Finite Group Databasebeta

← #6526555

Subgroups — #6526555

Order 512.

1748 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35317C211yesmaximal, characteristic
2#34978C211yesmaximal, characteristic, complemented
3#29354C212yesmaximal, complemented
4#29348C212yesmaximal, complemented
5#29226C212yesmaximal, complemented
6#29218C212yesmaximal, complemented
7#27822C212yesmaximal, complemented
8#27813C212yesmaximal, complemented
9#27736C212yesmaximal, complemented
10#27727C212yesmaximal, complemented
11#27719C214yesmaximal, complemented
12#27709C214yesmaximal, complemented
13#21893C211yesmaximal, characteristic
14#21834C211yesmaximal, characteristic, complemented
15#21390C211yesmaximal, characteristic
16#21313C211yesmaximal, characteristic, complemented
17#20550C211yesmaximal, characteristic, complemented
18C2×#536—21no—
19C2×#536—21no—
20C2×#532C2211yescharacteristic
21C22×C4⋊Q8C2211yescharacteristic
22C2×#431C2211yescharacteristic
23C2×#423C2211yescharacteristic
24#2868C2212yes—
25#2858C2212yes—
26#2658—22no—
27#2612—22no—
28#2571C2211yescharacteristic
29#2552C2211yescharacteristic
30#2552C2211yescharacteristic
31#2551C2211yescharacteristic
32#2542—21no—
33#2542—21no—
34#2542C2212yes—
35#2542C2212yes—
36#2501—22no—
37#2501C2214yes—
38#2441—21no—
39#2441C2212yes—
40#2439—21no—
41#2439C2212yes—
42#2396—22no—
43C4×C4⋊Q8—21no—
44C4×C4⋊Q8C2212yes—
45C4×#125—21no—
46C4×#125C2212yes—
47#2285C2211yescharacteristic
48C4×#117C2211yescharacteristic
49C4×#117C2211yescharacteristic
50#2100C2214yes—
51#2100C2214yes—
52#2099C2214yes—
53#2099C2214yes—
54#2073C2214yes—
55#2071C2214yes—
56#2025C2214yes—
57#2023C2214yes—
58#1992C2214yes—
59#1990C2214yes—
60#1990C2212yes—
61#1990C2212yes—
62#1988C2212yes—
63#1988C2212yes—
64#1987C2211yescharacteristic
65#1987C2212yes—
66#1987C2212yes—
67#1987C2212yes—
68#1987C2211yescharacteristic
69#1987C2212yes—
70#1987C2212yes—
71#1982C2212yes—
72#1982C2211yescharacteristic
73#1982C2211yescharacteristic
74#1936C2214yes—
75#1936C2214yes—
76#1934C2214yes—
77#1934C2214yes—
78#1895C2214yes—
79#1875C2212yes—
80#1875C2212yes—
81#1836C2214yes—
82#1834C2214yes—
83#1824C2214yes—
84#1802C2212yes—
85#1802C2212yes—
86#1770C2212yes—
87C8×C4⋊C4C2212yes—
88#1767C2212yes—
89C4×C4⋊C8C2212yes—
90C4×#102C2212yes—
91C4×#102C2212yes—
92#1753C2214yes—
93#1736C2212yes—
94#1736C2212yes—
95C22×#41—21no—
96C23×Q8—21no—
97C22×M16C2312yes—
98#536—48no—
99#536—48no—
100#532—24no—
101#532—24no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—22no—
104C2×C4⋊Q8—22no—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8—21no—
107C2×C4⋊Q8—24no—
108C2×C4⋊Q8—24no—
109C2×C4⋊Q8C2312yes—
110C2×C4⋊Q8C2311yescharacteristic
111C2×C4⋊Q8C2312yes—
112C2×C4⋊Q8C2312yes—
113C2×#127C2311yescharacteristic
114C2×#125—21no—
115C2×#125C2312yes—
116C2×#124—22no—
117C2×#124—22no—
118C2×#124—21no—
119C2×#123—24no—
120C2×#123—24no—
121C2×C22⋊Q8—22no—
122C2×C22⋊Q8—22no—
123C2×C22⋊Q8—22no—
124C2×C22⋊Q8—22no—
125C2×#121—21no—
126C2×#121C2312yes—
127C2×C4×Q8—22no—
128C2×C4×D4—21no—
129C2×C4×D4C2312yes—
130C2×#117—21no—
131C2×#117D411yescharacteristic
132C2×#117D411yescharacteristic
133C2×#117C2311yescharacteristic
134C2×#117C2311yescharacteristic
135C22×C4⋊C4—21no—
136C22×C4⋊C4C2312yes—
137C22×C42C2311yescharacteristic
138#431—24no—
139#423—24no—
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2311yescharacteristic
143C2×C4⋊C8C2311yescharacteristic
144C2×#102C2312yes—
145C2×#102C2312yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2312yes—
149C2×#102C2312yes—
150C2×C22⋊C8C2312yes—
151C2×#97C2314yes—
152C2×C4×C8C2312yes—
153C2×C4×C8C2314yes—
154#400—22no—
155#399—22no—
156#399—24no—
157#399—24no—
158#399—24no—
159#399—24no—
160#398—22no—
161#398—22no—
162#396—22no—
163#396—22no—
164#395—22no—
165#393—22no—
166#391—22no—
167#391—22no—
168#390—22no—
169#390—22no—
170#390C2314yes—
171#390C2314yes—
172#389—21no—
173#389—21no—
174#389—21no—
175#389—21no—
176#389—22no—
177#389—22no—
178#389C2312yes—
179#389C2312yes—
180#389C2312yes—
181#389C2312yes—
182#387—22no—
183#387—22no—
184#387—22no—
185#387—22no—
186#384—21no—
187#384—22no—
188#384—22no—
189#384C2312yes—
190#384C2311yescharacteristic
191#384C2311yescharacteristic
192#384C2314yes—
193#384C2314yes—
194#382—21no—
195#382—21no—
196#382C2312yes—
197#380—21no—
198#380C2312yes—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4D411yescharacteristic
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4C2311yescharacteristic
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2311yescharacteristic
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2311yescharacteristic
212C4×C22⋊C4—21no—
213C4×C22⋊C4—21no—
214C4×C22⋊C4D412yes—
215C4×C22⋊C4C2312yes—
216#376—21no—
217#376D412yes—
218C43—21no—
219C43C2312yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2312yes—
230#340C2312yes—
231#340C2314yes—
232#340C2312yes—
233#340C2314yes—
234#336C2314yes—
235#336C2314yes—
236C2×#41—44no—
237C2×#41—44no—
238C2×#41—44no—
239C22×Q8—22no—
240C22×Q8—22no—
241C22×Q8—21no—
242C22×Q8—22no—
243C22×Q8—44no—
244C22×Q8—44no—
245C22×Q8—44no—
246C22×D4—22no—
247C22×D4—21no—
248C22×D4—21no—
249C22×D4C2412yes—
250C23×C4—22no—
251C23×C4—22no—
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2411yescharacteristic
255C2×M16—24no—
256C2×M16—42no—
257C2×M16—42no—
258C22×C8C2414yes—
259C22×C8C2412yes—
260C22×C8C2412yes—
261C4⋊Q8—24no—
262C4⋊Q8—24no—
263C4⋊Q8—24no—
264C4⋊Q8—24no—
265C4⋊Q8—24no—
266C4⋊Q8—22no—
267C4⋊Q8—22no—
268C4⋊Q8—42no—
269C4⋊Q8—48no—
270C4⋊Q8—48no—
271C4⋊Q8—44no—
272C4⋊Q8—44no—
273C4⋊Q8—44no—
274#127—22no—
275#127—22no—
276#125—24no—
277#125—42no—
278#125—44no—
279#124—48no—
280#124—48no—
281#124—44no—
282#123—416no—
283#123—416no—
284C22⋊Q8—48no—
285C22⋊Q8—48no—
286C22⋊Q8—48no—
287C22⋊Q8—48no—
288#121—24no—
289#121—24no—
290#121—44no—
291C4×Q8—84no—
292C4×D4—24no—
293C4×D4—24no—
294C4×D4—44no—
295#117—24no—
296#117—24no—
297#117—22no—
298#117—22no—
299#117—41no—
300#117—41no—
301#117—82no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—44no—
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4C2×D411yescharacteristic
329C2×C4⋊C4C2×D411yescharacteristic
330C2×C4⋊C4C2414yes—
331C2×C4⋊C4C2411yescharacteristic
332C2×C4⋊C4C2411yescharacteristic
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2411yescharacteristic
337C2×C4⋊C4C2411yescharacteristic
338C2×C22⋊C4—21no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—24no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4C2×D412yes—
348C2×C22⋊C4C2×D412yes—
349C2×C42—22no—
350C2×C42—21no—
351C2×C42—21no—
352C2×C42—21no—
353C2×C42—21no—
354C2×C42—21no—
355C2×C42—21no—
356C2×C42C2×D412yes—
357C2×C42C2×D412yes—
358C2×C42C2×D412yes—
359C2×C42C2×D412yes—
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D411yescharacteristic
362C2×C42#4114yes—
363C2×C42#4114yes—
364C2×C42C2412yes—
365C2×C42C2412yes—
366C2×C42C2411yescharacteristic
367C2×C42C2411yescharacteristic
368C4⋊C8—24no—
369C4⋊C8—24no—
370C4⋊C8—22no—
371C4⋊C8—22no—
372#102—24no—
373#102—24no—
374#102—42no—
375#102—42no—
376#102—44no—
377#102—44no—
378C22⋊C8—24no—
379#97—44no—
380C4×C8—24no—
381C4×C8—44no—
382#95—21no—
383#95—21no—
384#95—21no—
385#95—21no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95C2×D412yes—
395#95C2×D412yes—
396#95C2×D412yes—
397#95C2×D412yes—
398C24—22no—
399C24C22×D411yescharacteristic
400#41—816no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8—44no—
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8—42no—
407C2×Q8—44no—
408C2×Q8—44no—
409C2×Q8—44no—
410C2×Q8—44no—
411C2×Q8—44no—
412C2×Q8—82no—
413C2×Q8—816no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—44no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4—42no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4#12414yes—
452C22×C4#12414yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C2×#4112yes—
463C22×C4C2×#4112yes—
464C22×C4C2511yescharacteristic
465M16—84no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—24no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—24no—
473C2×C8—22no—
474C2×C8—24no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—48no—
495C4⋊C4—48no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—42no—
500C4⋊C4—48no—
501C4⋊C4—48no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—42no—
505C4⋊C4—44no—
506C4⋊C4—42no—
507C4⋊C4—81no—
508C4⋊C4—81no—
509C4⋊C4—82no—
510C4⋊C4—82no—
511C4⋊C4—82no—
512C4⋊C4AGL(1,ℤ/8)14yes—
513C4⋊C4AGL(1,ℤ/8)14yes—
514C4⋊C42−1+412yes—
515C4⋊C42−1+412yes—
516C22⋊C4—24no—
517C22⋊C4—24no—
518C22⋊C4—42no—
519C22⋊C4—42no—
520C22⋊C4—48no—
521C22⋊C4—48no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—48no—
525C22⋊C4—48no—
526C22⋊C4—82no—
527C42—24no—
528C42—24no—
529C42—24no—
530C42—24no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—42no—
536C42—42no—
537C42—42no—
538C42—42no—
539C42—44no—
540C42—42no—
541C42—42no—
542C42#12414yes—
543C42#12414yes—
544C42C2×#4112yes—
545C42C2×#4112yes—
546C42C2×#4112yes—
547C42C2×#4112yes—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23C2×#12412yes—
562C23C23×D411yescharacteristic
563Q8—88no—
564Q8—88no—
565Q8—84no—
566Q8—88no—
567D4—42no—
568D4—42no—
569D4—42no—
570D4—42no—
571D4—84no—
572D4—88no—
573D4—84no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—28no—
585C2×C4—28no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—88no—
621C2×C4—88no—
622C2×C4C2×#12412yes—
623C2×C4C2×#12412yes—
624C2×C4C2×#12412yes—
625C2×C4C2×#12412yes—
626C2×C4C2×#12412yes—
627C2×C4C2×AGL(1,ℤ/8)14yes—
628C2×C4C2×AGL(1,ℤ/8)14yes—
629C2×C4C2×AGL(1,ℤ/8)12yes—
630C2×C4C2×AGL(1,ℤ/8)12yes—
631C2×C4#57712yes—
632C2×C4#57812yes—
633C2×C4C22×#4111yescharacteristic
634C2×C4C22×#4111yescharacteristic
635C2×C4C2×2−1+411yescharacteristic
636C2×C4C2×2−1+411yescharacteristic
637C2×C4#58511yescharacteristic
638C2×C4#58511yescharacteristic
639C8—42no—
640C8—42no—
641C8—44no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—41no—
646C22—41no—
647C22—41no—
648C22—41no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22C22×#12411yescharacteristic
657C22#348911yescharacteristic
658C22#348911yescharacteristic
659C22C22×AGL(1,ℤ/8)11yescharacteristic
660C22C22×AGL(1,ℤ/8)11yescharacteristic
661C22C2×#57711yescharacteristic
662C22C2×#57811yescharacteristic
663C4—22no—
664C4—24no—
665C4—24no—
666C4—22no—
667C4—22no—
668C4—42no—
669C4—42no—
670C4—42no—
671C4—42no—
672C4—82no—
673C4—81no—
674C4—82no—
675C4—84no—
676C4—81no—
677C4—82no—
678C4#313714yes—
679C4#313814yes—
680C4#350711yescharacteristic
681C4#350711yescharacteristic
682C4#350911yescharacteristic
683C4#351011yescharacteristic
684C2—42no—
685C2—162no—
686C2#3562811yescharacteristic
687C2C2×#313711yescharacteristic
688C2C2×#313811yescharacteristic
689C2#6212811yescharacteristic
690C2#6213211yescharacteristic
691C2#6214811yescharacteristic
692C2#6215011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
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
2. #34978C2semidirect
3. #29354C2semidirect
4. #29348C2semidirect
5. #29226C2semidirect
6. #29218C2semidirect
7. #27822C2semidirect
8. #27813C2semidirect
9. #27736C2semidirect
10. #27727C2semidirect
11. #27719C2semidirect
12. #27709C2semidirect
14. #21834C2semidirect
16. #21313C2semidirect
17. #20550C2semidirect

Maximal subgroups

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

subgroupindex
#353172
#349782
#293542
#293482
#292262
#292182
#278222
#278132
#277362
#277272
#277192
#277092
#218932
#218342
#213902
#213132
#205502