Finite Group Databasebeta

← #533378

Subgroups — #533378

Order 512.

3856 conjugacy classes of subgroups.

The subgroup lattice has 711 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2409C211yesmaximal, characteristic
2C2×#2362C212yesmaximal, complemented
3C2×Q8×C22⋊C4C212yesmaximal, complemented
4#14537C212yesmaximal, complemented
5#14409C212yesmaximal, complemented
6#14232C212yesmaximal, complemented
7#14214C214yesmaximal, complemented
8#14017C212yesmaximal, complemented
9#14012C214yesmaximal
10#13994C212yesmaximal, complemented
11#13969C212yesmaximal, complemented
12#13871C211yesmaximal, characteristic
13#13823C212yesmaximal, complemented
14C4×#391C212yesmaximal, complemented
15C4×#390C211yesmaximal, characteristic
16C24×Q8C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C22×C4×Q8C2212yes—
19C22×C4×Q8C2212yes—
20C22×C4×D4C2211yescharacteristic
21C22×#117C2211yescharacteristic
22#2409C4116yescomplemented
23C2×#395C2212yes—
24C2×#393C2212yes—
25#2362—216no—
26Q8×C22⋊C4—216no—
27C2×#391C2214yes—
28C2×#391C2214yes—
29C2×#391C2212yes—
30C2×#391C2212yes—
31C2×#391C2212yes—
32C2×#390C2212yes—
33C2×#389C2212yes—
34C2×#388C2214yes—
35C2×#388C2214yes—
36C2×#388C2214yes—
37C2×#387C2214yes—
38C2×#386C2212yes—
39C2×#385C2212yes—
40C2×#380C2212yes—
41C2×#380C2214yes—
42C2×#380C2211yescharacteristic
43C2×#380C2211yescharacteristic
44C2×C4×C4⋊C4C2214yes—
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2212yes—
48C2×C4×C22⋊C4C2214yes—
49C2×C4×C22⋊C4C2212yes—
50C2×C43C2211yescharacteristic
51#1444C2214yes—
52#1442C2214yes—
53#1442C2214yes—
54#1441C2214yes—
55#1440C2212yes—
56#1440C2214yes—
57#1439C2214yes—
58#1438C2212yes—
59#1437C2212yes—
60#1437C2211yescharacteristic
61#1437C2214yes—
62#1437C2211yescharacteristic
63#1437C2211yescharacteristic
64#1437C2211yescharacteristic
65#1436C2212yes—
66#1436C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75#1434C2214yes—
76C4×#95C2214yes—
77C4×#95C2214yes—
78C23×Q8—28no—
79C23×Q8—28no—
80C23×Q8—42no—
81C23×Q8—42no—
82C23×Q8C2×C418yes—
83C23×Q8C2312yes—
84C23×Q8C2312yes—
85C23×D4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2312yes—
89C2×C22⋊Q8—28no—
90C2×C22⋊Q8—28no—
91C2×C22⋊Q8C2×C4116yes—
92C2×C4×Q8—28no—
93C2×C4×Q8—28no—
94C2×C4×Q8—28no—
95C2×C4×Q8—28no—
96C2×C4×Q8—44no—
97C2×C4×Q8—44no—
98C2×C4×D4—42no—
99C2×C4×D4—42no—
100C2×C4×D4C2×C418yes—
101C2×#117—28no—
102C2×#117—42no—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2311yescharacteristic
123C22×C42C2312yes—
124C22×C42C2314yes—
125C22×C42C2312yes—
126C22×C42C2314yes—
127C22×C42C2314yes—
128C22×C42C2311yescharacteristic
129C22×C42C2312yes—
130C22×C42C2311yescharacteristic
131C22×C42C2312yes—
132C22×C42C2311yescharacteristic
133#395C2×C4116yes—
134#393C2×C4116yes—
135#391—28no—
136#391—216no—
137#391—216no—
138#391C2×C4116yes—
139#391D4116yescomplemented
140#390—28no—
141#389—28no—
142#388—216no—
143#388—216no—
144#388—216no—
145#387—216no—
146#386C2×C4116yes—
147#385C2×C4116yes—
148#380—28no—
149#380—216no—
150#380—42no—
151#380C2×C418yes—
152C4×C4⋊C4—28no—
153C4×C4⋊C4—216no—
154C4×C4⋊C4—42no—
155C4×C22⋊C4—28no—
156C4×C22⋊C4—28no—
157C4×C22⋊C4—216no—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2311yescharacteristic
174C2×#95C2312yes—
175C2×#95C2311yescharacteristic
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C43—42no—
182C25C2411yescharacteristic
183C25C2411yescharacteristic
184C22×Q8—264no—
185C22×Q8—24no—
186C22×Q8—28no—
187C22×Q8—24no—
188C22×Q8—28no—
189C22×Q8—24no—
190C22×Q8—24no—
191C22×Q8—42no—
192C22×Q8—42no—
193C22×Q8—416no—
194C22×Q8—416no—
195C22×Q8—432no—
196C22×Q8—416no—
197C22×Q8C22×C418yes—
198C22×Q8C22×C418yes—
199C22×Q8C2×D418yes—
200C22×Q8#4118yes—
201C22×D4—24no—
202C22×D4—24no—
203C22×D4—41no—
204C22×D4—41no—
205C22×D4C22×C414yes—
206C23×C4—24no—
207C23×C4—28no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—28no—
211C23×C4—24no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—42no—
217C23×C4—42no—
218C23×C4C22×C414yes—
219C23×C4C22×C414yes—
220C23×C4C22×C418yes—
221C23×C4C2412yes—
222C23×C4C2414yes—
223C23×C4C2414yes—
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2414yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2414yes—
231C23×C4C2411yescharacteristic
232C23×C4C2412yes—
233C23×C4C2411yescharacteristic
234C22⋊Q8—264no—
235C4×Q8—432no—
236C4×Q8—432no—
237C4×D4—416no—
238#117—416no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—41no—
258C2×C4⋊C4—41no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4C22×C418yes—
263C2×C4⋊C4C22×C4116yes—
264C2×C4⋊C4C22×C414yes—
265C2×C4⋊C4C2×D418yes—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—42no—
284C2×C22⋊C4—42no—
285C2×C22⋊C4—41no—
286C2×C22⋊C4—41no—
287C2×C22⋊C4—41no—
288C2×C22⋊C4—41no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—42no—
291C2×C22⋊C4C22×C4116yes—
292C2×C22⋊C4C22×C418yes—
293C2×C22⋊C4C22×C414yes—
294C2×C22⋊C4C22×C414yes—
295C2×C22⋊C4#4118yes—
296C2×C42—24no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—24no—
300C2×C42—28no—
301C2×C42—24no—
302C2×C42—28no—
303C2×C42—28no—
304C2×C42—24no—
305C2×C42—28no—
306C2×C42—28no—
307C2×C42—28no—
308C2×C42—28no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—28no—
314C2×C42—42no—
315C2×C42—44no—
316C2×C42—44no—
317C2×C42—41no—
318C2×C42—42no—
319C2×C42—41no—
320C2×C42—42no—
321C2×C42C22×C414yes—
322C2×C42C2×D418yes—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—28no—
328#95—28no—
329#95—28no—
330#95—28no—
331#95—28no—
332#95—28no—
333#95—28no—
334#95—28no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—41no—
339#95—41no—
340#95—42no—
341#95C22×C418yes—
342#95C22×C4116yes—
343#95C22×C4116yes—
344#95C2×D4116yes—
345#95C2×D4116yes—
346C24—22no—
347C24—22no—
348C24—21no—
349C24—21no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—21no—
357C24—21no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24C23×C412yes—
362C24C23×C412yes—
363C24C2×#4114yes—
364C24C2511yescharacteristic
365C2×Q8—216no—
366C2×Q8—232no—
367C2×Q8—216no—
368C2×Q8—216no—
369C2×Q8—216no—
370C2×Q8—216no—
371C2×Q8—216no—
372C2×Q8—48no—
373C2×Q8—48no—
374C2×Q8—416no—
375C2×Q8—48no—
376C2×Q8—4128no—
377C2×Q8C4×D4132yes—
378C2×D4—28no—
379C2×D4—28no—
380C2×D4—44no—
381C2×D4—48no—
382C2×D4—48no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—216no—
431C22×C4—232no—
432C22×C4—216no—
433C22×C4—28no—
434C22×C4—28no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—416no—
452C22×C4—48no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—48no—
456C22×C4—48no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4C23×C412yes—
460C22×C4C23×C418yes—
461C22×C4C23×C412yes—
462C22×C4C23×C412yes—
463C22×C4C23×C418yes—
464C22×C4C23×C412yes—
465C22×C4C23×C412yes—
466C22×C4C22×D414yes—
467C22×C4C22×D418yes—
468C22×C4C22×D418yes—
469C22×C4C22×D414yes—
470C22×C4C22×D414yes—
471C22×C4C2×#4114yes—
472C22×C4C2×#4118yes—
473C22×C4C2×#4114yes—
474C22×C4C2×#4114yes—
475C22×C4H(5,𝔽2)12yes—
476C22×C4H(5,𝔽2)12yes—
477C22×C42−1+414yes—
478C22×C42−1+414yes—
479C4⋊C4—216no—
480C4⋊C4—232no—
481C4⋊C4—216no—
482C4⋊C4—44no—
483C4⋊C4—48no—
484C4⋊C4—416no—
485C4⋊C4—416no—
486C4⋊C4—416no—
487C4⋊C4—48no—
488C22⋊C4—216no—
489C22⋊C4—232no—
490C22⋊C4—216no—
491C22⋊C4—44no—
492C22⋊C4—44no—
493C22⋊C4—48no—
494C22⋊C4—48no—
495C22⋊C4—48no—
496C22⋊C4—416no—
497C22⋊C4—48no—
498C22⋊C4—84no—
499C42—28no—
500C42—216no—
501C42—48no—
502C42—44no—
503C42—48no—
504C42—416no—
505C42—416no—
506C42—44no—
507C42—48no—
508C42—416no—
509C23—24no—
510C23—22no—
511C23—22no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—24no—
524C23—41no—
525C23—42no—
526C23—41no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—41no—
532C23—41no—
533C23—42no—
534C23—41no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—41no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—41no—
548C23—41no—
549C23—42no—
550C23C4×#4114yes—
551C23C24×C411yescharacteristic
552C23C23×D412yes—
553C23C22×#4112yes—
554C23C22×#4112yes—
555C23C2×H(5,𝔽2)11yescharacteristic
556C23C2×H(5,𝔽2)11yescharacteristic
557C23C2×2−1+412yes—
558C23C2×2−1+412yes—
559C23#58512yes—
560Q8—264no—
561Q8—432no—
562D4—416no—
563C2×C4—28no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—28no—
567C2×C4—28no—
568C2×C4—24no—
569C2×C4—28no—
570C2×C4—28no—
571C2×C4—28no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—28no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—28no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—28no—
588C2×C4—28no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—28no—
594C2×C4—24no—
595C2×C4—28no—
596C2×C4—24no—
597C2×C4—432no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—416no—
624C2×C4—416no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4C2×C4×D4116yes—
640C2×C4C2×C4×D418yes—
641C2×C4#51814yes—
642C2×C4#51914yes—
643C2×C4D4214yes—
644C2×C4#54618yes—
645C2×C4#54818yes—
646C2×C4#55114yes—
647C22—28no—
648C22—24no—
649C22—44no—
650C22—44no—
651C22—44no—
652C22—42no—
653C22—44no—
654C22—44no—
655C22—42no—
656C22—44no—
657C22—42no—
658C22—42no—
659C22—44no—
660C22—42no—
661C22—82no—
662C22—82no—
663C22—82no—
664C22—82no—
665C22C22×C4×D412yes—
666C22C2×C4×#4111yescharacteristic
667C22C2×#51811yescharacteristic
668C22C2×#51911yescharacteristic
669C22C4×2−1+412yes—
670C22C2×D4211yescharacteristic
671C22C2×#54612yes—
672C22C2×#54812yes—
673C22D4×#4112yes—
674C22C2×#55111yescharacteristic
675C22#348012yes—
676C22#348212yes—
677C22#349712yes—
678C22#349912yes—
679C22#351812yes—
680C22#352012yes—
681C22#352012yes—
682C22#352312yes—
683C22#353811yescharacteristic
684C22#354011yescharacteristic
685C22#354712yes—
686C4—28no—
687C4—216no—
688C4—24no—
689C4—48no—
690C4—44no—
691C4—42no—
692C4—42no—
693C4—44no—
694C4—42no—
695C4—48no—
696C4—82no—
697C4—84no—
698C4#235818yes—
699C2—44no—
700C2—82no—
701C2C2×#235811yescharacteristic
702C2C4×#54812yes—
703C2#3424511yescharacteristic
704C2#3431112yes—
705C2#3441211yescharacteristic
706C2#5965112yes—
707C2#5965412yes—
708C2#5975212yes—
709C2#5975512yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2362C2semidirect
3. C2×Q8×C22⋊C4C2semidirect
4. #14537C2semidirect
5. #14409C2semidirect
6. #14232C2semidirect
7. #14214C2semidirect
8. #14017C2semidirect
10. #13994C2semidirect
11. #13969C2semidirect
13. #13823C2semidirect
14. C4×#391C2semidirect
22. #2409C4semidirect
139. #391D4semidirect

Maximal subgroups

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

subgroupindex
C2×#24092
C2×#23622
C2×Q8×C22⋊C42
#145372
#144092
#142322
#142142
#140172
#140122
#139942
#139692
#138712
#138232
C4×#3912
C4×#3902