Finite Group Databasebeta

← #612816

Subgroups — #612816

Order 512.

3730 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic
2C2×#2857C212yesmaximal
3C2×#2546C211yesmaximal, characteristic, complemented
4C2×C4×C22⋊Q8C211yesmaximal, characteristic
5C22×C4×C4⋊C4C211yesmaximal, characteristic
6#15184C212yesmaximal, complemented
7#15180C212yesmaximal, complemented
8#14818C214yesmaximal, complemented
9#14766C214yesmaximal
10#14702C212yesmaximal
11#14155C211yesmaximal, characteristic, complemented
12#14046C214yesmaximal, complemented
13#13926C211yesmaximal, characteristic, complemented
14#13910C211yesmaximal, characteristic, complemented
15#13907C214yesmaximal
16C22×C4⋊Q8C2211yescharacteristic
17C22×C4⋊Q8C2212yes—
18C22×C22⋊Q8C2212yes—
19C22×C22⋊Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C23×C4⋊C4C2212yes—
22C23×C42C2212yes—
23C23×C42C2211yescharacteristic
24#2857—28no—
25#2857—216no—
26#2546—28no—
27C2×#398C2214yes—
28C4×C22⋊Q8—28no—
29C2×#391C2212yes—
30C2×#391C2212yes—
31C2×#391C2212yes—
32C2×#391C2211yescharacteristic
33C2×#391C2212yes—
34C2×#391C2212yes—
35C2×#389C2212yes—
36C2×#389C2218yes—
37C2×#389C2212yes—
38C2×#389C2211yescharacteristic
39C2×#389C2214yes—
40C2×#389C2214yes—
41C2×#387C2212yes—
42C2×#385C2212yes—
43C2×#384C2212yes—
44C2×#384C2211yescharacteristic
45C2×#380C2212yes—
46C2×#380C2218yes—
47C2×#380C2212yes—
48C2×#380C2211yescharacteristic
49C2×C4×C4⋊C4—28no—
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2212yes—
53C2×C4×C4⋊C4C2212yes—
54C2×C4×C4⋊C4C2212yes—
55C2×C4×C4⋊C4C2212yes—
56C2×C4×C4⋊C4C2211yescharacteristic
57C2×C4×C22⋊C4C2212yes—
58C22×#95C2212yes—
59#1445C2214yes—
60#1444C2214yes—
61#1442C2218yes—
62#1441C2218yes—
63#1441C2218yes—
64#1441C2218yes—
65#1441C2214yes—
66#1436C2218yes—
67C4⋊#95C2214yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2214yes—
71C22⋊#95C2218yes—
72C23×Q8C2312yes—
73C23×Q8C2311yescharacteristic
74C24×C4C2311yescharacteristic
75C24×C4C2314yes—
76C24×C4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C2×C4⋊Q8—28no—
79C2×C4⋊Q8—24no—
80C2×C4⋊Q8—28no—
81C2×C4⋊Q8—24no—
82C2×C4⋊Q8—24no—
83C2×C4⋊Q8—44no—
84C2×C22⋊Q8—24no—
85C2×C22⋊Q8—24no—
86C2×C22⋊Q8—28no—
87C2×C22⋊Q8—28no—
88C2×C22⋊Q8—28no—
89C2×C22⋊Q8D418yes—
90C2×C4×Q8—24no—
91C2×C4×Q8—24no—
92C2×C4×Q8—24no—
93C22×C4⋊C4—28no—
94C22×C4⋊C4—28no—
95C22×C4⋊C4Q8116yescomplemented
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2318yes—
105C22×C4⋊C4C2318yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C42—24no—
117C22×C42—24no—
118C22×C42—28no—
119C22×C42—28no—
120C22×C42—44no—
121C22×C42D418yes—
122C22×C42C2312yes—
123C22×C42C2312yes—
124C22×C42C2312yes—
125C22×C42C2314yes—
126C22×C42C2312yes—
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131#398—216no—
132#391—28no—
133#391—28no—
134#391—28no—
135#391—28no—
136#391—24no—
137#391—44no—
138#389—232no—
139#389—216no—
140#389—216no—
141#389—28no—
142#389—28no—
143#389—24no—
144#387—28no—
145#385—28no—
146#384—28no—
147#384—24no—
148#380—28no—
149#380—28no—
150#380—24no—
151#380—232no—
152C4×C4⋊C4—28no—
153C4×C4⋊C4—28no—
154C4×C4⋊C4—24no—
155C4×C4⋊C4—28no—
156C4×C4⋊C4—24no—
157C4×C4⋊C4—28no—
158C4×C4⋊C4D418yes—
159C4×C22⋊C4—28no—
160C2×#95—28no—
161C2×#95C2318yes—
162C2×#95C2318yes—
163C2×#95C2318yes—
164C2×#95C2318yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2318yes—
175C2×#95C2318yes—
176C2×#95C2312yes—
177C25C2411yescharacteristic
178C22×Q8—22no—
179C22×Q8—22no—
180C22×Q8—24no—
181C22×Q8—24no—
182C22×Q8—24no—
183C22×Q8—22no—
184C22×Q8—22no—
185C22×Q8—24no—
186C22×Q8—24no—
187C22×Q8—22no—
188C22×Q8—24no—
189C22×Q8—22no—
190C22×Q8—24no—
191C22×Q8C2×D414yes—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—24no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—28no—
200C23×C4—24no—
201C23×C4—28no—
202C23×C4—28no—
203C23×C4—22no—
204C23×C4—28no—
205C23×C4—22no—
206C23×C4—44no—
207C23×C4—42no—
208C23×C4—44no—
209C23×C4C2×D414yes—
210C23×C4C2×D414yes—
211C23×C4C2×D414yes—
212C23×C4C2×D414yes—
213C23×C4C2×Q8116yes—
214C23×C4C2×Q818yes—
215C23×C4#4114yes—
216C23×C4C2414yes—
217C23×C4C2414yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2411yescharacteristic
223C23×C4C2414yes—
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2414yes—
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C4⋊Q8—432no—
231C4⋊Q8—416no—
232C22⋊Q8—232no—
233C22⋊Q8—432no—
234C4×Q8—416no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—264no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—216no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—216no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—216no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—216no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—48no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—48no—
277C2×C4⋊C4C2×D414yes—
278C2×C4⋊C4C2×D418yes—
279C2×C4⋊C4C2×D418yes—
280C2×C4⋊C4C2×Q818yes—
281C2×C4⋊C4C2×Q8116yes—
282C2×C4⋊C4C2×Q818yes—
283C2×C4⋊C4#4118yes—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4C2×D418yes—
292C2×C42—24no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—24no—
300C2×C42—232no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—28no—
305C2×C42—24no—
306C2×C42—28no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—28no—
310C2×C42—24no—
311C2×C42—432no—
312C2×C42—44no—
313C2×C42C2×D418yes—
314C2×C42C2×D414yes—
315C2×C42C2×D414yes—
316C2×C42C2×D414yes—
317C2×C42C2×D414yes—
318C2×C42C2×D414yes—
319C2×C42#4114yes—
320#95—28no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—216no—
328#95—216no—
329#95—28no—
330#95—28no—
331#95—216no—
332#95—216no—
333#95—48no—
334#95—48no—
335#95C2×D418yes—
336C24—21no—
337C24—21no—
338C24—22no—
339C24—21no—
340C24—21no—
341C24—22no—
342C24C22×D412yes—
343C24C22×D412yes—
344C24C22×Q814yes—
345C24C2×#4112yes—
346C242−1+414yes—
347C24C2511yescharacteristic
348C2×Q8—28no—
349C2×Q8—28no—
350C2×Q8—28no—
351C2×Q8—44no—
352C2×Q8—48no—
353C2×Q8—48no—
354C2×Q8—44no—
355C2×Q8—48no—
356C2×Q8—48no—
357C2×Q8—48no—
358C2×Q8—44no—
359C2×Q8—48no—
360C2×Q8—44no—
361C2×Q8—48no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—24no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—28no—
376C22×C4—28no—
377C22×C4—216no—
378C22×C4—28no—
379C22×C4—216no—
380C22×C4—28no—
381C22×C4—28no—
382C22×C4—28no—
383C22×C4—28no—
384C22×C4—28no—
385C22×C4—28no—
386C22×C4—232no—
387C22×C4—21no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—28no—
408C22×C4—232no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—28no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—28no—
431C22×C4—44no—
432C22×C4—416no—
433C22×C4—416no—
434C22×C4—416no—
435C22×C4—416no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—416no—
439C22×C4#121116yes—
440C22×C4C4⋊Q8116yescomplemented
441C22×C4C22×D412yes—
442C22×C4C22×D414yes—
443C22×C4C22×D414yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D418yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×Q818yes—
456C22×C4C22×Q818yes—
457C22×C4C22×Q814yes—
458C22×C4C22×Q814yes—
459C22×C4C2×#4112yes—
460C22×C4C2×#4112yes—
461C22×C4C2×#4118yes—
462C22×C4C2×#4112yes—
463C22×C4C2×#4112yes—
464C22×C4C2×#4112yes—
465C22×C4C2×#4114yes—
466C22×C4H(5,𝔽2)12yes—
467C22×C42−1+412yes—
468C4⋊C4—28no—
469C4⋊C4—216no—
470C4⋊C4—232no—
471C4⋊C4—216no—
472C4⋊C4—432no—
473C4⋊C4—432no—
474C4⋊C4—48no—
475C4⋊C4—48no—
476C4⋊C4—48no—
477C4⋊C4—44no—
478C4⋊C4—48no—
479C4⋊C4—416no—
480C4⋊C4—48no—
481C4⋊C4—48no—
482C4⋊C4—48no—
483C4⋊C4—44no—
484C4⋊C4C22⋊Q8132yescomplemented
485C22⋊C4—216no—
486C22⋊C4—416no—
487C22⋊C4—48no—
488C42—28no—
489C42—28no—
490C42—28no—
491C42—216no—
492C42—416no—
493C42—48no—
494C42—48no—
495C42—48no—
496C42#121116yes—
497C23—22no—
498C23—24no—
499C23—22no—
500C23—24no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—24no—
507C23—22no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23C2×#12114yes—
516C23C2×C4⋊Q814yes—
517C23#53614yes—
518C23D4×Q818yes—
519C23#55718yes—
520C23C23×D411yescharacteristic
521C23C23×D411yescharacteristic
522C23C23×D411yescharacteristic
523C23C23×Q812yes—
524C23C22×#4111yescharacteristic
525C23C22×#4111yescharacteristic
526C23C22×#4112yes—
527C23C2×H(5,𝔽2)11yescharacteristic
528C23C2×2−1+412yes—
529C23C2×2−1+411yescharacteristic
530C23#58512yes—
531Q8—416no—
532Q8—816no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—28no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—28no—
546C2×C4—28no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—28no—
550C2×C4—24no—
551C2×C4—28no—
552C2×C4—28no—
553C2×C4—232no—
554C2×C4—232no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—28no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—28no—
564C2×C4—28no—
565C2×C4—28no—
566C2×C4—416no—
567C2×C4—464no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—48no—
597C2×C4—44no—
598C2×C4—48no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4C2×#12114yes—
604C2×C4C2×#12114yes—
605C2×C4C2×#12114yes—
606C2×C4C2×#12114yes—
607C2×C4C2×#12114yes—
608C2×C4C2×C22⋊Q8116yes—
609C2×C4C2×C22⋊Q818yes—
610C2×C4C2×C4⋊Q814yes—
611C2×C4C2×C4⋊Q814yes—
612C2×C4#53214yes—
613C2×C4#53318yes—
614C2×C4#53714yes—
615C2×C4D4214yes—
616C2×C4#54714yes—
617C2×C4#54814yes—
618C2×C4#55314yes—
619C22—24no—
620C22—28no—
621C22—24no—
622C22—24no—
623C22—24no—
624C22—24no—
625C22—28no—
626C22—24no—
627C22—28no—
628C22—28no—
629C22#254618yescomplemented
630C22C22×#12111yescharacteristic
631C22C22×#12111yescharacteristic
632C22C22×C22⋊Q812yes—
633C22C22×C4⋊Q811yescharacteristic
634C22C2×#53211yescharacteristic
635C22C2×#53312yes—
636C22C2×#53611yescharacteristic
637C22C2×#53711yescharacteristic
638C22#344911yescharacteristic
639C22C2×D4211yescharacteristic
640C22C2×#54711yescharacteristic
641C22C2×#54811yescharacteristic
642C22C2×D4×Q812yes—
643C22D4×#4112yes—
644C22C2×#55311yescharacteristic
645C22C2×#55712yes—
646C22#348012yes—
647C22#348912yes—
648C22#349512yes—
649C22#350112yes—
650C22#350212yes—
651C22#352012yes—
652C22#352312yes—
653C4—24no—
654C4—24no—
655C4—24no—
656C4—216no—
657C4—44no—
658C4—44no—
659C4—42no—
660C4—48no—
661C4—84no—
662C4—84no—
663C4—82no—
664C4—82no—
665C4#241618yes—
666C4#285718yescomplemented
667C2—28no—
668C2C2×#241611yescharacteristic
669C2C2×#254611yescharacteristic
670C2#3493711yescharacteristic
671C2#3517311yescharacteristic
672C2#3524111yescharacteristic
673C2#3530611yescharacteristic
674C2D4×C22⋊Q812yes—
675C2#3613312yes—
676C2#3657712yes—
677C2#3659912yes—
678C2C2×#285711yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#2546C2semidirect
6. #15184C2semidirect
7. #15180C2semidirect
8. #14818C2semidirect
11. #14155C2semidirect
12. #14046C2semidirect
13. #13926C2semidirect
14. #13910C2semidirect
95. C22×C4⋊C4Q8semidirect
440. C22×C4C4⋊Q8semidirect
484. C4⋊C4C22⋊Q8semidirect
629. C22#2546semidirect
666. C4#2857semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
C2×#25462
C2×C4×C22⋊Q82
C22×C4×C4⋊C42
#151842
#151802
#148182
#147662
#147022
#141552
#140462
#139262
#139102
#139072