Finite Group Databasebeta

← #6497997

Subgroups — #6497997

Order 512.

1436 conjugacy classes of subgroups.

The subgroup lattice has 717 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35294C211yesmaximal, characteristic
2#35281C211yesmaximal, characteristic
3#33294C211yesmaximal, characteristic, complemented
4#33293C211yesmaximal, characteristic, complemented
5#33290C211yesmaximal, characteristic, complemented
6#33289C211yesmaximal, characteristic, complemented
7#30230C211yesmaximal, characteristic, complemented
8#30229C211yesmaximal, characteristic, complemented
9#30229C211yesmaximal, characteristic, complemented
10#30228C211yesmaximal, characteristic, complemented
11#29229C214yesmaximal, complemented
12#29228C214yesmaximal, complemented
13#27971C214yesmaximal
14#27970C214yesmaximal
15#21835C211yesmaximal, characteristic
16#21835C211yesmaximal, characteristic
17#21834C211yesmaximal, characteristic
18#21833C211yesmaximal, characteristic
19#20462C211yesmaximal, characteristic
20C2×#533C2211yescharacteristic
21C22×C4⋊Q8C2211yescharacteristic
22C2×#431C2211yescharacteristic
23C2×#431C2211yescharacteristic
24#2868C2211yescharacteristic
25#2868C2211yescharacteristic
26#2868C2211yescharacteristic
27#2862C2211yescharacteristic
28#2862C2211yescharacteristic
29#2859C2211yescharacteristic
30#2859C2211yescharacteristic
31#2859C2211yescharacteristic
32#2660C2212yes—
33#2660C2212yes—
34#2629C2212yes—
35#2628C2212yes—
36#2570—21no—
37#2570—21no—
38#2567—21no—
39#2567—21no—
40#2550—21no—
41#2550—21no—
42#2546—21no—
43#2546—21no—
44#2502—21no—
45#2502—21no—
46#2441—21no—
47#2441—21no—
48#2439—21no—
49#2439—21no—
50#2434—21no—
51#2434—21no—
52#2331C2212yes—
53#2330C2212yes—
54#2317C2212yes—
55#2317C2212yes—
56#2285C2211yescharacteristic
57#2285C2211yescharacteristic
58C4×#117C2211yescharacteristic
59#2101C2214yes—
60#2101C2214yes—
61#2101C2214yes—
62#2101C2214yes—
63#2100C2214yes—
64#2100C2214yes—
65#2099C2214yes—
66#2099C2214yes—
67#1993C2214yes—
68#1992C2214yes—
69#1991C2214yes—
70#1990C2211yescharacteristic
71#1990C2211yescharacteristic
72#1990C2211yescharacteristic
73#1990C2211yescharacteristic
74#1990C2214yes—
75#1990C2211yescharacteristic
76#1990C2211yescharacteristic
77#1990C2211yescharacteristic
78#1990C2211yescharacteristic
79#1990C2211yescharacteristic
80#1990C2211yescharacteristic
81#1990C2211yescharacteristic
82#1990C2211yescharacteristic
83#1990C2211yescharacteristic
84#1990C2211yescharacteristic
85#1990C2211yescharacteristic
86#1990C2211yescharacteristic
87#1989C2214yes—
88#1989C2214yes—
89#1988C2214yes—
90#1987C2214yes—
91#1836C2214yes—
92#1836C2214yes—
93#1835C2214yes—
94#1834C2214yes—
95#1825C2214yes—
96#1825C2214yes—
97#1824C2214yes—
98#1823C2214yes—
99#1768C2212yes—
100#1768C2212yes—
101#1750C2212yes—
102#1750C2212yes—
103#1744C2211yescharacteristic
104#1744C2211yescharacteristic
105C4×#97C2211yescharacteristic
106C42×C8C2211yescharacteristic
107C23×Q8—21no—
108#533—24no—
109#533—24no—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8—21no—
112C2×C4⋊Q8—21no—
113C2×C4⋊Q8—21no—
114C2×C4⋊Q8—21no—
115C2×C4⋊Q8—21no—
116C2×C4⋊Q8—21no—
117C2×C4⋊Q8—21no—
118C2×C4⋊Q8—24no—
119C2×C4⋊Q8—24no—
120C2×C4⋊Q8C2312yes—
121C2×C4⋊Q8C2311yescharacteristic
122C2×C4⋊Q8C2311yescharacteristic
123C2×C4⋊Q8C2311yescharacteristic
124C2×C4⋊Q8C2311yescharacteristic
125C2×C4⋊Q8C2312yes—
126C2×#125—21no—
127C2×#125—21no—
128C2×#125C2312yes—
129C2×#125C2312yes—
130C2×C22⋊Q8—21no—
131C2×C22⋊Q8—21no—
132C2×C4×Q8—21no—
133C2×C4×Q8—21no—
134C2×#117—21no—
135C2×#117—21no—
136C2×#117C2312yes—
137C2×#117C2311yescharacteristic
138C2×#117C2311yescharacteristic
139C22×C4⋊C4—21no—
140C22×C4⋊C4C2312yes—
141C22×C42C2311yescharacteristic
142#431—24no—
143#431—24no—
144C2×C4⋊C8C2312yes—
145C2×C4⋊C8C2312yes—
146C2×C22⋊C8C2312yes—
147C2×C22⋊C8C2312yes—
148C2×#97C2312yes—
149C2×#97C2312yes—
150C2×C4×C8C2312yes—
151C2×C4×C8C2312yes—
152C2×C4×C8C2311yescharacteristic
153C2×C4×C8C2311yescharacteristic
154C2×C4×C8C2311yescharacteristic
155C2×C4×C8C2311yescharacteristic
156#400—22no—
157#399—22no—
158#398—22no—
159#398—22no—
160#398—22no—
161#398—22no—
162#398—22no—
163#398—22no—
164#396—22no—
165#396—22no—
166#395—22no—
167#395—22no—
168#395—22no—
169#395—22no—
170#395—22no—
171#393—22no—
172#391—21no—
173#391—21no—
174#391—21no—
175#391—21no—
176#391—21no—
177#391—21no—
178#391—21no—
179#391—21no—
180#391—22no—
181#391—22no—
182#389—22no—
183#389—22no—
184#389—22no—
185#389—22no—
186#389—21no—
187#389—21no—
188#389—21no—
189#389—21no—
190#389—21no—
191#389—21no—
192#389—21no—
193#389—21no—
194#389—22no—
195#389—22no—
196#389C2312yes—
197#389C2312yes—
198#389C2312yes—
199#389C2312yes—
200#387—22no—
201#387—22no—
202#387—22no—
203#387—22no—
204#385—22no—
205#385—22no—
206#385—22no—
207#385—22no—
208#384—21no—
209#384—21no—
210#384—21no—
211#384—21no—
212#384—21no—
213#384—21no—
214#384C2312yes—
215#384C2312yes—
216#384C2312yes—
217#384C2312yes—
218#384C2312yes—
219#384C2311yescharacteristic
220#384C2311yescharacteristic
221#384C2311yescharacteristic
222#384C2311yescharacteristic
223#382—21no—
224#382—21no—
225#382—21no—
226#382—21no—
227#382C2312yes—
228#382C2312yes—
229#382C2312yes—
230#382C2312yes—
231#382C2312yes—
232#380—21no—
233#380—21no—
234#380C2312yes—
235#380C2312yes—
236C4×C4⋊C4—21no—
237C4×C4⋊C4—21no—
238C4×C4⋊C4D411yescharacteristic
239C4×C4⋊C4D411yescharacteristic
240C4×C4⋊C4D411yescharacteristic
241C4×C4⋊C4D411yescharacteristic
242C4×C4⋊C4C2312yes—
243C4×C4⋊C4C2312yes—
244C4×C22⋊C4D411yescharacteristic
245C4×C22⋊C4D411yescharacteristic
246C4×C22⋊C4D411yescharacteristic
247C4×C22⋊C4D411yescharacteristic
248#376C2311yescharacteristic
249#376C2311yescharacteristic
250C43C2311yescharacteristic
251C43C2311yescharacteristic
252#340C2314yes—
253#340C2314yes—
254#340C2314yes—
255#340C2314yes—
256#340C2314yes—
257#340C2314yes—
258#340C2314yes—
259#340C2314yes—
260#340C2314yes—
261#340C2314yes—
262#340C2314yes—
263#340C2314yes—
264#340C2314yes—
265#340C2314yes—
266#340C2314yes—
267#340C2314yes—
268#336C2312yes—
269#336C2312yes—
270#336C2312yes—
271#336C2312yes—
272C22×Q8—21no—
273C22×Q8—21no—
274C22×Q8—21no—
275C22×Q8—21no—
276C22×Q8—22no—
277C22×Q8—22no—
278C22×Q8—44no—
279C22×Q8—82no—
280C22×Q8—82no—
281C23×C4—22no—
282C23×C4C2×D411yescharacteristic
283C23×C4C2×D411yescharacteristic
284C23×C4C2411yescharacteristic
285C22×C8C2412yes—
286C22×C8C2412yes—
287C22×C8C2412yes—
288C22×C8C2412yes—
289C4⋊Q8—22no—
290C4⋊Q8—22no—
291C4⋊Q8—22no—
292C4⋊Q8—22no—
293C4⋊Q8—24no—
294C4⋊Q8—24no—
295C4⋊Q8—22no—
296C4⋊Q8—22no—
297C4⋊Q8—22no—
298C4⋊Q8—22no—
299C4⋊Q8—28no—
300C4⋊Q8—44no—
301C4⋊Q8—44no—
302C4⋊Q8—82no—
303C4⋊Q8—82no—
304C4⋊Q8—82no—
305C4⋊Q8—82no—
306C4⋊Q8—82no—
307C4⋊Q8—82no—
308#125—24no—
309#125—24no—
310#125—28no—
311#125—82no—
312#125—82no—
313C22⋊Q8—44no—
314C22⋊Q8—44no—
315C4×Q8—44no—
316C4×Q8—44no—
317#117—24no—
318#117—24no—
319#117—24no—
320#117—24no—
321#117—82no—
322#117—82no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—44no—
359C2×C4⋊C4C2×D411yescharacteristic
360C2×C4⋊C4C2×D411yescharacteristic
361C2×C4⋊C4C2×D411yescharacteristic
362C2×C4⋊C4C2×D411yescharacteristic
363C2×C4⋊C4C2412yes—
364C2×C4⋊C4C2412yes—
365C2×C4⋊C4C2412yes—
366C2×C4⋊C4C2412yes—
367C2×C4⋊C4C2412yes—
368C2×C4⋊C4C2412yes—
369C2×C4⋊C4C2412yes—
370C2×C4⋊C4C2412yes—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4C2×D411yescharacteristic
375C2×C22⋊C4C2×D411yescharacteristic
376C2×C22⋊C4C2×D411yescharacteristic
377C2×C22⋊C4C2×D411yescharacteristic
378C2×C42—22no—
379C2×C42—22no—
380C2×C42—22no—
381C2×C42C2×D411yescharacteristic
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D411yescharacteristic
387C2×C42C2×D411yescharacteristic
388C2×C42C2×D411yescharacteristic
389C2×C42C2×D411yescharacteristic
390C2×C42C2×D411yescharacteristic
391C2×C42#4114yes—
392C2×C42#4114yes—
393C2×C42C2411yescharacteristic
394C2×C42C2411yescharacteristic
395C2×C42C2411yescharacteristic
396C2×C42C2411yescharacteristic
397C2×C42C2411yescharacteristic
398C2×C42C2411yescharacteristic
399C4⋊C8—24no—
400C4⋊C8—24no—
401C22⋊C8—24no—
402C22⋊C8—24no—
403#97—24no—
404#97—24no—
405C4×C8—22no—
406C4×C8—22no—
407C4×C8—22no—
408C4×C8—22no—
409C4×C8—24no—
410C4×C8—24no—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—22no—
420#95—22no—
421#95—22no—
422#95—22no—
423#95—22no—
424#95—22no—
425#95—22no—
426#95—22no—
427#95—22no—
428#95—22no—
429#95C2×D411yescharacteristic
430#95C2×D411yescharacteristic
431#95C2×D411yescharacteristic
432#95C2×D411yescharacteristic
433#95C2×D411yescharacteristic
434#95C2×D411yescharacteristic
435#95C2×D411yescharacteristic
436#95C2×D411yescharacteristic
437C24C22×D411yescharacteristic
438C2×Q8—42no—
439C2×Q8—42no—
440C2×Q8—44no—
441C2×Q8—42no—
442C2×Q8—42no—
443C2×Q8—44no—
444C2×Q8—42no—
445C2×Q8—42no—
446C2×Q8—42no—
447C2×Q8—42no—
448C2×Q8—42no—
449C2×Q8—42no—
450C2×Q8—42no—
451C2×Q8—42no—
452C2×Q8—82no—
453C2×Q8—82no—
454C2×Q8—82no—
455C2×Q8—82no—
456C2×Q8—816no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—44no—
472C22×C4—82no—
473C22×C4—82no—
474C22×C4#12414yes—
475C22×C4#12414yes—
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C2×#4112yes—
490C22×C4C2×#4112yes—
491C22×C4C2511yescharacteristic
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—24no—
517C4⋊C4—24no—
518C4⋊C4—24no—
519C4⋊C4—24no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—82no—
540C4⋊C4—82no—
541C4⋊C4—82no—
542C4⋊C4—82no—
543C4⋊C4—82no—
544C4⋊C4—82no—
545C4⋊C4—82no—
546C4⋊C4—82no—
547C4⋊C4—82no—
548C4⋊C4—82no—
549C4⋊C4—82no—
550C4⋊C4—82no—
551C4⋊C4—82no—
552C4⋊C4—82no—
553C22⋊C4—22no—
554C22⋊C4—22no—
555C22⋊C4—22no—
556C22⋊C4—22no—
557C22⋊C4—44no—
558C22⋊C4—82no—
559C22⋊C4—82no—
560C42—22no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—22no—
565C42—22no—
566C42—22no—
567C42—22no—
568C42—22no—
569C42—22no—
570C42—44no—
571C42—82no—
572C42—82no—
573C42#12414yes—
574C42#12414yes—
575C42C2×#4112yes—
576C42C2×#4112yes—
577C42C2×#4112yes—
578C42C2×#4112yes—
579C422−1+414yes—
580C422−1+414yes—
581C23—21no—
582C23—21no—
583C23—21no—
584C23—21no—
585C23—21no—
586C23—21no—
587C23C2×#12412yes—
588C23C23×D411yescharacteristic
589Q8—88no—
590Q8—84no—
591Q8—84no—
592Q8—84no—
593Q8—84no—
594Q8—164no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—28no—
610C2×C4—28no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—88no—
650C2×C4C2×#12412yes—
651C2×C4C2×#12412yes—
652C2×C4C2×#12412yes—
653C2×C4C2×#12412yes—
654C2×C4C2×#12412yes—
655C2×C4#53612yes—
656C2×C4#53612yes—
657C2×C4#53612yes—
658C2×C4#53612yes—
659C2×C4#57612yes—
660C2×C4#57712yes—
661C2×C4#57712yes—
662C2×C4#57812yes—
663C2×C4C22×#4111yescharacteristic
664C2×C4C22×#4111yescharacteristic
665C2×C4C2×2−1+412yes—
666C2×C4C2×2−1+412yes—
667C8—42no—
668C8—42no—
669C8—42no—
670C8—42no—
671C22—22no—
672C22—22no—
673C22—22no—
674C22—41no—
675C22—41no—
676C22—41no—
677C22—41no—
678C22C22×#12411yescharacteristic
679C22C2×#53611yescharacteristic
680C22C2×#53611yescharacteristic
681C22C2×#57611yescharacteristic
682C22C2×#57711yescharacteristic
683C22C2×#57711yescharacteristic
684C22C2×#57811yescharacteristic
685C4—22no—
686C4—22no—
687C4—22no—
688C4—22no—
689C4—22no—
690C4—22no—
691C4—22no—
692C4—41no—
693C4—41no—
694C4—41no—
695C4—41no—
696C4—82no—
697C4—82no—
698C4—82no—
699C4—82no—
700C4—82no—
701C4—82no—
702C4—82no—
703C4—82no—
704C4#313614yes—
705C4#313714yes—
706C4#350912yes—
707C4#350912yes—
708C2—42no—
709C2#3532011yescharacteristic
710C2C2×#313611yescharacteristic
711C2C2×#313711yescharacteristic
712C2#6087711yescharacteristic
713C2#6087811yescharacteristic
714C2#6088011yescharacteristic
715C2#6088111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33294C2semidirect
4. #33293C2semidirect
5. #33290C2semidirect
6. #33289C2semidirect
7. #30230C2semidirect
8. #30229C2semidirect
9. #30229C2semidirect
10. #30228C2semidirect
11. #29229C2semidirect
12. #29228C2semidirect

Maximal subgroups

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

subgroupindex
#352942
#352812
#332942
#332932
#332902
#332892
#302302
#302292
#302282
#292292
#292282
#279712
#279702
#218352
#218342
#218332
#204622