Finite Group Databasebeta

← #517290

Subgroups — #517290

Order 512.

3422 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic
2C2×#2323C212yesmaximal
3C2×C4×C4⋊Q8C211yesmaximal, characteristic, complemented
4C2×C4×#117C211yesmaximal, characteristic
5#14714C211yesmaximal, characteristic, complemented
6#14712C211yesmaximal, characteristic, complemented
7#14686C211yesmaximal, characteristic
8#14261C212yesmaximal, complemented
9#14258C212yesmaximal
10#14046C214yesmaximal, complemented
11#14042C214yesmaximal, complemented
12#13994C214yesmaximal
13#13974C214yesmaximal
14#13786C212yesmaximal, complemented
15C4×#384C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C22×C4×Q8C2212yes—
19C22×#117C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C23×C42C2211yescharacteristic
22#2857C4116yescomplemented
23#2323—216no—
24C4×C4⋊Q8—28no—
25C2×#391C2212yes—
26C2×#391C2212yes—
27C2×#391C2212yes—
28C2×#391C2212yes—
29C2×#389C2212yes—
30C2×#389C2214yes—
31C2×#389C2214yes—
32C2×#389C2214yes—
33C2×#389C2214yes—
34C2×#388C2214yes—
35C2×#388C2214yes—
36C2×#387C2214yes—
37C2×#384C2211yescharacteristic
38C2×#384C2211yescharacteristic
39C2×#384C2211yescharacteristic
40C2×#380C2212yes—
41C2×#380C2214yes—
42C2×#380C2214yes—
43C4×#117—28no—
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2214yes—
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C4⋊C4C2214yes—
49C2×C4×C22⋊C4C2212yes—
50C2×C4×C22⋊C4C2212yes—
51C2×#376C2211yescharacteristic
52C2×#376C2211yescharacteristic
53C2×C43C2211yescharacteristic
54C2×C43C2211yescharacteristic
55#1443C2218yes—
56#1442C2218yes—
57#1441C2218yes—
58#1440C2218yes—
59#1436C2218yes—
60C4⋊#95C2212yes—
61C4⋊#95C2214yes—
62C4⋊#95C2214yes—
63C4⋊#95C2212yes—
64C4⋊#95C2214yes—
65C4⋊#95C2214yes—
66C4⋊#95C2214yes—
67C4⋊#95C2212yes—
68#1434C2218yes—
69C23×Q8C2312yes—
70C24×C4C2312yes—
71C24×C4C2311yescharacteristic
72C2×C4⋊Q8—24no—
73C2×C4⋊Q8—24no—
74C2×C4⋊Q8C2×C418yes—
75C2×C22⋊Q8—28no—
76C2×C22⋊Q8—28no—
77C2×C22⋊Q8C2×C4116yes—
78C2×C4×Q8—28no—
79C2×C4×Q8—28no—
80C2×C4×Q8—28no—
81C2×#117—24no—
82C2×#117—24no—
83C2×#117—24no—
84C2×#117—24no—
85C2×#117—24no—
86C2×#117D418yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2314yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2311yescharacteristic
103C22×C22⋊C4C2311yescharacteristic
104C22×C22⋊C4C2314yes—
105C22×C42—24no—
106C22×C42—24no—
107C22×C42C2×C418yes—
108C22×C42C2314yes—
109C22×C42C2312yes—
110C22×C42C2312yes—
111C22×C42C2311yescharacteristic
112C22×C42C2312yes—
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115C22×C42C2312yes—
116C22×C42C2312yes—
117C22×C42C2311yescharacteristic
118C22×C42C2314yes—
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121#391—28no—
122#391—28no—
123#391—28no—
124#391C2×C4116yes—
125#389—216no—
126#389—216no—
127#389—28no—
128#389—48no—
129#389C2×C4132yes—
130#388—216no—
131#388—216no—
132#387—216no—
133#384—24no—
134#384—24no—
135#384C2×C418yes—
136#380—28no—
137#380—48no—
138#380C2×C4132yes—
139C4×C4⋊C4—216no—
140C4×C4⋊C4—28no—
141C4×C4⋊C4—28no—
142C4×C4⋊C4—28no—
143C4×C4⋊C4—48no—
144C4×C22⋊C4—28no—
145C4×C22⋊C4—28no—
146#376—24no—
147#376—24no—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2318yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2318yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2318yes—
160C2×#95C2318yes—
161C2×#95C2318yes—
162C2×#95C2318yes—
163C43—24no—
164C43D418yes—
165C25C2411yescharacteristic
166C22×Q8—24no—
167C22×Q8—24no—
168C22×Q8—24no—
169C22×Q8—24no—
170C22×Q8—24no—
171C22×Q8—24no—
172C22×Q8C22×C418yes—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4C22×C418yes—
184C23×C4C22×C414yes—
185C23×C4C2×D414yes—
186C23×C4#4114yes—
187C23×C4C2414yes—
188C23×C4C2414yes—
189C23×C4C2414yes—
190C23×C4C2414yes—
191C23×C4C2412yes—
192C23×C4C2411yescharacteristic
193C23×C4C2411yescharacteristic
194C23×C4C2412yes—
195C23×C4C2412yes—
196C23×C4C2411yescharacteristic
197C23×C4C2411yescharacteristic
198C23×C4C2411yescharacteristic
199C23×C4C2411yescharacteristic
200C23×C4C2412yes—
201C4⋊Q8—232no—
202C22⋊Q8—264no—
203C4×Q8—432no—
204#117—232no—
205#117—232no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—48no—
238C2×C4⋊C4C22×C418yes—
239C2×C4⋊C4C22×C4116yes—
240C2×C4⋊C4C22×C4116yes—
241C2×C4⋊C4C22×C4116yes—
242C2×C4⋊C4C2×D414yes—
243C2×C4⋊C4C2×D414yes—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—48no—
255C2×C22⋊C4C22×C4116yes—
256C2×C22⋊C4C2×D414yes—
257C2×C22⋊C4C2×D414yes—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—24no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—24no—
267C2×C42—22no—
268C2×C42—24no—
269C2×C42—24no—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—232no—
273C2×C42—28no—
274C2×C42—28no—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42—28no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—28no—
283C2×C42—48no—
284C2×C42C22×C418yes—
285C2×C42C22×C414yes—
286C2×C42C22×C414yes—
287C2×C42C2×D414yes—
288C2×C42C2×D414yes—
289C2×C42C2×D418yes—
290C2×C42C2×D418yes—
291C2×C42C2×D414yes—
292C2×C42C2×D414yes—
293C2×C42C2×D414yes—
294C2×C42#4114yes—
295C2×C42#4114yes—
296C2×C42#4114yes—
297#95—24no—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—28no—
302#95—216no—
303#95—28no—
304#95—28no—
305#95—28no—
306#95—28no—
307#95—216no—
308#95—48no—
309#95—48no—
310#95—48no—
311#95C22×C4132yes—
312C24—21no—
313C24—21no—
314C24—21no—
315C24—21no—
316C24—22no—
317C24—22no—
318C24—22no—
319C24—22no—
320C24C23×C412yes—
321C24C22×D412yes—
322C24C2×#4112yes—
323C24C2511yescharacteristic
324C2×Q8—216no—
325C2×Q8—216no—
326C2×Q8—216no—
327C2×Q8—48no—
328C2×Q8—48no—
329C2×Q8—48no—
330C2×Q8—48no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—28no—
340C22×C4—28no—
341C22×C4—28no—
342C22×C4—28no—
343C22×C4—28no—
344C22×C4—28no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—28no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—28no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—216no—
382C22×C4—216no—
383C22×C4—216no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—28no—
391C22×C4—28no—
392C22×C4—28no—
393C22×C4—28no—
394C22×C4—48no—
395C22×C4—48no—
396C22×C4—48no—
397C22×C4—48no—
398C22×C4C4×D4116yes—
399C22×C4C23×C414yes—
400C22×C4C23×C418yes—
401C22×C4C23×C418yes—
402C22×C4C23×C412yes—
403C22×C4C23×C412yes—
404C22×C4C23×C414yes—
405C22×C4C22×D412yes—
406C22×C4C22×D414yes—
407C22×C4C22×D412yes—
408C22×C4C22×D412yes—
409C22×C4C22×D412yes—
410C22×C4C22×D412yes—
411C22×C4C22×D412yes—
412C22×C4C22×D412yes—
413C22×C4C22×D412yes—
414C22×C4C22×D412yes—
415C22×C4C22×D414yes—
416C22×C4C2×#4114yes—
417C22×C4C2×#4112yes—
418C22×C4C2×#4112yes—
419C22×C4C2×#4112yes—
420C22×C4C2×#4112yes—
421C22×C4C2×#4112yes—
422C22×C4C2×#4114yes—
423C22×C4C2×#4112yes—
424C22×C4C2×#4112yes—
425C22×C4H(5,𝔽2)14yes—
426C22×C42−1+414yes—
427C22×C42−1+414yes—
428C22×C42−1+414yes—
429C4⋊C4—216no—
430C4⋊C4—216no—
431C4⋊C4—232no—
432C4⋊C4—232no—
433C4⋊C4—232no—
434C4⋊C4—28no—
435C4⋊C4—28no—
436C4⋊C4—416no—
437C4⋊C4—48no—
438C4⋊C4—48no—
439C4⋊C4—416no—
440C4⋊C4—416no—
441C4⋊C4—88no—
442C22⋊C4—216no—
443C22⋊C4—232no—
444C22⋊C4—28no—
445C22⋊C4—28no—
446C22⋊C4—88no—
447C42—216no—
448C42—28no—
449C42—28no—
450C42—28no—
451C42—28no—
452C42—216no—
453C42—216no—
454C42—416no—
455C42—48no—
456C42—48no—
457C42—88no—
458C42C4×D4116yes—
459C42#121116yes—
460C23—22no—
461C23—22no—
462C23—24no—
463C23—24no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—24no—
473C23—24no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23C2×C4×D414yes—
483C23C24×C411yescharacteristic
484C23C23×D411yescharacteristic
485C23C23×D411yescharacteristic
486C23C22×#4111yescharacteristic
487C23C22×#4111yescharacteristic
488C23C22×#4111yescharacteristic
489C23C22×#4111yescharacteristic
490C23C2×H(5,𝔽2)12yes—
491C23C2×2−1+412yes—
492C23C2×2−1+412yes—
493C23C2×2−1+412yes—
494Q8—432no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—232no—
510C2×C4—24no—
511C2×C4—28no—
512C2×C4—28no—
513C2×C4—28no—
514C2×C4—28no—
515C2×C4—24no—
516C2×C4—216no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—48no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—432no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—48no—
557C2×C4—48no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—48no—
563C2×C4—48no—
564C2×C4C2×C4×D414yes—
565C2×C4C2×C4×D418yes—
566C2×C4C2×C4×D414yes—
567C2×C4C2×C4×D414yes—
568C2×C4C4×#4114yes—
569C2×C4#51918yes—
570C2×C4#52018yes—
571C2×C4C2×#12114yes—
572C2×C4C2×#12114yes—
573C2×C4C2×#12114yes—
574C2×C4#52514yes—
575C2×C4#52914yes—
576C2×C4#53214yes—
577C2×C4#53618yes—
578C2×C4#53718yes—
579C2×C4#54118yes—
580C2×C4#54218yes—
581C22—24no—
582C22—24no—
583C22—24no—
584C22—44no—
585C22—44no—
586C22—42no—
587C22—42no—
588C22—44no—
589C22—44no—
590C22—42no—
591C22—42no—
592C22C22×C4×D411yescharacteristic
593C22C22×C4×D411yescharacteristic
594C22C2×C4×#4111yescharacteristic
595C22C2×#51912yes—
596C22C2×#52012yes—
597C22C22×#12111yescharacteristic
598C22C2×#52511yescharacteristic
599C22C2×#52911yescharacteristic
600C22C2×#53211yescharacteristic
601C22C2×#53612yes—
602C22C2×#53712yes—
603C22C2×#54112yes—
604C22C2×#54212yes—
605C22#349811yescharacteristic
606C22#349911yescharacteristic
607C22#349911yescharacteristic
608C22#349911yescharacteristic
609C22#350912yes—
610C22#350912yes—
611C22#350912yes—
612C22#351112yes—
613C22#351112yes—
614C22#351112yes—
615C4—24no—
616C4—24no—
617C4—24no—
618C4—44no—
619C4—48no—
620C4—42no—
621C4—44no—
622C4—44no—
623C4—48no—
624C4—42no—
625C4—84no—
626C4—84no—
627C4C4×#12118yes—
628C4#2323116yescomplemented
629C2—44no—
630C2C2×C4×#12111yescharacteristic
631C2C2×#232312yes—
632C2#3420711yescharacteristic
633C2#3420911yescharacteristic
634C2#3422512yes—
635C2#3527311yescharacteristic
636C2#3527711yescharacteristic
637C2#3529112yes—
638C2#3529712yes—
639C2#3531611yescharacteristic
640C2#3531911yescharacteristic
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
3. C2×C4×C4⋊Q8C2semidirect
5. #14714C2semidirect
6. #14712C2semidirect
8. #14261C2semidirect
10. #14046C2semidirect
11. #14042C2semidirect
14. #13786C2semidirect
15. C4×#384C2semidirect
22. #2857C4semidirect
628. C4#2323semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
C2×#23232
C2×C4×C4⋊Q82
C2×C4×#1172
#147142
#147122
#146862
#142612
#142582
#140462
#140422
#139942
#139742
#137862
C4×#3842