Finite Group Databasebeta

← #988797

Subgroups — #988797

Order 512.

4070 conjugacy classes of subgroups.

The subgroup lattice has 728 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2434C212yesmaximal, complemented
2C2×#2434C212yesmaximal, complemented
3C2×#2409C212yesmaximal
4#17085C212yesmaximal, complemented
5#17072C212yesmaximal, complemented
6#16318C214yesmaximal, complemented
7#15961C212yesmaximal
8#15622C214yesmaximal, complemented
9#15188C212yesmaximal, complemented
10#15180C211yesmaximal, characteristic, complemented
11#15179C212yesmaximal, complemented
12#14535C212yesmaximal
13#14155C211yesmaximal, characteristic, complemented
14#13997C212yesmaximal, complemented
15C4×#394C211yesmaximal, characteristic
16C24×Q8C2212yes—
17C22×C4⋊Q8C2211yescharacteristic
18C22×C4⋊Q8C2212yes—
19C22×C4⋊Q8C2211yescharacteristic
20C22×C22⋊Q8C2212yes—
21C22×C22⋊Q8C2212yes—
22C22×C4×D4C2211yescharacteristic
23#2434—216no—
24#2434—216no—
25#2409—216no—
26C2×#395C2214yes—
27C2×#395C2214yes—
28C2×#395C2214yes—
29C2×#395C2214yes—
30C2×#395C2214yes—
31C2×#394C2211yescharacteristic
32C2×#393C2214yes—
33C2×#391C2212yes—
34C2×#391C2212yes—
35C2×#391C2212yes—
36C2×#391C2212yes—
37C2×#391C2212yes—
38C2×#391C2212yes—
39C2×#389C2212yes—
40C2×#386C2214yes—
41C2×#385C2214yes—
42C2×#385C2214yes—
43C2×#385C2214yes—
44C2×#385C2214yes—
45C2×#385C2214yes—
46C2×#380C2212yes—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4C2212yes—
50C2×C4×C22⋊C4C2212yes—
51#1445C2212yes—
52#1445C2212yes—
53#1445C2214yes—
54#1442C2214yes—
55#1442C2214yes—
56#1442C2211yescharacteristic
57#1442C2212yes—
58#1442C2212yes—
59#1442C2214yes—
60#1441C2212yes—
61#1441C2214yes—
62#1441C2211yescharacteristic
63#1441C2214yes—
64#1441C2212yes—
65#1441C2214yes—
66#1440C2212yes—
67#1440C2214yes—
68#1439C2212yes—
69#1439C2214yes—
70#1439C2214yes—
71#1438C2211yescharacteristic
72#1437C2214yes—
73C4⋊#95C2212yes—
74C4⋊#95C2214yes—
75C4⋊#95C2212yes—
76C4⋊#95C2211yescharacteristic
77C4×#95C2211yescharacteristic
78C23×Q8—28no—
79C23×Q8—28no—
80C23×Q8—28no—
81C23×Q8—44no—
82C23×Q8—44no—
83C23×Q8—44no—
84C23×Q8—44no—
85C23×Q8C2312yes—
86C23×Q8C2312yes—
87C23×Q8C2314yes—
88C23×D4C2311yescharacteristic
89C24×C4C2312yes—
90C24×C4C2314yes—
91C2×C4⋊Q8—28no—
92C2×C4⋊Q8—28no—
93C2×C4⋊Q8—28no—
94C2×C4⋊Q8—42no—
95C2×C4⋊Q8—42no—
96C2×C4⋊Q8—42no—
97C2×C4⋊Q8—44no—
98C2×C4⋊Q8D418yescomplemented
99C2×C22⋊Q8—28no—
100C2×C22⋊Q8—28no—
101C2×C22⋊Q8—44no—
102C2×C22⋊Q8—44no—
103C2×C22⋊Q8—44no—
104C2×C22⋊Q8D4116yes—
105C2×C4×D4—28no—
106C2×C4×D4—42no—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2311yescharacteristic
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2314yes—
129C22×C42C2311yescharacteristic
130C22×C42C2312yes—
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2312yes—
134#395—216no—
135#395—216no—
136#395—216no—
137#395—216no—
138#395—216no—
139#394Q818yescomplemented
140#393—216no—
141#391—28no—
142#391—28no—
143#391—28no—
144#391—28no—
145#391—28no—
146#391—28no—
147#389—44no—
148#386—216no—
149#385—216no—
150#385—216no—
151#385—216no—
152#385—216no—
153#385—216no—
154#380—28no—
155C4×C4⋊C4—42no—
156C4×C4⋊C4—42no—
157C4×C22⋊C4—28no—
158C4×C22⋊C4—28no—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2311yescharacteristic
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2311yescharacteristic
182C2×#95C2312yes—
183C2×#95C2311yescharacteristic
184C2×#95C2311yescharacteristic
185C25C2412yes—
186C22×Q8—24no—
187C22×Q8—24no—
188C22×Q8—24no—
189C22×Q8—24no—
190C22×Q8—28no—
191C22×Q8—28no—
192C22×Q8—28no—
193C22×Q8—24no—
194C22×Q8—24no—
195C22×Q8—24no—
196C22×Q8—24no—
197C22×Q8—24no—
198C22×Q8—24no—
199C22×Q8—24no—
200C22×Q8—24no—
201C22×Q8—44no—
202C22×Q8—44no—
203C22×Q8—44no—
204C22×Q8—44no—
205C22×Q8—432no—
206C22×Q8—432no—
207C22×Q8—432no—
208C22×Q8—432no—
209C22×Q8—432no—
210C22×Q8—432no—
211C22×Q8—432no—
212C22×Q8C2×D418yes—
213C22×Q8C2×D418yes—
214C22×D4—24no—
215C22×D4—24no—
216C22×D4—41no—
217C22×D4—41no—
218C22×D4C2×Q814yes—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—28no—
226C23×C4—28no—
227C23×C4—28no—
228C23×C4—44no—
229C23×C4—44no—
230C23×C4—44no—
231C23×C4—44no—
232C23×C4C2×D418yes—
233C23×C4C2414yes—
234C23×C4C2411yescharacteristic
235C23×C4C2414yes—
236C23×C4C2411yescharacteristic
237C23×C4C2412yes—
238C23×C4C2414yes—
239C23×C4C2414yes—
240C23×C4C2411yescharacteristic
241C23×C4C2412yes—
242C23×C4C2411yescharacteristic
243C23×C4C2412yes—
244C23×C4C2411yescharacteristic
245C23×C4C2411yescharacteristic
246C23×C4C2411yescharacteristic
247C4⋊Q8—416no—
248C4⋊Q8—416no—
249C4⋊Q8—432no—
250C22⋊Q8—432no—
251C22⋊Q8—432no—
252C4×D4—416no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—41no—
271C2×C4⋊C4—41no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—41no—
284C2×C4⋊C4—41no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4C2×D418yes—
288C2×C4⋊C4C2×D4116yes—
289C2×C4⋊C4C2×D418yes—
290C2×C4⋊C4C2×D418yes—
291C2×C4⋊C4C2×Q814yescomplemented
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—28no—
307C2×C22⋊C4—28no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—42no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4C2×D4116yes—
314C2×C22⋊C4C2×Q818yescomplemented
315C2×C22⋊C4C2×Q818yescomplemented
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—42no—
327C2×C42—42no—
328C2×C42C2×D414yes—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—28no—
334#95—28no—
335#95—28no—
336#95—28no—
337#95—28no—
338#95—28no—
339#95—28no—
340#95—28no—
341#95—28no—
342#95—28no—
343#95—28no—
344#95—28no—
345#95—28no—
346#95—24no—
347#95—24no—
348#95—42no—
349#95—41no—
350#95—41no—
351#95—41no—
352#95—42no—
353#95—44no—
354#95C2×Q814yescomplemented
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24C22×D414yes—
368C24C22×Q814yes—
369C242−1+414yes—
370C24C2511yescharacteristic
371C2×Q8—216no—
372C2×Q8—216no—
373C2×Q8—216no—
374C2×Q8—216no—
375C2×Q8—216no—
376C2×Q8—216no—
377C2×Q8—416no—
378C2×Q8—416no—
379C2×Q8—416no—
380C2×Q8—416no—
381C2×Q8—416no—
382C2×Q8—416no—
383C2×Q8—4256no—
384C2×Q8—416no—
385C2×Q8—48no—
386C2×Q8—48no—
387C2×Q8—48no—
388C2×Q8—48no—
389C2×Q8—48no—
390C2×Q8—48no—
391C2×Q8—48no—
392C2×Q8—48no—
393C2×D4—216no—
394C2×D4—44no—
395C2×D4—44no—
396C2×D4—48no—
397C2×D4—44no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
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—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—24no—
454C22×C4—216no—
455C22×C4—216no—
456C22×C4—216no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—21no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—21no—
466C22×C4—24no—
467C22×C4—21no—
468C22×C4—24no—
469C22×C4—416no—
470C22×C4—416no—
471C22×C4—416no—
472C22×C4—416no—
473C22×C4—416no—
474C22×C4—416no—
475C22×C4—416no—
476C22×C4—48no—
477C22×C4—48no—
478C22×C4—48no—
479C22×C4—48no—
480C22×C4C22×D418yes—
481C22×C4C22×D412yes—
482C22×C4C22×D418yes—
483C22×C4C22×D414yes—
484C22×C4C22×D418yes—
485C22×C4C22×D412yes—
486C22×C4C22×D414yes—
487C22×C4C22×D412yes—
488C22×C4C22×Q814yes—
489C22×C4C22×Q812yes—
490C22×C4C22×Q812yes—
491C22×C4C22×Q812yes—
492C22×C4H(5,𝔽2)12yes—
493C22×C4H(5,𝔽2)12yes—
494C22×C4H(5,𝔽2)14yes—
495C22×C42−1+412yes—
496C22×C42−1+414yes—
497C22×C42−1+412yes—
498C4⋊C4—216no—
499C4⋊C4—416no—
500C4⋊C4—48no—
501C4⋊C4—48no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—416no—
506C4⋊C4—48no—
507C4⋊C4—48no—
508C4⋊C4—416no—
509C4⋊C4—416no—
510C4⋊C4—48no—
511C4⋊C4—88no—
512C22⋊C4—216no—
513C22⋊C4—216no—
514C22⋊C4—216no—
515C22⋊C4—216no—
516C22⋊C4—416no—
517C22⋊C4—416no—
518C22⋊C4—48no—
519C22⋊C4—416no—
520C42—28no—
521C42—44no—
522C42—44no—
523C42—48no—
524C42—84no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23D4×Q818yes—
560C23C23×D411yescharacteristic
561C23C23×D412yes—
562C23C23×Q811yescharacteristic
563C23C2×H(5,𝔽2)11yescharacteristic
564C23C2×H(5,𝔽2)11yescharacteristic
565C23C2×H(5,𝔽2)12yes—
566C23C2×2−1+412yes—
567C23C2×2−1+411yescharacteristic
568C23C2×2−1+411yescharacteristic
569C23C2×2−1+412yes—
570C23#58511yescharacteristic
571Q8—464no—
572Q8—432no—
573Q8—432no—
574D4—416no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—28no—
583C2×C4—28no—
584C2×C4—24no—
585C2×C4—28no—
586C2×C4—24no—
587C2×C4—28no—
588C2×C4—24no—
589C2×C4—28no—
590C2×C4—28no—
591C2×C4—28no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—28no—
601C2×C4—28no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—24no—
605C2×C4—22no—
606C2×C4—24no—
607C2×C4—464no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—432no—
645C2×C4—41no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—42no—
653C2×C4—41no—
654C2×C4—41no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4#53414yes—
658C2×C4#53418yes—
659C2×C4D4214yes—
660C2×C4D4218yes—
661C2×C4D4×Q814yes—
662C2×C4H(3,𝔽4)14yes—
663C22—28no—
664C22—28no—
665C22—28no—
666C22—44no—
667C22—44no—
668C22—44no—
669C22—44no—
670C22—44no—
671C22—44no—
672C22—44no—
673C22—44no—
674C22—44no—
675C22—44no—
676C22—44no—
677C22—44no—
678C22C2×#53411yescharacteristic
679C22C2×#53412yes—
680C22C2×D4211yescharacteristic
681C22C2×D4212yes—
682C22C2×D4×Q812yes—
683C22C2×D4×Q811yescharacteristic
684C22#348612yes—
685C22#348612yes—
686C22#348912yes—
687C22#349711yescharacteristic
688C22#349912yes—
689C22#350111yescharacteristic
690C22#350311yescharacteristic
691C22#350312yes—
692C22C2×H(3,𝔽4)11yescharacteristic
693C22#353811yescharacteristic
694C22#353811yescharacteristic
695C22#355011yescharacteristic
696C22#355212yes—
697C22#355212yes—
698C22#355212yes—
699C22#356111yescharacteristic
700C22#356112yes—
701C4—24no—
702C4—42no—
703C4—42no—
704C4—42no—
705C4—42no—
706C4—48no—
707C4—44no—
708C4—48no—
709C4—48no—
710C4—44no—
711C4—44no—
712C4—81no—
713C4—84no—
714C4#267918yes—
715C2—48no—
716C2#3959312yes—
717C2#3961511yescharacteristic
718C2#4280711yescharacteristic
719C2#4291212yes—
720C2#4524812yes—
721C2#4532111yescharacteristic
722C2#4600411yescharacteristic
723C2C2×#267911yescharacteristic
724C2#5303211yescharacteristic
725C2#5899811yescharacteristic
726C2#5912712yes—
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×#2434C2semidirect
2. C2×#2434C2semidirect
4. #17085C2semidirect
5. #17072C2semidirect
6. #16318C2semidirect
8. #15622C2semidirect
9. #15188C2semidirect
10. #15180C2semidirect
11. #15179C2semidirect
13. #14155C2semidirect
14. #13997C2semidirect
98. C2×C4⋊Q8D4semidirect
139. #394Q8semidirect
291. C2×C4⋊C4C2×Q8semidirect
314. C2×C22⋊C4C2×Q8semidirect
315. C2×C22⋊C4C2×Q8semidirect
354. #95C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#24342
C2×#24092
#170852
#170722
#163182
#159612
#156222
#151882
#151802
#151792
#145352
#141552
#139972
C4×#3942