Finite Group Databasebeta

← #519558

Subgroups — #519558

Order 512.

3334 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊Q8C212yesmaximal
2C2×C4×C22⋊Q8C211yesmaximal, characteristic
3#14697C212yesmaximal
4#14682C211yesmaximal, characteristic
5#14113C214yesmaximal, complemented
6#14044C214yesmaximal
7#14006C212yesmaximal, complemented
8#13978C212yesmaximal, complemented
9#13949C212yesmaximal, complemented
10#13935C211yesmaximal, characteristic, complemented
11#13918C211yesmaximal, characteristic, complemented
12#13896C212yesmaximal
13#13821C211yesmaximal, characteristic, complemented
14#13814C212yesmaximal
15Q8×#95C211yesmaximal, characteristic, complemented
16#13804C212yesmaximal, complemented
17C2×C4×#95C211yesmaximal, characteristic
18C22×C22⋊Q8C2211yescharacteristic
19C22×C4×Q8C2212yes—
20C22×C4×Q8C2211yescharacteristic
21C23×C42C2212yes—
22C23×C42C2211yescharacteristic
23C4×C22⋊Q8—216no—
24C4×C22⋊Q8C4116yescomplemented
25C2×#391C2212yes—
26C2×#391C2211yescharacteristic
27C2×#389C2212yes—
28C2×#389C2212yes—
29C2×#389C2212yes—
30C2×#389C2211yescharacteristic
31C2×#387C2214yes—
32C2×#387C2212yes—
33C2×#385C2214yes—
34C2×#385C2214yes—
35C2×#385C2212yes—
36C2×#382C2212yes—
37C2×#382C2212yes—
38C2×#380C2212yes—
39C2×#380C2211yescharacteristic
40C2×C4×C4⋊C4C2214yes—
41C2×C4×C4⋊C4C2212yes—
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C22⋊C4C2214yes—
46C2×C4×C22⋊C4C2212yes—
47C22×#95C2211yescharacteristic
48C22×#95C2212yes—
49C22×#95C2211yescharacteristic
50#1442C2214yes—
51#1442C2214yes—
52#1442C2214yes—
53#1442C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1436C2214yes—
57#1436C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62C4⋊#95C2214yes—
63C4⋊#95C2212yes—
64C4⋊#95C2211yescharacteristic
65C4⋊#95C2211yescharacteristic
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2214yes—
70#1434C2212yes—
71#1434C2212yes—
72#1434C2212yes—
73#1434C2212yes—
74#1434C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2214yes—
77C22⋊#95C2212yes—
78C4×#95C2212yes—
79C4×#95C2211yescharacteristic
80C4×#95C2211yescharacteristic
81C4×#95C2212yes—
82C4×#95C2211yescharacteristic
83C4×#95C2211yescharacteristic
84C4×#95C2212yes—
85C23×Q8C2311yescharacteristic
86C24×C4C2312yes—
87C24×C4C2311yescharacteristic
88C24×C4C2312yes—
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C2×C22⋊Q8C2×C418yes—
92C2×C22⋊Q8D418yes—
93C2×C22⋊Q8Q818yes—
94C2×C4×Q8—28no—
95C2×C4×Q8—28no—
96C2×C4×Q8—44no—
97C2×C4×Q8—44no—
98C2×C4×Q8C2×C418yescomplemented
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C42—28no—
115C22×C42—28no—
116C22×C42—44no—
117C22×C42—44no—
118C22×C42C2×C418yes—
119C22×C42C2312yes—
120C22×C42C2312yes—
121C22×C42C2312yes—
122C22×C42C2314yes—
123C22×C42C2314yes—
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2312yes—
127C22×C42C2312yes—
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2312yes—
131#391—28no—
132#391C2×C418yescomplemented
133#389—28no—
134#389—28no—
135#389—44no—
136#389C2×C418yescomplemented
137#387—216no—
138#387C2×C4116yescomplemented
139#385—216no—
140#385—216no—
141#385C2×C4116yes—
142#382—28no—
143#382—44no—
144#380—28no—
145#380C2×C418yes—
146C4×C4⋊C4—216no—
147C4×C4⋊C4—28no—
148C4×C4⋊C4—28no—
149C4×C4⋊C4C2×C4116yescomplemented
150C4×C4⋊C4C2×C418yescomplemented
151C4×C22⋊C4—216no—
152C4×C22⋊C4C2×C4116yes—
153C2×#95—28no—
154C2×#95—24no—
155C2×#95—24no—
156C2×#95C2311yescharacteristic
157C2×#95C2311yescharacteristic
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2311yescharacteristic
187C2×#95C2311yescharacteristic
188C25C2411yescharacteristic
189C22×Q8—24no—
190C22×Q8—24no—
191C22×Q8C22×C414yes—
192C22×Q8C2×D414yes—
193C22×Q8C2×Q814yes—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—42no—
210C23×C4—41no—
211C23×C4—41no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4C22×C414yes—
218C23×C4C22×C414yes—
219C23×C4C22×C414yes—
220C23×C4C2×D414yes—
221C23×C4C2×Q814yes—
222C23×C4#4118yes—
223C23×C4C2414yes—
224C23×C4C2414yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C23×C4C2412yes—
238C23×C4C2411yescharacteristic
239C22⋊Q8C4⋊C4164yescomplemented
240C4×Q8—416no—
241C4×Q8—432no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4C22×C414yes—
266C2×C4⋊C4C22×C414yes—
267C2×C4⋊C4C22×C418yes—
268C2×C4⋊C4C22×C418yes—
269C2×C4⋊C4C22×C418yes—
270C2×C4⋊C4C22×C414yes—
271C2×C4⋊C4C2×D418yes—
272C2×C4⋊C4C2×D414yes—
273C2×C4⋊C4C2×Q814yes—
274C2×C4⋊C4C2×Q818yes—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4C22×C418yes—
280C2×C22⋊C4C22×C418yes—
281C2×C22⋊C4C2×D418yes—
282C2×C22⋊C4C2×Q818yes—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—28no—
290C2×C42—28no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—28no—
295C2×C42—28no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—416no—
301C2×C42—432no—
302C2×C42—44no—
303C2×C42—44no—
304C2×C42—42no—
305C2×C42—42no—
306C2×C42—42no—
307C2×C42—42no—
308C2×C42—44no—
309C2×C42C22×C414yes—
310C2×C42C22×C418yes—
311C2×C42C22×C414yes—
312C2×C42C22×C418yes—
313C2×C42C22×C414yes—
314C2×C42C22×C414yes—
315C2×C42#4118yes—
316#95—28no—
317#95—28no—
318#95—28no—
319#95—28no—
320#95—28no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—22no—
325#95—22no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—28no—
331#95—24no—
332#95—28no—
333#95—28no—
334#95—28no—
335#95—44no—
336#95—42no—
337#95—42no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95C22×C418yes—
342#95C22×C418yes—
343#95C22×C418yes—
344#95C22×C418yes—
345#95C22×C418yes—
346#95#4114yes—
347#95#4114yes—
348C24—21no—
349C24—22no—
350C24—21no—
351C24—22no—
352C24—21no—
353C24—21no—
354C24C23×C412yes—
355C24C22×D412yes—
356C24C22×Q812yes—
357C24C2×#4114yes—
358C242−1+414yes—
359C24C2511yescharacteristic
360C2×Q8—28no—
361C2×Q8—28no—
362C2×Q8—216no—
363C2×Q8—216no—
364C2×Q8C2×C4⋊C4116yes—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—28no—
380C22×C4—28no—
381C22×C4—216no—
382C22×C4—216no—
383C22×C4—216no—
384C22×C4—28no—
385C22×C4—28no—
386C22×C4—216no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—28no—
426C22×C4—216no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4C2×C4⋊C4116yes—
460C22×C4C23×C412yes—
461C22×C4C23×C414yes—
462C22×C4C23×C412yes—
463C22×C4C23×C412yes—
464C22×C4C23×C414yes—
465C22×C4C23×C412yes—
466C22×C4C23×C414yes—
467C22×C4C23×C414yes—
468C22×C4C23×C412yes—
469C22×C4C23×C412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D414yes—
472C22×C4C22×D414yes—
473C22×C4C22×D412yes—
474C22×C4C22×Q814yes—
475C22×C4C22×Q814yes—
476C22×C4C22×Q812yes—
477C22×C4C22×Q812yes—
478C22×C4C2×#4114yes—
479C22×C4C2×#4114yes—
480C22×C4C2×#4114yes—
481C22×C4C2×#4112yes—
482C22×C4C2×#4112yes—
483C22×C4C2×#4114yes—
484C22×C4C2×#4114yes—
485C22×C4C2×#4114yes—
486C22×C4C2×#4114yes—
487C22×C4H(5,𝔽2)14yes—
488C4⋊C4—28no—
489C4⋊C4—28no—
490C4⋊C4—48no—
491C4⋊C4—48no—
492C4⋊C4—416no—
493C4⋊C4—416no—
494C4⋊C4—416no—
495C4⋊C4—48no—
496C4⋊C4—48no—
497C4⋊C4—88no—
498C4⋊C4C2×C4⋊C4116yes—
499C4⋊C4C2×C4⋊C4132yes—
500C22⋊C4—216no—
501C22⋊C4—416no—
502C22⋊C4C2×C4⋊C4132yes—
503C42—216no—
504C42—28no—
505C42—28no—
506C42—216no—
507C42—44no—
508C42—44no—
509C42—48no—
510C42—416no—
511C42—48no—
512C42—48no—
513C42—416no—
514C42—48no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—22no—
523C23—22no—
524C23—24no—
525C23—22no—
526C23—22no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23C22×C4⋊C414yes—
534C23C4×#4114yes—
535C23#51914yes—
536C23#54818yes—
537C23#55718yes—
538C23C24×C411yescharacteristic
539C23C23×D411yescharacteristic
540C23C23×Q811yescharacteristic
541C23C22×#4111yescharacteristic
542C23C22×#4112yes—
543C23C22×#4112yes—
544C23C22×#4111yescharacteristic
545C23C2×H(5,𝔽2)12yes—
546C23C2×2−1+412yes—
547C23#58511yescharacteristic
548C23#58511yescharacteristic
549Q8—232no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—28no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—28no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—232no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—432no—
600C2×C4—416no—
601C2×C4—432no—
602C2×C4—416no—
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—44no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4C22×C4⋊C418yes—
640C2×C4C22×C4⋊C418yes—
641C2×C4C22×C4⋊C414yes—
642C2×C4C22×C4⋊C414yes—
643C2×C4C4×#4114yes—
644C2×C4C4×#4114yes—
645C2×C4#51814yes—
646C2×C4#52918yes—
647C2×C4#54618yes—
648C2×C4#55418yes—
649C22—28no—
650C22—28no—
651C22—24no—
652C22—24no—
653C22—24no—
654C22—24no—
655C22—24no—
656C22—28no—
657C22—28no—
658C22—24no—
659C22#235118yes—
660C22C23×C4⋊C411yescharacteristic
661C22C2×C4×#4111yescharacteristic
662C22C2×C4×#4111yescharacteristic
663C22C2×C4×#4111yescharacteristic
664C22C2×#51811yescharacteristic
665C22C2×#51911yescharacteristic
666C22#342811yescharacteristic
667C22C2×#52912yes—
668C22#344412yes—
669C22C2×#54612yes—
670C22C2×#54812yes—
671C22D4×#4111yescharacteristic
672C22D4×#4111yescharacteristic
673C22C2×#55412yes—
674C22C2×#55712yes—
675C22Q8×#4111yescharacteristic
676C22Q8×#4111yescharacteristic
677C22#349512yes—
678C22#349512yes—
679C22#350512yes—
680C22#350612yes—
681C22#352012yes—
682C22#352012yes—
683C4—28no—
684C4—24no—
685C4—28no—
686C4—28no—
687C4—24no—
688C4—44no—
689C4—44no—
690C4—44no—
691C4—44no—
692C4—42no—
693C4—42no—
694C4—42no—
695C4—44no—
696C4—84no—
697C4—84no—
698C4#234918yes—
699C2—28no—
700C2C4×#52911yescharacteristic
701C2C2×#234911yescharacteristic
702C2C2×#235111yescharacteristic
703C2C4⋊C4×#4111yescharacteristic
704C2#3422111yescharacteristic
705C2#3422311yescharacteristic
706C2#3424511yescharacteristic
707C2#3610812yes—
708C2#3614812yes—
709C2#3655212yes—
710C2#3657112yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC25C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #14113C2semidirect
7. #14006C2semidirect
8. #13978C2semidirect
9. #13949C2semidirect
10. #13935C2semidirect
11. #13918C2semidirect
13. #13821C2semidirect
15. Q8×#95C2semidirect
16. #13804C2semidirect
24. C4×C22⋊Q8C4semidirect
98. C2×C4×Q8C2×C4semidirect
132. #391C2×C4semidirect
136. #389C2×C4semidirect
138. #387C2×C4semidirect
149. C4×C4⋊C4C2×C4semidirect
150. C4×C4⋊C4C2×C4semidirect
239. C22⋊Q8C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊Q82
#146972
#146822
#141132
#140442
#140062
#139782
#139492
#139352
#139182
#138962
#138212
#138142
Q8×#952
#138042
C2×C4×#952