Finite Group Databasebeta

← #760003

Subgroups — #760003

Order 512.

2322 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2567C211yesmaximal, characteristic
2C2×C4×C4⋊Q8C211yesmaximal, characteristic
3#16611C214yesmaximal
4#15225C212yesmaximal
5#15186C212yesmaximal
6#15146C212yesmaximal
7#15139C212yesmaximal
8#14876C214yesmaximal
9#14161C211yesmaximal, characteristic
10#14133C212yesmaximal
11#14030C212yesmaximal
12#14021C212yesmaximal
13#13926C211yesmaximal, characteristic
14#13911C211yesmaximal, characteristic
15#13884C212yesmaximal
16C4×#391C211yesmaximal, characteristic
17C4×#389C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×C4×Q8C2212yes—
21#2567—28no—
22C2×#400C2214yes—
23C2×#395C2214yes—
24C4×C4⋊Q8—28no—
25C2×#391C2211yescharacteristic
26C2×#391C2212yes—
27C2×#391C2212yes—
28C2×#391C2211yescharacteristic
29C2×#391C2212yes—
30C2×#389C2212yes—
31C2×#389C2212yes—
32C2×#389C2212yes—
33C2×#389C2211yescharacteristic
34C2×#389C2212yes—
35C2×#389C2211yescharacteristic
36C2×#387C2214yes—
37C2×#384C2212yes—
38C2×#384C2211yescharacteristic
39C2×#382C2212yes—
40C2×C4×C4⋊C4C2212yes—
41C2×C4×C4⋊C4C2212yes—
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C43C2211yescharacteristic
47#1446C2214yes—
48#1445C2214yes—
49#1443C2214yes—
50#1443C2214yes—
51#1443C2214yes—
52#1442C2214yes—
53#1442C2214yes—
54#1442C2214yes—
55#1442C2214yes—
56#1442C2214yes—
57#1442C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1436C2212yes—
62#1436C2212yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2212yes—
66#1436C2212yes—
67C4⋊#95C2211yescharacteristic
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2211yescharacteristic
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2211yescharacteristic
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2211yescharacteristic
81#1434C2214yes—
82C4×#95C2212yes—
83C4×#95C2212yes—
84C4×#95C2212yes—
85C23×Q8C2311yescharacteristic
86C23×Q8C2312yes—
87C2×C4⋊Q8—24no—
88C2×C4⋊Q8—24no—
89C2×C4⋊Q8D418yes—
90C2×C4×Q8—28no—
91C2×C4×Q8—24no—
92C2×C4×Q8—24no—
93C2×C4×Q8—24no—
94C2×C4×Q8—44no—
95C2×C4×Q8—44no—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2312yes—
114C22×C42C2311yescharacteristic
115C22×C42C2312yes—
116C22×C42C2312yes—
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119C22×C42C2312yes—
120C22×C42C2312yes—
121C22×C42C2314yes—
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126#400—48no—
127#395—216no—
128#391—28no—
129#391—24no—
130#391—28no—
131#391—28no—
132#391—24no—
133#389—28no—
134#389—28no—
135#389—28no—
136#389—28no—
137#389—24no—
138#389Q818yescomplemented
139#387—48no—
140#384—28no—
141#384—24no—
142#382—28no—
143C4×C4⋊C4—28no—
144C4×C4⋊C4—28no—
145C4×C4⋊C4—24no—
146C4×C4⋊C4—28no—
147C4×C4⋊C4—24no—
148C4×C4⋊C4—24no—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2312yes—
182C43—24no—
183C22×Q8—22no—
184C22×Q8—22no—
185C22×Q8—22no—
186C22×Q8—24no—
187C22×Q8—24no—
188C22×Q8—22no—
189C22×Q8—22no—
190C22×Q8—22no—
191C22×Q8—42no—
192C22×Q8—42no—
193C22×Q8—42no—
194C22×Q8—42no—
195C22×Q8C2×D418yes—
196C22×Q8#4114yes—
197C23×C4C2412yes—
198C23×C4C2414yes—
199C23×C4C2412yes—
200C23×C4C2412yes—
201C23×C4C2412yes—
202C23×C4C2414yes—
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2411yescharacteristic
206C23×C4C2412yes—
207C23×C4C2411yescharacteristic
208C23×C4C2412yes—
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212C23×C4C2411yescharacteristic
213C23×C4C2411yescharacteristic
214C4⋊Q8—232no—
215C4×Q8—432no—
216C4×Q8—416no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4C2×D418yes—
266C2×C4⋊C4C2×D418yes—
267C2×C4⋊C4C2×Q814yes—
268C2×C4⋊C4C2×Q814yes—
269C2×C4⋊C4C2×Q818yes—
270C2×C4⋊C4#4114yes—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—28no—
279C2×C42—24no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—22no—
292C2×C42—24no—
293C2×C42—22no—
294C2×C42—28no—
295C2×C42—24no—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—24no—
299C2×C42—44no—
300C2×C42—41no—
301C2×C42—41no—
302C2×C42C2×D414yes—
303C2×C42C2×Q814yes—
304C2×C42#4114yes—
305C2×C42#4114yes—
306C2×C42#4114yes—
307#95—28no—
308#95—28no—
309#95—24no—
310#95—28no—
311#95—28no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—28no—
317#95—28no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—28no—
322#95—24no—
323#95—24no—
324#95—28no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—42no—
329#95—42no—
330#95—42no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95C2×Q818yes—
340C24C2511yescharacteristic
341C2×Q8—216no—
342C2×Q8—28no—
343C2×Q8—28no—
344C2×Q8—28no—
345C2×Q8—48no—
346C2×Q8—48no—
347C2×Q8—48no—
348C2×Q8—48no—
349C2×Q8—48no—
350C2×Q8—48no—
351C2×Q8—44no—
352C2×Q8—44no—
353C2×Q8—44no—
354C2×Q8—44no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4C22×D418yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×Q812yes—
451C22×C4C22×Q814yes—
452C22×C4C22×Q814yes—
453C22×C4C22×Q812yes—
454C22×C4C22×Q812yes—
455C22×C4C2×#4114yes—
456C22×C4C2×#4114yes—
457C22×C4C2×#4112yes—
458C22×C4C2×#4112yes—
459C22×C4C2×#4112yes—
460C22×C4C2×#4112yes—
461C22×C4C2×#4112yes—
462C22×C4C2×#4112yes—
463C22×C4C2×#4112yes—
464C22×C4C2×#4112yes—
465C22×C4C2×#4112yes—
466C22×C4C2×#4112yes—
467C22×C4C2×#4112yes—
468C22×C4H(5,𝔽2)12yes—
469C22×C4H(5,𝔽2)12yes—
470C22×C42−1+412yes—
471C22×C42−1+412yes—
472C22×C42−1+412yes—
473C22×C42−1+412yes—
474C4⋊C4—216no—
475C4⋊C4—216no—
476C4⋊C4—28no—
477C4⋊C4—28no—
478C4⋊C4—28no—
479C4⋊C4—216no—
480C4⋊C4—48no—
481C4⋊C4—416no—
482C4⋊C4—48no—
483C4⋊C4—48no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—48no—
487C4⋊C4—48no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—88no—
491C4⋊C4—88no—
492C42—28no—
493C42—28no—
494C42—28no—
495C42—416no—
496C42—44no—
497C42—48no—
498C42—48no—
499C42—48no—
500C42—44no—
501C42—44no—
502C42—48no—
503C42#123116yes—
504C23C23×D411yescharacteristic
505C23C23×Q811yescharacteristic
506C23C22×#4111yescharacteristic
507C23C22×#4111yescharacteristic
508C23C22×#4111yescharacteristic
509C23C22×#4111yescharacteristic
510C23C22×#4111yescharacteristic
511C23C2×H(5,𝔽2)11yescharacteristic
512C23C2×H(5,𝔽2)11yescharacteristic
513C23C2×2−1+411yescharacteristic
514C23C2×2−1+411yescharacteristic
515C23C2×2−1+411yescharacteristic
516C23C2×2−1+411yescharacteristic
517C23#58511yescharacteristic
518C23#58511yescharacteristic
519Q8—432no—
520Q8—416no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—28no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—28no—
550C2×C4—28no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4C2×#12314yes—
625C2×C4C2×#12314yes—
626C2×C4C2×#12314yes—
627C2×C4#52914yes—
628C2×C4#53314yes—
629C2×C4#54114yes—
630C2×C4#54114yes—
631C2×C4#54814yes—
632C2×C4#54814yes—
633C2×C4#54814yes—
634C2×C4D4×Q814yes—
635C2×C4#55214yes—
636C2×C4#55414yes—
637C2×C4#55614yes—
638C2×C4#55614yes—
639C2×C4#55714yes—
640C22C22×#12311yescharacteristic
641C22C2×#52911yescharacteristic
642C22C2×#53311yescharacteristic
643C22C2×#54111yescharacteristic
644C22C2×#54111yescharacteristic
645C22#346011yescharacteristic
646C22#346011yescharacteristic
647C22C2×#54811yescharacteristic
648C22C2×#54811yescharacteristic
649C22C2×#54811yescharacteristic
650C22C2×D4×Q811yescharacteristic
651C22C2×#55211yescharacteristic
652C22C2×#55411yescharacteristic
653C22C2×#55611yescharacteristic
654C22C2×#55611yescharacteristic
655C22#347511yescharacteristic
656C22C2×#55711yescharacteristic
657C22Q8×#4111yescharacteristic
658C22#348311yescharacteristic
659C22#348411yescharacteristic
660C22#348511yescharacteristic
661C22#350211yescharacteristic
662C22#350311yescharacteristic
663C22#350611yescharacteristic
664C22#350611yescharacteristic
665C22#350711yescharacteristic
666C22#351111yescharacteristic
667C22#351511yescharacteristic
668C22#351811yescharacteristic
669C22#352211yescharacteristic
670C22#352211yescharacteristic
671C22#354411yescharacteristic
672C22#355011yescharacteristic
673C22#355511yescharacteristic
674C22#355511yescharacteristic
675C4—24no—
676C4—24no—
677C4—44no—
678C4—48no—
679C4—44no—
680C4—44no—
681C4—42no—
682C4—44no—
683C4—42no—
684C4—42no—
685C4—82no—
686C4—82no—
687C4—82no—
688C4—82no—
689C4—84no—
690C4#242118yes—
691C4#244618yes—
692C2C2×#244611yescharacteristic
693C2C2×#242111yescharacteristic
694C2#3494111yescharacteristic
695C2#3515211yescharacteristic
696C2#3518511yescharacteristic
697C2#3520811yescharacteristic
698C2#3763811yescharacteristic
699C2#5907911yescharacteristic
700C2#5908111yescharacteristic
701C2#5908411yescharacteristic
702C2#5913611yescharacteristic
703C2#5942611yescharacteristic
704C2#5943511yescharacteristic
705C2#5944511yescharacteristic
706C2#5987011yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, prime radical, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
138. #389Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25672
C2×C4×C4⋊Q82
#166112
#152252
#151862
#151462
#151392
#148762
#141612
#141332
#140302
#140212
#139262
#139112
#138842
C4×#3912
C4×#3892