Finite Group Databasebeta

← #704322

Subgroups — #704322

Order 512.

3510 conjugacy classes of subgroups.

The subgroup lattice has 786 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2432C211yesmaximal, characteristic
2C2×#2625C211yesmaximal, characteristic, complemented
3C2×#2339C211yesmaximal, characteristic
4C2×#2311C211yesmaximal, characteristic
5#16394C212yesmaximal, complemented
6#16393C212yesmaximal, complemented
7#15494C214yesmaximal, complemented
8#15390C214yesmaximal, complemented
9#15188C212yesmaximal, complemented
10#15186C212yesmaximal, complemented
11#14826C212yesmaximal, complemented
12#14766C212yesmaximal, complemented
13#14139C211yesmaximal, characteristic, complemented
14#14130C211yesmaximal, characteristic, complemented
15#14027C212yesmaximal, complemented
16#13920C212yesmaximal, complemented
17#13853C211yesmaximal, characteristic, complemented
18C23×#41C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×#124C2211yescharacteristic
21C22×C22⋊Q8C2211yescharacteristic
22C22×C4×D4C2212yes—
23C22×#117C2211yescharacteristic
24C23×C4⋊C4C2211yescharacteristic
25#2625—28no—
26#2432—28no—
27C2×#400C2212yes—
28C2×#398C2212yes—
29C2×#397C2214yes—
30C2×#394C2212yes—
31C2×#394C2212yes—
32#2339—28no—
33#2311—28no—
34C2×#391C2212yes—
35C2×#391C2212yes—
36C2×#391C2211yescharacteristic
37C2×#391C2212yes—
38C2×#391C2212yes—
39C2×#390C2211yescharacteristic
40C2×#390C2212yes—
41C2×#389C2212yes—
42C2×#389C2212yes—
43C2×#386C2214yes—
44C2×#386C2214yes—
45C2×#385C2212yes—
46C2×#385C2212yes—
47C2×#385C2212yes—
48C2×#381C2212yes—
49C2×#380C2212yes—
50C2×#380C2212yes—
51C2×#380C2214yes—
52C2×#380C2211yescharacteristic
53C2×C4×C22⋊C4C2211yescharacteristic
54C2×C4×C22⋊C4C2211yescharacteristic
55C22×#95C2212yes—
56#1445C2211yescharacteristic
57#1445C2214yes—
58#1445C2211yescharacteristic
59#1445C2211yescharacteristic
60#1445C2211yescharacteristic
61#1444C2212yes—
62#1443C2214yes—
63#1443C2214yes—
64#1442C2212yes—
65#1442C2214yes—
66#1442C2214yes—
67#1442C2214yes—
68#1441C2212yes—
69#1441C2214yes—
70#1441C2214yes—
71#1441C2212yes—
72#1440C2214yes—
73#1440C2214yes—
74#1440C2212yes—
75#1440C2212yes—
76#1439C2214yes—
77#1439C2214yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C22⋊#95C2214yes—
81C22⋊#95C2214yes—
82C22⋊#95C2212yes—
83C22⋊#95C2214yes—
84C22⋊#95C2214yes—
85C4×#95C2212yes—
86C4×#95C2212yes—
87C22×#41—28no—
88C22×#41—24no—
89C22×#41—24no—
90C22×#41—44no—
91C22×#41D418yes—
92C23×Q8C2311yescharacteristic
93C23×Q8C2311yescharacteristic
94C23×D4C2312yes—
95C23×D4C2311yescharacteristic
96C24×C4C2312yes—
97C24×C4C2311yescharacteristic
98C24×C4C2314yes—
99C24×C4C2312yes—
100C2×C4⋊Q8—24no—
101C2×C4⋊Q8—44no—
102C2×#124—24no—
103C2×#124—28no—
104C2×C22⋊Q8—24no—
105C2×C22⋊Q8—24no—
106C2×C22⋊Q8—24no—
107C2×C4×D4—28no—
108C2×C4×D4—28no—
109C2×C4×D4—44no—
110C2×#117—24no—
111C2×#117—24no—
112C2×#117—24no—
113C22×C4⋊C4—24no—
114C22×C4⋊C4—44no—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2314yes—
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2314yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2311yescharacteristic
131C22×C22⋊C4C2311yescharacteristic
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2312yes—
134C22×C22⋊C4C2314yes—
135C22×C22⋊C4C2312yes—
136C22×C22⋊C4C2314yes—
137C22×C22⋊C4C2314yes—
138C22×C22⋊C4C2312yes—
139C22×C42C2312yes—
140C22×C42C2312yes—
141C22×C42C2311yescharacteristic
142C22×C42C2311yescharacteristic
143#400—28no—
144#398—28no—
145#397—216no—
146#394—28no—
147#394—28no—
148#391—28no—
149#391—24no—
150#391—28no—
151#391—28no—
152#391—44no—
153#390—28no—
154#390—24no—
155#389—28no—
156#389—44no—
157#386—216no—
158#386—216no—
159#385—28no—
160#385—28no—
161#385—44no—
162#381—28no—
163#380—28no—
164#380—28no—
165#380—48no—
166#380D418yes—
167C4×C22⋊C4—24no—
168C4×C22⋊C4—24no—
169C2×#95—28no—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2311yescharacteristic
180C2×#95C2311yescharacteristic
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2311yescharacteristic
186C2×#95C2311yescharacteristic
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2312yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2314yes—
197C2×#95C2314yes—
198C2×#95C2312yes—
199C25C2411yescharacteristic
200C25C2412yes—
201C2×#41—232no—
202C2×#41—232no—
203C2×#41—416no—
204C2×#41—432no—
205C2×#41—432no—
206C22×Q8—22no—
207C22×Q8—22no—
208C22×Q8—22no—
209C22×Q8—22no—
210C22×Q8—24no—
211C22×Q8—22no—
212C22×Q8—24no—
213C22×Q8—24no—
214C22×Q8—24no—
215C22×Q8C2×D414yes—
216C22×D4—24no—
217C22×D4—24no—
218C22×D4—28no—
219C22×D4—22no—
220C22×D4—22no—
221C22×D4—24no—
222C22×D4—44no—
223C22×D4C2×D418yes—
224C22×D4C2×D414yes—
225C22×D4#4118yes—
226C23×C4—24no—
227C23×C4—24no—
228C23×C4—28no—
229C23×C4—24no—
230C23×C4—24no—
231C23×C4—28no—
232C23×C4—24no—
233C23×C4—28no—
234C23×C4—28no—
235C23×C4—44no—
236C23×C4—44no—
237C23×C4—44no—
238C23×C4—44no—
239C23×C4—44no—
240C23×C4—44no—
241C23×C4—42no—
242C23×C4—44no—
243C23×C4—42no—
244C23×C4C2×D418yes—
245C23×C4C2×D414yes—
246C23×C4C2×D414yes—
247C23×C4#4114yes—
248C23×C4C2412yes—
249C23×C4C2412yes—
250C23×C4C2412yes—
251C23×C4C2412yes—
252C23×C4C2411yescharacteristic
253C23×C4C2411yescharacteristic
254C23×C4C2412yes—
255C23×C4C2414yes—
256C23×C4C2414yes—
257C23×C4C2412yes—
258C23×C4C2411yescharacteristic
259C23×C4C2411yescharacteristic
260C23×C4C2412yes—
261C23×C4C2411yescharacteristic
262C23×C4C2411yescharacteristic
263C4⋊Q8—416no—
264#124—416no—
265C22⋊Q8—416no—
266C4×D4—432no—
267#117—416no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—28no—
289C2×C4⋊C4—28no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—416no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4C2×D418yes—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—28no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—28no—
314C2×C22⋊C4—28no—
315C2×C22⋊C4—28no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—28no—
322C2×C22⋊C4—28no—
323C2×C22⋊C4—42no—
324C2×C22⋊C4—41no—
325C2×C22⋊C4—42no—
326C2×C22⋊C4—41no—
327C2×C22⋊C4—42no—
328C2×C22⋊C4—42no—
329C2×C22⋊C4—44no—
330C2×C22⋊C4—42no—
331C2×C22⋊C4—44no—
332C2×C22⋊C4—44no—
333C2×C22⋊C4—44no—
334C2×C22⋊C4—44no—
335C2×C22⋊C4C2×D418yes—
336C2×C42—24no—
337C2×C42—24no—
338C2×C42—24no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—24no—
342C2×C42—24no—
343C2×C42—24no—
344C2×C42—24no—
345C2×C42—24no—
346#95—22no—
347#95—28no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95—28no—
352#95—28no—
353#95—28no—
354#95—24no—
355#95—24no—
356#95—28no—
357#95—28no—
358#95—28no—
359#95—24no—
360#95—24no—
361#95—24no—
362#95—24no—
363#95—42no—
364#95—44no—
365#95—42no—
366#95—42no—
367#95—42no—
368#95—44no—
369#95—44no—
370#95—44no—
371#95—44no—
372#95—44no—
373#95—44no—
374#95C2×D418yes—
375C24—22no—
376C24—22no—
377C24—22no—
378C24—22no—
379C24—21no—
380C24—24no—
381C24—22no—
382C24—22no—
383C24—21no—
384C24—22no—
385C24—24no—
386C24—22no—
387C24—21no—
388C24—24no—
389C24—22no—
390C24—21no—
391C24—22no—
392C24C22×D412yes—
393C24C22×D412yes—
394C24C22×D414yes—
395C24C2×#4112yes—
396C24C2×#4118yes—
397C24C2511yescharacteristic
398#41—4128no—
399C2×Q8—28no—
400C2×Q8—28no—
401C2×Q8—28no—
402C2×Q8—44no—
403C2×Q8—44no—
404C2×Q8—44no—
405C2×Q8—48no—
406C2×Q8—48no—
407C2×Q8—48no—
408C2×Q8—48no—
409C2×D4—216no—
410C2×D4—216no—
411C2×D4—28no—
412C2×D4—28no—
413C2×D4—216no—
414C2×D4—216no—
415C2×D4—48no—
416C2×D4—416no—
417C2×D4—416no—
418C2×D4#123116yes—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—28no—
422C22×C4—28no—
423C22×C4—216no—
424C22×C4—216no—
425C22×C4—216no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—216no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—24no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—21no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—24no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—21no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—21no—
491C22×C4—22no—
492C22×C4—21no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—24no—
497C22×C4—24no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—24no—
501C22×C4—48no—
502C22×C4—416no—
503C22×C4—416no—
504C22×C4—416no—
505C22×C4—416no—
506C22×C4—416no—
507C22×C4—48no—
508C22×C4—88no—
509C22×C4—88no—
510C22×C4—88no—
511C22×C4—84no—
512C22×C4—84no—
513C22×C4—88no—
514C22×C4—88no—
515C22×C4—88no—
516C22×C4#121116yes—
517C22×C4C22×D414yes—
518C22×C4C22×D418yes—
519C22×C4C22×D412yes—
520C22×C4C22×D412yes—
521C22×C4C22×D412yes—
522C22×C4C22×D412yes—
523C22×C4C2×#4112yes—
524C22×C4C2×#4112yes—
525C22×C4C2×#4114yes—
526C22×C4H(5,𝔽2)12yes—
527C22×C4H(5,𝔽2)14yes—
528C22×C42−1+412yes—
529C22×C42−1+414yes—
530C4⋊C4—216no—
531C4⋊C4—48no—
532C4⋊C4—48no—
533C4⋊C4—48no—
534C4⋊C4—44no—
535C4⋊C4—44no—
536C4⋊C4—44no—
537C4⋊C4—48no—
538C4⋊C4—48no—
539C4⋊C4—88no—
540C4⋊C4—88no—
541C22⋊C4—48no—
542C22⋊C4—48no—
543C22⋊C4—44no—
544C22⋊C4—44no—
545C22⋊C4—48no—
546C22⋊C4—48no—
547C22⋊C4—416no—
548C22⋊C4—48no—
549C22⋊C4—48no—
550C22⋊C4—48no—
551C22⋊C4—48no—
552C22⋊C4—88no—
553C22⋊C4—88no—
554C42—48no—
555C42—44no—
556C42—44no—
557C42—48no—
558C23—24no—
559C23—22no—
560C23—22no—
561C23—24no—
562C23—24no—
563C23—28no—
564C23—24no—
565C23—24no—
566C23—28no—
567C23—24no—
568C23—22no—
569C23—24no—
570C23—24no—
571C23—22no—
572C23—22no—
573C23—22no—
574C23—24no—
575C23—22no—
576C23—24no—
577C23—22no—
578C23—22no—
579C23—22no—
580C23—28no—
581C23—28no—
582C23—28no—
583C23—28no—
584C23—44no—
585C23—44no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23—42no—
593C23—42no—
594C23—44no—
595C23—42no—
596C23—42no—
597C23—44no—
598C23—44no—
599C23—42no—
600C23—42no—
601C23—44no—
602C23C2×#12114yes—
603C23C2×#12318yes—
604C23C23×D411yescharacteristic
605C23C23×D411yescharacteristic
606C23C22×#4111yescharacteristic
607C23C22×#4112yes—
608C23C2×H(5,𝔽2)12yes—
609C23C2×H(5,𝔽2)11yescharacteristic
610C23C2×2−1+412yes—
611C23C2×2−1+411yescharacteristic
612C23#58512yes—
613C23#58511yescharacteristic
614C23#58511yescharacteristic
615Q8—416no—
616Q8—88no—
617D4—232no—
618D4—432no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—232no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—28no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—24no—
631C2×C4—28no—
632C2×C4—28no—
633C2×C4—24no—
634C2×C4—24no—
635C2×C4—24no—
636C2×C4—24no—
637C2×C4—432no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—42no—
644C2×C4—41no—
645C2×C4—41no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—44no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—41no—
659C2×C4—41no—
660C2×C4—42no—
661C2×C4—44no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—41no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—44no—
672C2×C4—42no—
673C2×C4—41no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—41no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—44no—
687C2×C4—41no—
688C2×C4—42no—
689C2×C4—42no—
690C2×C4—42no—
691C2×C4—42no—
692C2×C4—42no—
693C2×C4—42no—
694C2×C4—42no—
695C2×C4—42no—
696C2×C4—42no—
697C2×C4—42no—
698C2×C4—42no—
699C2×C4—44no—
700C2×C4—44no—
701C2×C4—42no—
702C2×C4—44no—
703C2×C4—44no—
704C2×C4—832no—
705C2×C4—816no—
706C2×C4C2×#12114yes—
707C2×C4C2×#12114yes—
708C2×C4C2×#12314yes—
709C2×C4#53414yes—
710C2×C4#54618yes—
711C2×C4#55118yes—
712C22—24no—
713C22—28no—
714C22—28no—
715C22—28no—
716C22—24no—
717C22—24no—
718C22—48no—
719C22—44no—
720C22—44no—
721C22—48no—
722C22—48no—
723C22—44no—
724C22—44no—
725C22—48no—
726C22—42no—
727C22—42no—
728C22—44no—
729C22—42no—
730C22—42no—
731C22—44no—
732C22—44no—
733C22—44no—
734C22C22×#12111yescharacteristic
735C22C22×#12311yescharacteristic
736C22C2×#53411yescharacteristic
737C22#346112yes—
738C22C2×#54612yes—
739C22C2×#55112yes—
740C22#348811yescharacteristic
741C22#348912yes—
742C22#348911yescharacteristic
743C22#349611yescharacteristic
744C22#349711yescharacteristic
745C22#349911yescharacteristic
746C22#350011yescharacteristic
747C22#351111yescharacteristic
748C22#351211yescharacteristic
749C22#351512yes—
750C22#351512yes—
751C22#352012yes—
752C22#352112yes—
753C22#355212yes—
754C22#355812yes—
755C22#356112yes—
756C22#356212yes—
757C4—24no—
758C4—48no—
759C4—44no—
760C4—44no—
761C4—42no—
762C4—42no—
763C4—82no—
764C4—82no—
765C4—81no—
766C4—81no—
767C4—84no—
768C4—82no—
769C4—82no—
770C4—82no—
771C4#241218yes—
772C2—44no—
773C2—48no—
774C2C2×#241211yescharacteristic
775C2#3530811yescharacteristic
776C2#3556411yescharacteristic
777C2#3599811yescharacteristic
778C2#3606711yescharacteristic
779C2#3824812yes—
780C2#3906812yes—
781C2#4291211yescharacteristic
782C2#4297911yescharacteristic
783C2#4510812yes—
784C2#4517512yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×#41C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2625C2semidirect
5. #16394C2semidirect
6. #16393C2semidirect
7. #15494C2semidirect
8. #15390C2semidirect
9. #15188C2semidirect
10. #15186C2semidirect
11. #14826C2semidirect
12. #14766C2semidirect
13. #14139C2semidirect
14. #14130C2semidirect
15. #14027C2semidirect
16. #13920C2semidirect
17. #13853C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24322
C2×#26252
C2×#23392
C2×#23112
#163942
#163932
#154942
#153902
#151882
#151862
#148262
#147662
#141392
#141302
#140272
#139202
#138532