Finite Group Databasebeta

← #616158

Subgroups — #616158

Order 512.

3734 conjugacy classes of subgroups.

The subgroup lattice has 726 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic, complemented
2C2×#2857C211yesmaximal, characteristic, complemented
3C2×#2415C212yesmaximal
4C2×#2285C211yesmaximal, characteristic, complemented
5C2×C42×D4C211yesmaximal, characteristic
6#15237C212yesmaximal, complemented
7#15236C212yesmaximal, complemented
8#15132C214yesmaximal, complemented
9#15129C214yesmaximal
10#14714C211yesmaximal, characteristic, complemented
11#14712C211yesmaximal, characteristic, complemented
12#14686C211yesmaximal, characteristic, complemented
13#14138C212yesmaximal, complemented
14#14046C214yesmaximal, complemented
15#13974C214yesmaximal, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C22×C22⋊Q8C2212yes—
19C22×C4×D4C2212yes—
20C22×C4×D4C2211yescharacteristic
21C22×#117C2212yes—
22C23×C42C2211yescharacteristic
23C23×C42C2211yescharacteristic
24#2857—28no—
25#2857—28no—
26#2415—216no—
27C2×#397C2214yes—
28C2×#394C2214yes—
29C2×#391C2212yes—
30C2×#391C2212yes—
31C2×#391C2212yes—
32C2×#391C2212yes—
33C2×#389C2214yes—
34C2×#389C2214yes—
35C2×#389C2214yes—
36C2×#389C2212yes—
37C2×#389C2214yes—
38C2×#388C2214yes—
39#2285—28no—
40C2×#384C2211yescharacteristic
41C2×#384C2211yescharacteristic
42C2×#384C2211yescharacteristic
43C2×#382C2212yes—
44C2×#380C2214yes—
45C2×#380C2214yes—
46C2×#380C2214yes—
47C2×#380C2214yes—
48C2×#380C2212yes—
49C42×D4—28no—
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2214yes—
52C2×C4×C4⋊C4C2212yes—
53C2×C4×C22⋊C4C2212yes—
54C2×C4×C22⋊C4C2211yescharacteristic
55C2×C4×C22⋊C4C2211yescharacteristic
56C2×C4×C22⋊C4C2212yes—
57C2×C43C2211yescharacteristic
58#1443C2218yes—
59#1442C2218yes—
60#1441C2218yes—
61#1440C2218yes—
62#1440C2218yes—
63#1436C2218yes—
64C4⋊#95C2214yes—
65C4⋊#95C2214yes—
66C4⋊#95C2214yes—
67C4⋊#95C2214yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C23×Q8C2312yes—
71C23×D4C2311yescharacteristic
72C24×C4C2311yescharacteristic
73C24×C4C2312yes—
74C24×C4C2312yes—
75C24×C4C2311yescharacteristic
76C2×C4⋊Q8—24no—
77C2×C4⋊Q8—24no—
78C2×C4⋊Q8—24no—
79C2×C22⋊Q8—28no—
80C2×C22⋊Q8—28no—
81C2×C22⋊Q8—28no—
82C2×C22⋊Q8—28no—
83C2×C22⋊Q8—28no—
84C2×C22⋊Q8—44no—
85C2×C4×D4—28no—
86C2×C4×D4—24no—
87C2×C4×D4—28no—
88C2×C4×D4—24no—
89C2×C4×D4—28no—
90C2×C4×D4D418yes—
91C2×#117—28no—
92C2×#117—28no—
93C2×#117—44no—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2311yescharacteristic
113C22×C22⋊C4C2311yescharacteristic
114C22×C42—24no—
115C22×C42—24no—
116C22×C42—24no—
117C22×C42—24no—
118C22×C42—24no—
119C22×C42D418yes—
120C22×C42C2312yes—
121C22×C42C2311yescharacteristic
122C22×C42C2312yes—
123C22×C42C2311yescharacteristic
124C22×C42C2314yes—
125C22×C42C2312yes—
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2312yes—
129C22×C42C2311yescharacteristic
130#397—216no—
131#394—216no—
132#391—28no—
133#391—28no—
134#391—28no—
135#391—44no—
136#389—216no—
137#389—216no—
138#389—216no—
139#389—28no—
140#389—216no—
141#388—216no—
142#384—24no—
143#384—24no—
144#384D418yescomplemented
145#382—28no—
146#380—216no—
147#380—28no—
148#380—216no—
149#380—216no—
150#380—216no—
151C4×C4⋊C4—216no—
152C4×C4⋊C4—28no—
153C4×C4⋊C4—24no—
154C4×C22⋊C4—28no—
155C4×C22⋊C4—24no—
156C4×C22⋊C4—28no—
157C4×C22⋊C4—24no—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2318yes—
163C2×#95C2318yes—
164C2×#95C2312yes—
165C2×#95C2318yes—
166C2×#95C2318yes—
167C2×#95C2318yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2318yes—
171C43—24no—
172C25C2411yescharacteristic
173C25C2411yescharacteristic
174C22×Q8—24no—
175C22×Q8—24no—
176C22×Q8—24no—
177C22×Q8—24no—
178C22×Q8—24no—
179C22×Q8—24no—
180C22×Q8—24no—
181C22×D4—22no—
182C22×D4—24no—
183C22×D4—22no—
184C22×D4—24no—
185C22×D4C2×D414yes—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—24no—
192C23×C4—22no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—22no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—22no—
201C23×C4—24no—
202C23×C4—22no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—22no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4C2×D418yes—
209C23×C4C2×D414yes—
210C23×C4C2×D414yes—
211C23×C4C2×D414yes—
212C23×C4#4118yes—
213C23×C4#4114yes—
214C23×C4C2414yes—
215C23×C4C2414yes—
216C23×C4C2414yes—
217C23×C4C2414yes—
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2412yes—
227C4⋊Q8—416no—
228C22⋊Q8—432no—
229C22⋊Q8—432no—
230C4×D4—264no—
231C4×D4—232no—
232#117—432no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—216no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4C2×D418yes—
266C2×C4⋊C4C2×D414yes—
267C2×C4⋊C4C2×D4116yes—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—216no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4C2×D414yes—
290C2×C22⋊C4C2×D414yes—
291C2×C42—22no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—22no—
299C2×C42—232no—
300C2×C42—28no—
301C2×C42—28no—
302C2×C42—24no—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—24no—
310C2×C42—22no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—22no—
314C2×C42—44no—
315C2×C42—416no—
316C2×C42C2×D414yes—
317C2×C42C2×D418yes—
318C2×C42C2×D414yes—
319C2×C42C2×D414yes—
320C2×C42C2×D414yes—
321C2×C42#4118yes—
322C2×C42#4114yes—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—216no—
327#95—216no—
328#95—216no—
329#95—28no—
330#95—28no—
331#95—28no—
332#95—28no—
333#95—216no—
334#95—216no—
335#95—48no—
336C24—21no—
337C24—21no—
338C24—21no—
339C24—21no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—21no—
344C24—21no—
345C24—21no—
346C24—21no—
347C24—22no—
348C24—22no—
349C24—21no—
350C24—21no—
351C24—22no—
352C24—22no—
353C24C22×D412yes—
354C24C22×D412yes—
355C24C22×D412yes—
356C24C2×#4112yes—
357C24C2×#4114yes—
358C24C2511yescharacteristic
359C2×Q8—48no—
360C2×Q8—48no—
361C2×Q8—48no—
362C2×Q8—48no—
363C2×Q8—48no—
364C2×Q8—48no—
365C2×Q8—48no—
366C2×D4—216no—
367C2×D4—216no—
368C2×D4—28no—
369C2×D4—28no—
370C2×D4—28no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—28no—
379C22×C4—28no—
380C22×C4—28no—
381C22×C4—28no—
382C22×C4—28no—
383C22×C4—28no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—21no—
392C22×C4—28no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—28no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—216no—
417C22×C4—28no—
418C22×C4—216no—
419C22×C4—216no—
420C22×C4—216no—
421C22×C4—28no—
422C22×C4—28no—
423C22×C4—216no—
424C22×C4—216no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—28no—
433C22×C4—28no—
434C22×C4—28no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—28no—
441C22×C4—24no—
442C22×C4—48no—
443C22×C4—44no—
444C22×C4—48no—
445C22×C4—44no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—44no—
450C22×C4—48no—
451C22×C4—44no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4#121116yes—
455C22×C4C22×D412yes—
456C22×C4C22×D414yes—
457C22×C4C22×D418yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D414yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D414yes—
467C22×C4C2×#4114yes—
468C22×C4C2×#4114yes—
469C22×C4C2×#4112yes—
470C22×C4C2×#4114yes—
471C22×C4C2×#4112yes—
472C22×C4C2×#4112yes—
473C22×C4C2×#4114yes—
474C22×C4C2×#4114yes—
475C22×C4C2×#4114yes—
476C22×C4H(5,𝔽2)12yes—
477C22×C42−1+414yes—
478C22×C42−1+414yes—
479C22×C42−1+412yes—
480C4⋊C4—216no—
481C4⋊C4—28no—
482C4⋊C4—216no—
483C4⋊C4—232no—
484C4⋊C4—416no—
485C4⋊C4—48no—
486C4⋊C4—416no—
487C4⋊C4—416no—
488C4⋊C4—416no—
489C4⋊C4—416no—
490C4⋊C4—48no—
491C4⋊C4—416no—
492C22⋊C4—216no—
493C22⋊C4—216no—
494C22⋊C4—28no—
495C22⋊C4—28no—
496C22⋊C4—416no—
497C22⋊C4—416no—
498C22⋊C4—416no—
499C22⋊C4—416no—
500C42—216no—
501C42—28no—
502C42—28no—
503C42—216no—
504C42—416no—
505C42—48no—
506C42—44no—
507C42—48no—
508C42—44no—
509C42#121116yes—
510C23—22no—
511C23—22no—
512C23—24no—
513C23—24no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—24no—
526C23—22no—
527C23—24no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—41no—
537C23—42no—
538C23—41no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—41no—
543C23—42no—
544C23—41no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23C2×#12114yes—
555C23#53214yes—
556C23C23×D411yescharacteristic
557C23C23×D411yescharacteristic
558C23C23×D411yescharacteristic
559C23C22×#4111yescharacteristic
560C23C22×#4111yescharacteristic
561C23C22×#4112yes—
562C23C22×#4112yes—
563C23C2×H(5,𝔽2)11yescharacteristic
564C23C2×2−1+412yes—
565C23C2×2−1+412yes—
566C23C2×2−1+411yescharacteristic
567Q8—816no—
568D4—232no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—232no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—28no—
588C2×C4—22no—
589C2×C4—216no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—48no—
617C2×C4—42no—
618C2×C4—432no—
619C2×C4—432no—
620C2×C4—416no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—48no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—48no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—48no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—48no—
638C2×C4C2×#12114yes—
639C2×C4C2×#12118yes—
640C2×C4C2×#12114yes—
641C2×C4C2×#12114yes—
642C2×C4#52518yes—
643C2×C4#53214yes—
644C2×C4#53618yes—
645C2×C4#53714yes—
646C2×C4D4214yes—
647C2×C4#54718yes—
648C2×C4#54714yes—
649C2×C4#54818yes—
650C2×C4#54814yes—
651C2×C4#55218yes—
652C2×C4#55314yes—
653C2×C4#55618yes—
654C22—24no—
655C22—24no—
656C22—24no—
657C22—24no—
658C22—24no—
659C22—24no—
660C22—42no—
661C22—42no—
662C22—42no—
663C22—44no—
664C22—44no—
665C22—42no—
666C22—42no—
667C22—42no—
668C22—42no—
669C22—44no—
670C22—44no—
671C22—44no—
672C22—44no—
673C22—44no—
674C22—44no—
675C22—42no—
676C22C22×#12111yescharacteristic
677C22C22×#12111yescharacteristic
678C22C2×#52512yes—
679C22C2×#53211yescharacteristic
680C22C2×#53211yescharacteristic
681C22#344412yes—
682C22C2×#53612yes—
683C22C2×#53711yescharacteristic
684C22C2×D4211yescharacteristic
685C22C2×#54712yes—
686C22C2×#54711yescharacteristic
687C22C2×#54812yes—
688C22C2×#54811yescharacteristic
689C22C2×#55212yes—
690C22C2×#55311yescharacteristic
691C22C2×#55612yes—
692C22#349911yescharacteristic
693C22#349911yescharacteristic
694C22#350912yes—
695C22#350912yes—
696C22#350912yes—
697C22#351112yes—
698C22#351112yes—
699C4—24no—
700C4—24no—
701C4—48no—
702C4—42no—
703C4—44no—
704C4—44no—
705C4—44no—
706C4—42no—
707C4—84no—
708C4—84no—
709C4—84no—
710C4#2415116yescomplemented
711C4#241618yes—
712C2—44no—
713C2—44no—
714C2C2×#241611yescharacteristic
715C2C2×#241512yes—
716C2#3529712yes—
717C2#3531611yescharacteristic
718C2#3612211yescharacteristic
719C2#3612711yescharacteristic
720C2#3615112yes—
721C2#3618111yescharacteristic
722C2#3618811yescharacteristic
723C2#3621012yes—
724C2#3662411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2857C2semidirect
2. C2×#2857C2semidirect
4. C2×#2285C2semidirect
6. #15237C2semidirect
7. #15236C2semidirect
8. #15132C2semidirect
10. #14714C2semidirect
11. #14712C2semidirect
12. #14686C2semidirect
13. #14138C2semidirect
14. #14046C2semidirect
15. #13974C2semidirect
144. #384D4semidirect
710. C4#2415semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
C2×#24152
C2×#22852
C2×C42×D42
#152372
#152362
#151322
#151292
#147142
#147122
#146862
#141382
#140462
#139742