q8D412yes—
80C22×C4⋊C4C2×C412yes—
81C22×C4⋊C4C2×C412yes—
82C22×C4⋊C4C2×C411yescharacteristic
83C22×C4⋊C4C2×C412yes—
84C22×C4⋊C4C2×C412yes—
85C22×C4⋊C4C2×C414yes—
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2312yes—
88C22×C42C2×C412yes—
89C22×C42C2312yes—
90C22×C42C2311yescharacteristic
91#446—216no—
92#446—28no—
93C8×Q8—28no—
94#423—24no—
95#423—24no—
96#423—22no—
97#423—22no—
98#423—22no—
99#423D414yes—
100C2×C4⋊C8—22no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—22no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8—21no—
105C2×C4⋊C8—21no—
106C2×#101—22no—
107C2×#101—22no—
108C2×#101—22no—
109C2×#101—22no—
110C2×#101—22no—
111C2×#101—24no—
112C2×#101—24no—
113C2×#101—42no—
114C2×#101Q814yescomplemented
115C4×M16—24no—
116C4×M16—24no—
117C4×M16—22no—
118C4×M16—22no—
119C4×M16—22no—
120C4×M16D414yes—
121C2×#97—22no—
122C2×#97—21no—
123C2×#97—21no—
124C2×C4×C8—22no—
125C2×C4×C8—21no—
126C2×C4×C8—21no—
127#398—24no—
128#398—24no—
129#398—24no—
130#398—24no—
131#391—22no—
132#391—24no—
133#391—22no—
134#391—22no—
135#391—22no—
136#391—22no—
137#391—22no—
138#389—24no—
139#389—22no—
140#389—22no—
141#389—22no—
142#389—22no—
143#389—22no—
144#389—22no—
145#384—21no—
146#384—21no—
147#384—21no—
148#384—21no—
149C2×#95C2×C414yes—
150C2×#95C2×C414yes—
151C2×#95C2×C414yes—
152#343—24no—
153#343—22no—
154#343—22no—
155#343—24no—
156#343—24no—
157#343—42no—
158#343—42no—
159#336—22no—
160#336—22no—
161#336—22no—
162#336—22no—
163C22×Q8—24no—
164C22×Q8—22no—
165C22×Q8—22no—
166C22×Q8—21no—
167C22×Q8—21no—
168C22×Q8—21no—
169C22×Q8—21no—
170C22×Q8—21no—
171C22×Q8—21no—
172C22×Q8—21no—
173C22×Q8—21no—
174C22×Q8—21no—
175C22×Q8—21no—
176C22×Q8—21no—
177C22×Q8—21no—
178C22×Q8—42no—
179C22×Q8C22⋊C411yescharacteristic
180C22×Q8C22⋊C412yes—
181C22×Q8C22⋊C411yescharacteristic
182C22×Q8C2×Q812yes—
183C22×Q8#4112yes—
184C23×C4C22×C412yes—
185C23×C4C22×C412yes—
186C23×C4C22×C411yescharacteristic
187C23×C4C22×C414yes—
188C23×C4C22×C412yes—
189C23×C4C22×C412yes—
190C23×C4C22×C411yescharacteristic
191C23×C4C2411yescharacteristic
192C2×M16—21no—
193C2×M16—21no—
194C2×M16—21no—
195C2×M16—22no—
196C2×M16—21no—
197C2×M16—21no—
198C2×M16—21no—
199C2×M16—21no—
200C2×M16—21no—
201C2×M16—22no—
202C2×M16—21no—
203C2×M16—21no—
204C2×M16—21no—
205C2×M16—21no—
206C2×M16—22no—
207C2×M16—21no—
208C2×M16—21no—
209C2×M16—21no—
210C2×M16—21no—
211C2×M16—22no—
212C2×M16—21no—
213C2×M16—22no—
214C2×M16—21no—
215C2×M16—22no—
216C2×M16—22no—
217C2×M16—22no—
218C2×M16—22no—
219C2×M16—22no—
220C2×M16—22no—
221C2×M16C2×D414yes—
222C2×M16C2×D412yes—
223C2×M16C2×D412yes—
224C2×M16C2×Q812yes—
225C2×M16C2×Q812yes—
226C2×M16#4112yes—
227C2×M16#4112yes—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—22no—
232C22×C8—21no—
233C22×C8—22no—
234C4×Q8—22no—
235C4×Q8—24no—
236C4×Q8—22no—
237C4×Q8—22no—
238C4×Q8—22no—
239C4×Q8—24no—
240C4×Q8—22no—
241C4×Q8—22no—
242C4×Q8—24no—
243C4×Q8—22no—
244C4×Q8—24no—
245C4×Q8—22no—
246C4×Q8—42no—
247C4×Q8—42no—
248C4×Q8—42no—
249C4×Q8—42no—
250C4×Q8—42no—
251C4×Q8—42no—
252C4×Q8—42no—
253C4×Q8—42no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—82no—
282C2×C4⋊C4—84no—
283C2×C4⋊C4C22⋊C412yes—
284C2×C4⋊C4C22⋊C411yescharacteristic
285C2×C4⋊C4C22⋊C411yescharacteristic
286C2×C4⋊C4C22⋊C412yes—
287C2×C4⋊C4C22⋊C412yes—
288C2×C4⋊C4C22⋊C412yes—
289C2×C4⋊C4C22⋊C412yes—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42C2×D412yes—
299C2×C42C2×D412yes—
300C2×C42C2×D412yes—
301C2×C42C2×D412yes—
302C2×C42C2×D412yes—
303C2×C42C2×D412yes—
304C2×C42C2×D411yescharacteristic
305C2×C42C2×D411yescharacteristic
306C2×C42C2×D411yescharacteristic
307C2×C42C2×D411yescharacteristic
308C2×C42#4112yes—
309C2×C42#4112yes—
310C4⋊C8—28no—
311C4⋊C8—28no—
312C4⋊C8—24no—
313C4⋊C8—24no—
314C4⋊C8—42no—
315C4⋊C8—42no—
316#101—28no—
317#101—28no—
318#101—48no—
319#97—28no—
320#97—24no—
321#97—42no—
322C4×C8—28no—
323C4×C8—24no—
324C4×C8—42no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—24no—
331C24C23×C411yescharacteristic
332C2×Q8—22no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8—22no—
336C2×Q8—22no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—22no—
343C2×Q8—22no—
344C2×Q8—22no—
345C2×Q8—24no—
346C2×Q8—22no—
347C2×Q8—24no—
348C2×Q8—22no—
349C2×Q8—22no—
350C2×Q8—22no—
351C2×Q8—42no—
352C2×Q8—48no—
353C2×Q8—48no—
354C2×Q8—41no—
355C2×Q8—41no—
356C2×Q8—41no—
357C2×Q8—41no—
358C2×Q8—41no—
359C2×Q8—41no—
360C2×Q8—41no—
361C2×Q8—41no—
362C2×Q8—41no—
363C2×Q8—41no—
364C2×Q8—41no—
365C2×Q8—41no—
366C2×Q8—41no—
367C2×Q8—41no—
368C2×Q8—41no—
369C2×Q8—41no—
370C2×Q8#10114yescomplemented
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4C2×C22⋊C412yes—
400C22×C4C2×C22⋊C412yes—
401C22×C4C2×C22⋊C412yes—
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C2×C22⋊C411yescharacteristic
404C22×C4C2×C22⋊C411yescharacteristic
405C22×C4C2×C22⋊C412yes—
406C22×C4C2×C22⋊C411yescharacteristic
407C22×C4C4×D412yes—
408C22×C4C4×D412yes—
409C22×C4C4×D412yes—
410C22×C4C4×D412yes—
411C22×C4C4×Q814yes—
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C22×C4C22×Q811yescharacteristic
417C22×C4C2×#4111yescharacteristic
418C22×C4C2×#4111yescharacteristic
419C22×C4C2×#4111yescharacteristic
420C22×C4C2×#4111yescharacteristic
421C22×C4C2×#4111yescharacteristic
422C22×C4H(5,𝔽2)11yescharacteristic
423C22×C42−1+411yescharacteristic
424C22×C42−1+411yescharacteristic
425C22×C42−1+411yescharacteristic
426M16—22no—
427M16—22no—
428M16—22no—
429M16—22no—
430M16—22no—
431M16—22no—
432M16—22no—
433M16—24no—
434M16—24no—
435M16—22no—
436M16—22no—
437M16—42no—
438M16—41no—
439M16—42no—
440M16—41no—
441M16—41no—
442M16—41no—
443M16—41no—
444M16—41no—
445M16—42no—
446M16—42no—
447M16—42no—
448M16—42no—
449M16C22⋊Q814yes—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—24no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—41no—
458C2×C8—41no—
459C2×C8—41no—
460C2×C8—42no—
461C2×C8—41no—
462C2×C8—42no—
463C2×C8—41no—
464C2×C8—41no—
465C2×C8—42no—
466C2×C8—42no—
467C2×C8—42no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—44no—
476C4⋊C4—41no—
477C4⋊C4—41no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—82no—
488C4⋊C4—82no—
489C4⋊C4—84no—
490C4⋊C4—84no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—44no—
500C42—44no—
501C42#12012yes—
502C42#12012yes—
503C42#12112yes—
504C42#12112yes—
505C42#12112yes—
506C42#12112yes—
507C42#12312yes—
508C42#12312yes—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23C22×C22⋊C411yescharacteristic
514C23C2×C4×D411yescharacteristic
515C23C2×C4×D411yescharacteristic
516C23C2×C4×Q811yescharacteristic
517C23C4×#4111yescharacteristic
518C23#51911yescharacteristic
519C23#52011yescharacteristic
520Q8—28no—
521Q8—48no—
522Q8—42no—
523Q8—42no—
524Q8—44no—
525Q8—42no—
526Q8—42no—
527Q8—42no—
528Q8—42no—
529Q8—42no—
530Q8—42no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—84no—
572C2×C4—84no—
573C2×C4#38612yes—
574C2×C4#38612yes—
575C2×C4#38612yes—
576C2×C4#38612yes—
577C2×C4C2×#10114yes—
578C2×C4C2×#10112yes—
579C2×C4#41312yes—
580C2×C4C2×#12011yescharacteristic
581C2×C4C2×#12111yescharacteristic
582C2×C4C2×#12111yescharacteristic
583C2×C4C2×C22⋊Q811yescharacteristic
584C2×C4C2×#12311yescharacteristic
585C2×C4#52511yescharacteristic
586C2×C4#52911yescharacteristic
587C2×C4#53311yescharacteristic
588C2×C4#53611yescharacteristic
589C2×C4#53711yescharacteristic
590C2×C4#54111yescharacteristic
591C2×C4#54211yescharacteristic
592C2×C4#54611yescharacteristic
593C2×C4#54711yescharacteristic
594C2×C4#54811yescharacteristic
595C2×C4#54811yescharacteristic
596C2×C4#54811yescharacteristic
597C2×C4#54811yescharacteristic
598C2×C4#54811yescharacteristic
599C2×C4D4×Q811yescharacteristic
600C2×C4D4×Q811yescharacteristic
601C2×C4D4×Q811yescharacteristic
602C2×C4#55011yescharacteristic
603C2×C4#55011yescharacteristic
604C2×C4#55211yescharacteristic
605C2×C4#55211yescharacteristic
606C2×C4#55411yescharacteristic
607C2×C4#55611yescharacteristic
608C8—24no—
609C8—42no—
610C8—44no—
611C8—42no—
612C8—42no—
613C8—82no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—21no—
618C22—21no—
619C22—21no—
620C22—21no—
621C22—21no—
622C22—21no—
623C22—21no—
624C22—21no—
625C22—21no—
626C22C2×#38611yescharacteristic
627C22C4×C22⋊Q811yescharacteristic
628C22#232311yescharacteristic
629C22Q8×C22⋊C411yescharacteristic
630C22#234111yescharacteristic
631C22#236011yescharacteristic
632C22#236211yescharacteristic
633C22C22×#10111yescharacteristic
634C22C2×#41311yescharacteristic
635C22#288811yescharacteristic
636C22#288911yescharacteristic
637C4—22no—
638C4—22no—
639C4—22no—
640C4—24no—
641C4—42no—
642C4—42no—
643C4—44no—
644C4—42no—
645C4—82no—
646C4—84no—
647C4#190114yes—
648C4#240711yescharacteristic
649C4#240911yescharacteristic
650C4#242111yescharacteristic
651C4#242311yescharacteristic
652C4#242611yescharacteristic
653C4#242911yescharacteristic
654C4#253611yescharacteristic
655C4#253811yescharacteristic
656C2—21no—
657C2—21no—
658C2—21no—
659C2—21no—
660C2#1396911yescharacteristic
661C2C2×#190111yescharacteristic
662C2#2099211yescharacteristic
663C2Q8×#10111yescharacteristic
664C2#2251711yescharacteristic
665C2#2253811yescharacteristic
666C2#2253911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutator, prime radicalC24C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
114. C2×#101Q8semidirect
370. C2×Q8#101semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×M162
C2×#25822
#213032
C2×#18012
C2×C4×#1012
#112042
#110452
#109632
#109132
#108572