Finite Group Databasebeta

← #6630258

Subgroups — #6630258

Order 512.

2200 conjugacy classes of subgroups.

The subgroup lattice has 759 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3116C211yesmaximal, characteristic
2C2×#3107C211yesmaximal, characteristic
3#35322C211yesmaximal, characteristic, complemented
4C4×#520C211yesmaximal, characteristic
5#33202C211yesmaximal, characteristic, complemented
6#33201C211yesmaximal, characteristic, complemented
7#33163C211yesmaximal, characteristic, complemented
8#33162C211yesmaximal, characteristic, complemented
9#33117C214yesmaximal, complemented
10#33033C211yesmaximal, characteristic, complemented
11#33032C211yesmaximal, characteristic, complemented
12#33027C211yesmaximal, characteristic, complemented
13#33026C211yesmaximal, characteristic, complemented
14#32916C214yesmaximal
15#21516C214yesmaximal, complemented
16#21514C214yesmaximal, complemented
17#21394C211yesmaximal, characteristic, complemented
18#21385C211yesmaximal, characteristic, complemented
19#20815C211yesmaximal, characteristic, complemented
20C2×#533C2211yescharacteristic
21C2×#533C2211yescharacteristic
22C2×#520C2212yes—
23C2×C4×#41C2211yescharacteristic
24#3116—28no—
25#3107—28no—
26C2×#485C2212yes—
27C2×#485C2212yes—
28C2×#477C2211yescharacteristic
29C2×#475C2211yescharacteristic
30C2×#476C2211yescharacteristic
31C2×#474C2211yescharacteristic
32C2×#462C2211yescharacteristic
33C2×#462C2211yescharacteristic
34C2×#462C2214yes—
35C2×#461C2211yescharacteristic
36C2×#461C2211yescharacteristic
37C2×#461C2214yes—
38C2×#452C2212yes—
39C2×#452C2212yes—
40C2×#441C2212yes—
41C2×#440C2212yes—
42C2×C4×Q16C2212yes—
43C2×C4×SD16C2212yes—
44C2×#432C2211yescharacteristic
45C2×#431C2211yescharacteristic
46#2868C2211yescharacteristic
47#2862C2211yescharacteristic
48#2862C2211yescharacteristic
49#2862C2211yescharacteristic
50#2621C2214yes—
51#2565—21no—
52#2565—21no—
53#2565—22no—
54#2499—21no—
55#2499—21no—
56#2499—22no—
57#2327C2214yes—
58#2276C2212yes—
59#2275—22no—
60#2275C2212yes—
61#2274—22no—
62#2274C2212yes—
63C42×Q8C2211yescharacteristic
64C42×Q8C2211yescharacteristic
65C42×D4—21no—
66C42×D4—21no—
67C42×D4C2211yescharacteristic
68C42×D4C2211yescharacteristic
69C4×#117—21no—
70C4×#117C2211yescharacteristic
71C2×C4×C4⋊C4—21no—
72C2×C4×C4⋊C4C2211yescharacteristic
73#1988C2211yescharacteristic
74#1988C2211yescharacteristic
75#1987C2211yescharacteristic
76#1987C2211yescharacteristic
77#1981C2211yescharacteristic
78#1981C2211yescharacteristic
79#1981C2211yescharacteristic
80#1981C2211yescharacteristic
81#1981C2214yes—
82#1981C2214yes—
83#1949C2214yes—
84#1949C2214yes—
85#1928C2214yes—
86#1928C2214yes—
87#1927C2214yes—
88#1927C2214yes—
89#1878C2214yes—
90#1878C2214yes—
91#1877C2214yes—
92#1877C2214yes—
93#1872C2214yes—
94#1872C2214yes—
95#1842C2211yescharacteristic
96#1842C2211yescharacteristic
97#1831C2211yescharacteristic
98#1831C2211yescharacteristic
99#1805C2214yes—
100#1804C2214yes—
101#1798C2214yes—
102#1768C2214yes—
103#1763C2214yes—
104C4×#103C2214yes—
105C22×#41C2311yescharacteristic
106C22×Q16C2312yes—
107C22×Q16C2312yes—
108C22×SD16C2312yes—
109C22×SD16C2312yes—
110#533—24no—
111#533—24no—
112#533—24no—
113#533—24no—
114C2×C4⋊Q8C2314yes—
115C2×C4⋊Q8C2311yescharacteristic
116C2×C4⋊Q8C2311yescharacteristic
117C2×C4⋊Q8C2311yescharacteristic
118C2×C4⋊Q8C2311yescharacteristic
119C2×#125—21no—
120C2×#125—21no—
121C2×C22⋊Q8—21no—
122C2×C22⋊Q8—21no—
123C2×C22⋊Q8C2312yes—
124C2×C22⋊Q8C2312yes—
125#520—28no—
126#520—28no—
127C4×#41—24no—
128C4×#41D418yes—
129C2×C4×Q8—21no—
130C2×C4×Q8—21no—
131C2×C4×Q8C2312yes—
132C2×C4×Q8C2312yes—
133C2×C4×Q8C2312yes—
134C2×C4×Q8C2311yescharacteristic
135C2×C4×Q8C2311yescharacteristic
136C2×C4×Q8C2312yes—
137C2×C4×D4—21no—
138C2×C4×D4—21no—
139C2×C4×D4—22no—
140C2×C4×D4—22no—
141C2×C4×D4C2312yes—
142C2×C4×D4C2311yescharacteristic
143C2×C4×D4C2311yescharacteristic
144C2×C4×D4C2312yes—
145C2×#117—21no—
146C2×#117—21no—
147C2×#117—22no—
148C2×#117—21no—
149C2×#117C2311yescharacteristic
150C2×#117C2312yes—
151C22×C4⋊C4—22no—
152C22×C4⋊C4C2312yes—
153C22×C42—21no—
154C22×C42—21no—
155C22×C42—21no—
156C22×C42—21no—
157C22×C42C2311yescharacteristic
158#485—28no—
159#485—28no—
160#477—24no—
161#476—24no—
162#475—24no—
163#474—24no—
164#462—24no—
165#462—24no—
166#462—216no—
167#461—24no—
168#461—24no—
169#461—216no—
170#452—28no—
171#452—28no—
172#441—28no—
173#440—28no—
174C4×Q16—28no—
175C4×SD16—28no—
176#432—24no—
177#431—24no—
178C2×#107C2312yes—
179C2×#107C2312yes—
180C2×#106C2312yes—
181C2×#106C2312yes—
182C2×C4⋊C8C2311yescharacteristic
183C2×C4⋊C8C2311yescharacteristic
184C2×C4⋊C8C2314yes—
185C2×C4⋊C8C2314yes—
186C2×C4⋊C8C2311yescharacteristic
187C2×C4⋊C8C2311yescharacteristic
188C2×#103C2312yes—
189C2×#103C2312yes—
190C2×#103C2312yes—
191C2×#103C2312yes—
192C2×#103C2312yes—
193C2×#103C2312yes—
194C2×#103C2314yes—
195C2×#103C2314yes—
196C2×#103C2314yes—
197C2×#103C2314yes—
198C2×#102C2312yes—
199C2×#102C2312yes—
200C2×C22⋊C8C2312yes—
201C2×C22⋊C8C2312yes—
202C2×#97C2312yes—
203C2×C4×C8C2312yes—
204#398—24no—
205#398—24no—
206#397—24no—
207#397—24no—
208#391—21no—
209#391—21no—
210#391—22no—
211#391—21no—
212#391—21no—
213#391—22no—
214#391C2314yes—
215#391C2314yes—
216#387—24no—
217#387—24no—
218#387—24no—
219#387—24no—
220#384—21no—
221#384—21no—
222#384—22no—
223#382—21no—
224#382—21no—
225#382—22no—
226#382C2314yes—
227C4×C4⋊C4—22no—
228C4×C4⋊C4—22no—
229C4×C4⋊C4—21no—
230C4×C4⋊C4—21no—
231C4×C4⋊C4—21no—
232C4×C4⋊C4—21no—
233C4×C4⋊C4—22no—
234C4×C4⋊C4—22no—
235C4×C4⋊C4—22no—
236C4×C4⋊C4—22no—
237C4×C4⋊C4—22no—
238C4×C4⋊C4D411yescharacteristic
239C4×C4⋊C4D411yescharacteristic
240C4×C4⋊C4D412yes—
241C4×C4⋊C4C2312yes—
242C4×C4⋊C4C2312yes—
243C4×C4⋊C4C2312yes—
244C4×C4⋊C4C2312yes—
245C4×C4⋊C4C2311yescharacteristic
246C4×C4⋊C4C2311yescharacteristic
247C4×C4⋊C4C2312yes—
248C4×C22⋊C4—22no—
249C4×C22⋊C4—22no—
250C4×C22⋊C4—21no—
251C4×C22⋊C4—21no—
252C4×C22⋊C4—22no—
253C4×C22⋊C4—22no—
254C4×C22⋊C4—22no—
255C4×C22⋊C4—22no—
256C4×C22⋊C4—22no—
257C4×C22⋊C4D411yescharacteristic
258C4×C22⋊C4D411yescharacteristic
259C4×C22⋊C4D412yes—
260#376—22no—
261#376C2312yes—
262C2×#95—21no—
263C2×#95—22no—
264C43—21no—
265C43—21no—
266C43C2311yescharacteristic
267C43C2311yescharacteristic
268#340C2314yes—
269#340C2314yes—
270#340C2314yes—
271#340C2314yes—
272#336C2314yes—
273#336C2314yes—
274C2×#41—24no—
275C2×#41—24no—
276C2×#41—22no—
277C2×#41C2×D414yes—
278C22×Q8—21no—
279C22×Q8—21no—
280C22×Q8C2412yes—
281C22×Q8C2412yes—
282C22×Q8C2411yescharacteristic
283C22×D4—21no—
284C22×D4C2411yescharacteristic
285C23×C4—21no—
286C23×C4—22no—
287C23×C4—21no—
288C23×C4—22no—
289C23×C4—22no—
290C23×C4—21no—
291C23×C4C2×D411yescharacteristic
292C23×C4C2×D411yescharacteristic
293C23×C4C2411yescharacteristic
294C2×Q16—24no—
295C2×Q16—24no—
296C2×Q16—28no—
297C2×Q16—28no—
298C2×SD16—24no—
299C2×SD16—24no—
300C2×SD16—28no—
301C2×SD16—28no—
302C22×C8C2412yes—
303C22×C8C2412yes—
304C22×C8C2412yes—
305C22×C8C2412yes—
306C4⋊Q8—22no—
307C4⋊Q8—22no—
308C4⋊Q8—22no—
309C4⋊Q8—22no—
310C4⋊Q8—22no—
311C4⋊Q8—22no—
312C4⋊Q8—22no—
313C4⋊Q8—22no—
314C4⋊Q8—28no—
315C4⋊Q8—28no—
316#125—44no—
317#125—44no—
318C22⋊Q8—24no—
319C22⋊Q8—24no—
320C22⋊Q8—24no—
321C22⋊Q8—24no—
322C22⋊Q8—44no—
323C22⋊Q8—44no—
324C4×Q8—24no—
325C4×Q8—24no—
326C4×Q8—22no—
327C4×Q8—22no—
328C4×Q8—24no—
329C4×Q8—24no—
330C4×Q8—24no—
331C4×Q8—24no—
332C4×Q8—24no—
333C4×Q8—44no—
334C4×Q8—44no—
335C4×Q8C2×D414yes—
336C4×Q8C2×D414yes—
337C4×Q8#4118yes—
338C4×D4—216no—
339C4×D4—216no—
340C4×D4—24no—
341C4×D4—28no—
342C4×D4—28no—
343C4×D4—24no—
344C4×D4—22no—
345C4×D4—22no—
346C4×D4—24no—
347C4×D4C2×D414yes—
348C4×D4C2×D414yes—
349C4×D4#4118yes—
350#117—216no—
351#117—22no—
352#117—28no—
353#117—24no—
354#117—24no—
355#117—44no—
356#117—44no—
357#117C2×D414yes—
358C2×C4⋊C4—216no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—24no—
365C2×C4⋊C4—24no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—22no—
374C2×C4⋊C4—22no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—24no—
377C2×C4⋊C4—24no—
378C2×C4⋊C4C2×D412yes—
379C2×C4⋊C4C2×D412yes—
380C2×C4⋊C4C2412yes—
381C2×C4⋊C4C2412yes—
382C2×C4⋊C4C2411yescharacteristic
383C2×C4⋊C4C2411yescharacteristic
384C2×C4⋊C4C2412yes—
385C2×C4⋊C4C2412yes—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—22no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—22no—
391C2×C22⋊C4—22no—
392C2×C22⋊C4—22no—
393C2×C22⋊C4C2×D412yes—
394C2×C22⋊C4C2×D412yes—
395C2×C42—21no—
396C2×C42—22no—
397C2×C42—22no—
398C2×C42—21no—
399C2×C42—21no—
400C2×C42—28no—
401C2×C42—22no—
402C2×C42—22no—
403C2×C42—21no—
404C2×C42—21no—
405C2×C42—22no—
406C2×C42—22no—
407C2×C42—22no—
408C2×C42—22no—
409C2×C42—22no—
410C2×C42—82no—
411C2×C42—82no—
412C2×C42—82no—
413C2×C42C2×D411yescharacteristic
414C2×C42C2×D411yescharacteristic
415C2×C42C2×D411yescharacteristic
416C2×C42C2×D411yescharacteristic
417C2×C42C2×D412yes—
418C2×C42C2×D412yes—
419C2×C42C2×D411yescharacteristic
420C2×C42C2×D411yescharacteristic
421C2×C42C2×D414yes—
422C2×C42#4114yes—
423C2×C42C2412yes—
424C2×C42C2411yescharacteristic
425C2×C42C2411yescharacteristic
426C2×C42C2412yes—
427#107—24no—
428#107—24no—
429#106—24no—
430#106—24no—
431C4⋊C8—22no—
432C4⋊C8—22no—
433C4⋊C8—22no—
434C4⋊C8—22no—
435C4⋊C8—28no—
436C4⋊C8—28no—
437#103—24no—
438#103—24no—
439#103—24no—
440#103—24no—
441#103—24no—
442#103—24no—
443#103—28no—
444#103—28no—
445#103—28no—
446#103—28no—
447#102—24no—
448#102—24no—
449C22⋊C8—24no—
450C22⋊C8—24no—
451#97—24no—
452C4×C8—24no—
453#95—22no—
454#95—22no—
455#95—22no—
456#95—21no—
457#95—21no—
458#95—22no—
459#95—22no—
460#95—24no—
461#95—24no—
462#95—24no—
463#95—24no—
464#95—24no—
465#95—24no—
466#95C2×D412yes—
467#95C2×D412yes—
468#95C2×D412yes—
469#95C2×D412yes—
470C24—21no—
471C24C22×D411yescharacteristic
472#41—216no—
473#41—28no—
474#41—28no—
475C2×Q8—21no—
476C2×Q8—22no—
477C2×Q8—22no—
478C2×Q8—22no—
479C2×Q8—24no—
480C2×Q8—24no—
481C2×Q8—24no—
482C2×Q8—24no—
483C2×Q8—22no—
484C2×Q8—22no—
485C2×Q8—44no—
486C2×Q8—42no—
487C2×Q8—42no—
488C2×Q8—42no—
489C2×Q8—42no—
490C2×Q8C22×D412yes—
491C2×Q8C2×#4114yes—
492C2×D4—28no—
493C2×D4—24no—
494C2×D4—22no—
495C2×D4—21no—
496C2×D4C22×D412yes—
497C2×D4C2×#4114yes—
498C22×C4—28no—
499C22×C4—21no—
500C22×C4—24no—
501C22×C4—22no—
502C22×C4—22no—
503C22×C4—22no—
504C22×C4—22no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—22no—
508C22×C4—22no—
509C22×C4—28no—
510C22×C4—28no—
511C22×C4—24no—
512C22×C4—24no—
513C22×C4—28no—
514C22×C4—22no—
515C22×C4—22no—
516C22×C4—22no—
517C22×C4—22no—
518C22×C4—42no—
519C22×C4—42no—
520C22×C4—42no—
521C22×C4—82no—
522C22×C4—82no—
523C22×C4—82no—
524C22×C4—82no—
525C22×C4—82no—
526C22×C4—82no—
527C22×C4—82no—
528C22×C4#12114yes—
529C22×C4#12114yes—
530C22×C4C22×D412yes—
531C22×C4C22×D412yes—
532C22×C4C22×D411yescharacteristic
533C22×C4C22×D411yescharacteristic
534C22×C4C22×D411yescharacteristic
535C22×C4C22×D411yescharacteristic
536C22×C4C22×D412yes—
537C22×C4C22×D412yes—
538C22×C4C22×D412yes—
539C22×C4C22×D411yescharacteristic
540C22×C4C2×#4112yes—
541C22×C4C2511yescharacteristic
542Q16—48no—
543Q16—48no—
544SD16—48no—
545SD16—48no—
546C2×C8—22no—
547C2×C8—22no—
548C2×C8—22no—
549C2×C8—22no—
550C2×C8—24no—
551C2×C8—24no—
552C2×C8—24no—
553C2×C8—24no—
554C4⋊C4—28no—
555C4⋊C4—28no—
556C4⋊C4—24no—
557C4⋊C4—24no—
558C4⋊C4—24no—
559C4⋊C4—28no—
560C4⋊C4—24no—
561C4⋊C4—22no—
562C4⋊C4—22no—
563C4⋊C4—21no—
564C4⋊C4—21no—
565C4⋊C4—22no—
566C4⋊C4—22no—
567C4⋊C4—22no—
568C4⋊C4—22no—
569C4⋊C4—42no—
570C4⋊C4—42no—
571C4⋊C4—42no—
572C4⋊C4—42no—
573C4⋊C4—44no—
574C4⋊C4—44no—
575C4⋊C4—44no—
576C4⋊C4—44no—
577C4⋊C4—42no—
578C4⋊C4—42no—
579C4⋊C4—48no—
580C4⋊C4—48no—
581C4⋊C4C22×D412yes—
582C4⋊C4C22×D412yes—
583C4⋊C4C2×#4114yes—
584C4⋊C4C2×#4114yes—
585C22⋊C4—28no—
586C22⋊C4—24no—
587C22⋊C4—24no—
588C22⋊C4—24no—
589C22⋊C4—28no—
590C22⋊C4—24no—
591C22⋊C4—24no—
592C22⋊C4—44no—
593C22⋊C4—44no—
594C42—28no—
595C42—24no—
596C42—24no—
597C42—24no—
598C42—28no—
599C42—22no—
600C42—22no—
601C42—22no—
602C42—22no—
603C42—22no—
604C42—22no—
605C42—42no—
606C42—42no—
607C42—44no—
608C42—44no—
609C42—42no—
610C42—42no—
611C42—82no—
612C42—82no—
613C42—82no—
614C42—82no—
615C42—82no—
616C42#12114yes—
617C42#12114yes—
618C42C22×D412yes—
619C42C22×D412yes—
620C42C2×#4112yes—
621C42C2×#4112yes—
622C42C2×#4114yes—
623C42C2×#4114yes—
624C23—22no—
625C23—22no—
626C23—22no—
627C23—24no—
628C23—21no—
629C23—21no—
630C23—42no—
631C23—41no—
632C23—41no—
633C23C2×#12112yes—
634C23C23×D411yescharacteristic
635Q8—22no—
636Q8—24no—
637Q8—42no—
638Q8—42no—
639Q8—42no—
640Q8—42no—
641Q8—44no—
642Q8—44no—
643Q8—84no—
644Q8—84no—
645Q8#53214yes—
646D4—216no—
647D4—22no—
648D4—24no—
649D4#53214yes—
650C2×C4—216no—
651C2×C4—22no—
652C2×C4—22no—
653C2×C4—22no—
654C2×C4—22no—
655C2×C4—22no—
656C2×C4—24no—
657C2×C4—22no—
658C2×C4—22no—
659C2×C4—21no—
660C2×C4—22no—
661C2×C4—21no—
662C2×C4—22no—
663C2×C4—22no—
664C2×C4—24no—
665C2×C4—28no—
666C2×C4—22no—
667C2×C4—22no—
668C2×C4—22no—
669C2×C4—22no—
670C2×C4—22no—
671C2×C4—22no—
672C2×C4—24no—
673C2×C4—24no—
674C2×C4—22no—
675C2×C4—22no—
676C2×C4—44no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—41no—
683C2×C4—41no—
684C2×C4—48no—
685C2×C4—44no—
686C2×C4—42no—
687C2×C4—42no—
688C2×C4—42no—
689C2×C4—42no—
690C2×C4—44no—
691C2×C4—44no—
692C2×C4—42no—
693C2×C4—42no—
694C2×C4—84no—
695C2×C4—88no—
696C2×C4—88no—
697C2×C4—88no—
698C2×C4C2×#12112yes—
699C2×C4C2×#12112yes—
700C2×C4C2×#12112yes—
701C2×C4C2×#12112yes—
702C2×C4C2×#12112yes—
703C2×C4#52514yes—
704C2×C4#52514yes—
705C2×C4#57712yes—
706C2×C4#57812yes—
707C2×C4C23×D411yescharacteristic
708C2×C4C22×#4112yes—
709C2×C4C22×#4112yes—
710C2×C4C22×#4111yescharacteristic
711C8—42no—
712C8—42no—
713C8—42no—
714C8—42no—
715C22—22no—
716C22—22no—
717C22—22no—
718C22—24no—
719C22—42no—
720C22—42no—
721C22—42no—
722C22—44no—
723C22—42no—
724C22—44no—
725C22C22×#12111yescharacteristic
726C22C2×#52512yes—
727C22C2×#57711yescharacteristic
728C22C2×#57811yescharacteristic
729C22#358612yes—
730C4—22no—
731C4—22no—
732C4—22no—
733C4—22no—
734C4—24no—
735C4—24no—
736C4—24no—
737C4—42no—
738C4—42no—
739C4—42no—
740C4—44no—
741C4—44no—
742C4—42no—
743C4—82no—
744C4—82no—
745C4—82no—
746C4—82no—
747C4#304614yes—
748C4C2×#53211yescharacteristic
749C4C2×#53211yescharacteristic
750C4#344412yes—
751C2—42no—
752C2—44no—
753C2#3662411yescharacteristic
754C2C2×#304611yescharacteristic
755C2#6042711yescharacteristic
756C2#6059112yes—
757C2#6059412yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
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. #35322C2semidirect
5. #33202C2semidirect
6. #33201C2semidirect
7. #33163C2semidirect
8. #33162C2semidirect
9. #33117C2semidirect
10. #33033C2semidirect
11. #33032C2semidirect
12. #33027C2semidirect
13. #33026C2semidirect
15. #21516C2semidirect
16. #21514C2semidirect
17. #21394C2semidirect
18. #21385C2semidirect
19. #20815C2semidirect

Maximal subgroups

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

subgroupindex
C2×#31162
C2×#31072
#353222
C4×#5202
#332022
#332012
#331632
#331622
#331172
#330332
#330322
#330272
#330262
#329162
#215162
#215142
#213942
#213852
#208152