Finite Group Databasebeta

← #678380

Subgroups — #678380

Order 512.

5094 conjugacy classes of subgroups.

The subgroup lattice has 724 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#547C211yesmaximal, characteristic
2C2×#2568C211yesmaximal, characteristic, complemented
3C2×#2538C211yesmaximal, characteristic, complemented
4C2×C4×#124C211yesmaximal, characteristic, complemented
5#15667C218yesmaximal, complemented
6#15219C212yesmaximal, complemented
7#15215C212yesmaximal, complemented
8#15179C214yesmaximal, complemented
9#14714C211yesmaximal, characteristic, complemented
10#14139C212yesmaximal, complemented
11#13937C214yesmaximal, complemented
12#13906C211yesmaximal, characteristic, complemented
13#13881C212yesmaximal, complemented
14C4×#390C211yesmaximal, characteristic, complemented
15C23×#41C2212yes—
16C2×#547—28no—
17C2×#547—28no—
18C2×#547—28no—
19C22×C4⋊Q8C2211yescharacteristic
20C22×#124C2211yescharacteristic
21C22×#123C2214yes—
22C22×C22⋊Q8C2212yes—
23C22×C22⋊Q8C2212yes—
24C22×#121C2212yes—
25C22×C4×Q8C2211yescharacteristic
26C22×C4×D4C2211yescharacteristic
27C22×C4×D4C2211yescharacteristic
28C23×C4⋊C4C2212yes—
29#2568—28no—
30#2538—28no—
31C2×#400C2214yes—
32C2×#396C2214yes—
33C4×#124—28no—
34C2×#391C2211yescharacteristic
35C2×#391C2212yes—
36C2×#391C2214yes—
37C2×#391C2214yes—
38C2×#390C2212yes—
39C2×#390C2212yes—
40C2×#390C2211yescharacteristic
41C2×#390C2212yes—
42C2×#389C2212yes—
43C2×#388C2214yes—
44C2×#384C2211yescharacteristic
45C2×#382C2211yescharacteristic
46C2×#382C2212yes—
47C2×#380C2212yes—
48C2×#380C2214yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C22⋊C4C2212yes—
51C2×C4×C22⋊C4C2214yes—
52C2×C4×C22⋊C4C2212yes—
53C2×C43C2211yescharacteristic
54#1442C2218yes—
55#1441C2218yes—
56#1440C2218yes—
57#1439C2214yes—
58#1439C2218yes—
59#1438C2218yes—
60#1438C2218yes—
61#1437C2214yes—
62C4⋊#95C2214yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C22⋊#95C2218yes—
68C22⋊#95C2218yes—
69C22×#41—28no—
70C22×#41—28no—
71C22×#41—216no—
72C22×#41—28no—
73C22×#41—48no—
74C23×Q8C2312yes—
75C23×Q8C2311yescharacteristic
76C23×D4C2311yescharacteristic
77C23×D4C2314yes—
78C23×D4C2311yescharacteristic
79C24×C4C2312yes—
80C24×C4C2312yes—
81C24×C4C2314yes—
82#547—2128no—
83C2×C4⋊Q8—24no—
84C2×C4⋊Q8—24no—
85C2×C4⋊Q8D418yes—
86C2×#124—24no—
87C2×#124—24no—
88C2×#124—24no—
89C2×#123—216no—
90C2×#123—216no—
91C2×#123—216no—
92C2×C22⋊Q8—28no—
93C2×C22⋊Q8—28no—
94C2×C22⋊Q8—28no—
95C2×C22⋊Q8—28no—
96C2×C22⋊Q8—28no—
97C2×C22⋊Q8—44no—
98C2×#121—28no—
99C2×#121—28no—
100C2×#121—28no—
101C2×C4×Q8—24no—
102C2×C4×Q8—24no—
103C2×C4×Q8—24no—
104C2×C4×D4—24no—
105C2×C4×D4—24no—
106C2×C4×D4—24no—
107C2×C4×D4—24no—
108C2×C4×D4D418yes—
109C2×C4×D4D418yes—
110C22×C4⋊C4—28no—
111C22×C4⋊C4—28no—
112C22×C4⋊C4—28no—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2314yes—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2314yes—
134C22×C22⋊C4C2318yes—
135C22×C42C2312yes—
136C22×C42C2311yescharacteristic
137C22×C42C2312yes—
138C22×C42C2312yes—
139C22×C42C2312yes—
140C22×C42C2311yescharacteristic
141C22×C42C2311yescharacteristic
142#400—216no—
143#396—216no—
144#391—216no—
145#391—28no—
146#391—24no—
147#391—48no—
148#390—28no—
149#390—24no—
150#390—28no—
151#390—44no—
152#389—28no—
153#388—216no—
154#384—24no—
155#382—28no—
156#382—24no—
157#380—28no—
158#380—48no—
159C4×C4⋊C4—24no—
160C4×C22⋊C4—216no—
161C4×C22⋊C4—28no—
162C4×C22⋊C4—44no—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2318yes—
166C2×#95C2318yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2318yes—
171C2×#95C2318yes—
172C2×#95C2318yes—
173C2×#95C2314yes—
174C43—24no—
175C25C2412yes—
176C25C2412yes—
177C2×#41—264no—
178C2×#41—464no—
179C2×#41—464no—
180C2×#41—432no—
181C2×#41—432no—
182C22×Q8—22no—
183C22×Q8—22no—
184C22×Q8—22no—
185C22×Q8—24no—
186C22×Q8—24no—
187C22×Q8—22no—
188C22×Q8—28no—
189C22×Q8—22no—
190C22×Q8—28no—
191C22×Q8—22no—
192C22×Q8C2×D418yes—
193C22×Q8#4114yescomplemented
194C22×D4—28no—
195C22×D4—28no—
196C22×D4—216no—
197C22×D4—22no—
198C22×D4—22no—
199C22×D4—22no—
200C22×D4—22no—
201C22×D4—28no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—24no—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4—48no—
208C22×D4C2×D414yes—
209C22×D4C2×D414yes—
210C22×D4#4118yes—
211C23×C4—28no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—28no—
215C23×C4—28no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—28no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—28no—
223C23×C4—24no—
224C23×C4—216no—
225C23×C4—48no—
226C23×C4C2×D418yes—
227C23×C4C2×D418yes—
228C23×C4C2414yes—
229C23×C4C2412yes—
230C23×C4C2414yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2414yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239C23×C4C2411yescharacteristic
240C4⋊Q8—232no—
241#124—232no—
242#123—2128no—
243C22⋊Q8—264no—
244C22⋊Q8—432no—
245#121—264no—
246C4×Q8—416no—
247C4×D4—232no—
248C4×D4—232no—
249C2×C4⋊C4—264no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4C2×D414yes—
281C2×C4⋊C4C2×D418yes—
282C2×C4⋊C4C2×D414yes—
283C2×C4⋊C4C2×D418yes—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—48no—
308C2×C22⋊C4—48no—
309C2×C22⋊C4—48no—
310C2×C22⋊C4—48no—
311C2×C22⋊C4C2×D418yes—
312C2×C22⋊C4C2×D418yes—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—22no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—22no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—22no—
329C2×C42—42no—
330C2×C42C2×D414yes—
331C2×C42C2×D414yes—
332C2×C42C2×D414yes—
333C2×C42#4114yescomplemented
334#95—28no—
335#95—28no—
336#95—24no—
337#95—28no—
338#95—216no—
339#95—216no—
340#95—28no—
341#95—44no—
342#95—48no—
343#95—48no—
344#95—48no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—24no—
351C24—24no—
352C24—22no—
353C24—24no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—24no—
359C24—22no—
360C24—24no—
361C24—24no—
362C24—22no—
363C24—24no—
364C24C22×D414yes—
365C24C22×D414yes—
366C24C2×#4118yes—
367C24C2511yescharacteristic
368#41—4256no—
369C2×Q8—216no—
370C2×Q8—28no—
371C2×Q8—28no—
372C2×Q8—232no—
373C2×Q8—28no—
374C2×Q8—44no—
375C2×Q8—48no—
376C2×Q8—48no—
377C2×Q8—44no—
378C2×Q8—416no—
379C2×Q8—44no—
380C2×Q8—44no—
381C2×D4—232no—
382C2×D4—28no—
383C2×D4—28no—
384C2×D4—28no—
385C2×D4—28no—
386C2×D4—28no—
387C2×D4—216no—
388C2×D4—216no—
389C2×D4—432no—
390C2×D4—432no—
391C2×D4—416no—
392C2×D4—416no—
393C2×D4—44no—
394C2×D4—44no—
395C2×D4—44no—
396C2×D4—44no—
397C2×D4#124116yescomplemented
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—232no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—28no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—28no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—216no—
434C22×C4—216no—
435C22×C4—216no—
436C22×C4—232no—
437C22×C4—216no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—28no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—28no—
455C22×C4—432no—
456C22×C4—432no—
457C22×C4—416no—
458C22×C4—416no—
459C22×C4—48no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—416no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4C22×D418yes—
467C22×C4C22×D414yes—
468C22×C4C22×D414yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C2×#4114yes—
479C22×C4C2×#4114yes—
480C22×C4C2×#4112yes—
481C22×C4C2×#4112yes—
482C22×C4C2×#4114yes—
483C22×C4H(5,𝔽2)14yes—
484C22×C4H(5,𝔽2)12yes—
485C22×C42−1+414yes—
486C22×C42−1+412yes—
487C4⋊C4—232no—
488C4⋊C4—28no—
489C4⋊C4—216no—
490C4⋊C4—28no—
491C4⋊C4—216no—
492C4⋊C4—416no—
493C4⋊C4—48no—
494C4⋊C4—44no—
495C4⋊C4—48no—
496C4⋊C4—48no—
497C4⋊C4—48no—
498C4⋊C4—84no—
499C22⋊C4—232no—
500C22⋊C4—232no—
501C22⋊C4—216no—
502C22⋊C4—216no—
503C22⋊C4—416no—
504C22⋊C4—416no—
505C22⋊C4—416no—
506C22⋊C4—416no—
507C22⋊C4—88no—
508C22⋊C4—816no—
509C42—28no—
510C42—48no—
511C42—48no—
512C42—44no—
513C42—48no—
514C42—48no—
515C42#120116yes—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—28no—
520C23—28no—
521C23—24no—
522C23—28no—
523C23—28no—
524C23—24no—
525C23—24no—
526C23—28no—
527C23—28no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—28no—
532C23—28no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—42no—
544C23—42no—
545C23—44no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—42no—
554C23—42no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23C2×#12418yes—
562C23C23×D411yescharacteristic
563C23C23×D411yescharacteristic
564C23C23×D411yescharacteristic
565C23C22×#4111yescharacteristic
566C23C22×#4111yescharacteristic
567C23C22×#4112yes—
568C23C2×H(5,𝔽2)12yes—
569C23C2×H(5,𝔽2)11yescharacteristic
570C23C2×2−1+411yescharacteristic
571C23C2×2−1+412yes—
572C23#58512yes—
573Q8—416no—
574Q8—432no—
575D4—232no—
576D4—464no—
577D4—416no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—28no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—28no—
590C2×C4—216no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—464no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—48no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—432no—
622C2×C4—432no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—48no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—48no—
640C2×C4C2×#12014yes—
641C2×C4C2×#12014yes—
642C2×C4C2×#12014yes—
643C2×C4#52514yes—
644C2×C4C2×#12414yes—
645C2×C4#53514yes—
646C2×C4#53714yes—
647C2×C4#54614yes—
648C2×C4#54618yes—
649C2×C4#54618yes—
650C2×C4#54714yes—
651C2×C4#54814yes—
652C2×C4#55014yes—
653C22—28no—
654C22—28no—
655C22—28no—
656C22—48no—
657C22—48no—
658C22—44no—
659C22—48no—
660C22—44no—
661C22—44no—
662C22—44no—
663C22—48no—
664C22—48no—
665C22—44no—
666C22—44no—
667C22—44no—
668C22—48no—
669C22—84no—
670C22—84no—
671C22—82no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22C22×#12011yescharacteristic
676C22C2×#52511yescharacteristic
677C22C22×#12411yescharacteristic
678C22C2×#53511yescharacteristic
679C22C2×#53711yescharacteristic
680C22C2×#54612yes—
681C22C2×#54611yescharacteristic
682C22C2×#54612yes—
683C22C2×#54711yescharacteristic
684C22C2×#54811yescharacteristic
685C22C2×#55011yescharacteristic
686C22#347512yes—
687C22#348212yes—
688C22#348912yes—
689C22#349712yes—
690C22#349911yescharacteristic
691C22#350011yescharacteristic
692C22#351112yes—
693C22#351512yes—
694C22#351712yes—
695C22#352012yes—
696C22#354612yes—
697C22#355212yes—
698C4—24no—
699C4—44no—
700C4—44no—
701C4—48no—
702C4—44no—
703C4—44no—
704C4—44no—
705C4—84no—
706C4—84no—
707C4—82no—
708C4#240718yes—
709C4#243518yes—
710C2—48no—
711C2—84no—
712C2C2×#243511yescharacteristic
713C2C2×#240711yescharacteristic
714C2#3566011yescharacteristic
715C2#3616411yescharacteristic
716C2#3630311yescharacteristic
717C2#3640412yes—
718C2#3890812yes—
719C2#3895611yescharacteristic
720C2#3948811yescharacteristic
721C2#5941012yes—
722C2#5941812yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#547C2
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×#2568C2semidirect
3. C2×#2538C2semidirect
4. C2×C4×#124C2semidirect
5. #15667C2semidirect
6. #15219C2semidirect
7. #15215C2semidirect
8. #15179C2semidirect
9. #14714C2semidirect
10. #14139C2semidirect
11. #13937C2semidirect
12. #13906C2semidirect
13. #13881C2semidirect
14. C4×#390C2semidirect
193. C22×Q8#41semidirect
333. C2×C42#41semidirect
397. C2×D4#124semidirect

Maximal subgroups

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

subgroupindex
C22×#5472
C2×#25682
C2×#25382
C2×C4×#1242
#156672
#152192
#152152
#151792
#147142
#141392
#139372
#139062
#138812
C4×#3902