Finite Group Databasebeta

← #719147

Subgroups — #719147

Order 512.

5754 conjugacy classes of subgroups.

The subgroup lattice has 651 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×Q8C211yesmaximal, characteristic, complemented
2C2×#2542C211yesmaximal, characteristic, complemented
3C2×#2536C211yesmaximal, characteristic, complemented
4C2×#2409C214yesmaximal, complemented
5C2×#2397C211yesmaximal, characteristic
6#16511C214yesmaximal, complemented
7#15453C214yesmaximal, complemented
8#15430C214yesmaximal, complemented
9#15177C214yesmaximal, complemented
10#14796C214yesmaximal, complemented
11#14274C211yesmaximal, characteristic, complemented
12#13905C211yesmaximal, characteristic, complemented
13#13786C211yesmaximal, characteristic, complemented
14C24×Q8C2212yes—
15C2×D4×Q8—28no—
16C2×D4×Q8—28no—
17C2×D4×Q8—28no—
18C22×C4⋊Q8C2212yes—
19C22×C4⋊Q8C2211yescharacteristic
20C22×C22⋊Q8C2214yes—
21C22×C22⋊Q8C2212yes—
22C22×C22⋊Q8C2212yes—
23C22×C22⋊Q8C2214yes—
24C22×C4×Q8C2211yescharacteristic
25C22×C4×D4C2211yescharacteristic
26C22×C4×D4C2212yes—
27C22×C4×D4C2214yes—
28C22×C4×D4C2211yescharacteristic
29#2542—28no—
30#2536—28no—
31#2409—232no—
32#2397—28no—
33C2×#399C2212yes—
34C2×#399C2214yes—
35C2×#399C2212yes—
36C2×#398C2212yes—
37C2×#395C2218yes—
38C2×#394C2212yes—
39C2×#393C2218yes—
40C2×#392C2212yes—
41C2×#392C2212yes—
42C2×#391C2214yes—
43C2×#391C2212yes—
44C2×#390C2212yes—
45C2×#388C2214yes—
46C2×#386C2218yes—
47C2×#385C2218yes—
48C2×#384C2212yes—
49C2×#382C2211yescharacteristic
50C2×#380C2212yes—
51C2×#380C2214yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×#376C2211yescharacteristic
54#1445C2214yes—
55#1445C2212yes—
56#1445C2214yes—
57#1444C2218yes—
58#1439C2218yes—
59#1439C2218yes—
60#1438C2218yes—
61#1438C2212yes—
62#1437C2214yes—
63#1436C2214yes—
64C4⋊#95C2212yes—
65#1434C2214yes—
66C23×Q8—28no—
67C23×Q8—28no—
68C23×Q8—216no—
69C23×Q8—28no—
70C23×Q8—48no—
71C23×Q8C2312yes—
72C23×Q8C2314yes—
73C23×Q8C2311yescharacteristic
74C23×D4C2311yescharacteristic
75C23×D4C2311yescharacteristic
76C23×D4C2314yes—
77C24×C4C2312yes—
78C24×C4C2312yes—
79C24×C4C2314yes—
80D4×Q8—2128no—
81C2×C4⋊Q8—28no—
82C2×C4⋊Q8—28no—
83C2×C4⋊Q8—28no—
84C2×C4⋊Q8—24no—
85C2×C4⋊Q8—24no—
86C2×C4⋊Q8D418yescomplemented
87C2×C22⋊Q8—28no—
88C2×C22⋊Q8—28no—
89C2×C22⋊Q8—216no—
90C2×C22⋊Q8—28no—
91C2×C22⋊Q8—216no—
92C2×C22⋊Q8—216no—
93C2×C22⋊Q8—28no—
94C2×C22⋊Q8—216no—
95C2×C22⋊Q8—216no—
96C2×C22⋊Q8—216no—
97C2×C22⋊Q8—48no—
98C2×C4×Q8—24no—
99C2×C4×Q8—24no—
100C2×C4×Q8D418yescomplemented
101C2×C4×D4—28no—
102C2×C4×D4—216no—
103C2×C4×D4—28no—
104C2×C4×D4—28no—
105C2×C4×D4—24no—
106C2×C4×D4—24no—
107C2×C4×D4—24no—
108C2×C4×D4—216no—
109C2×C4×D4—24no—
110C2×C4×D4—24no—
111C2×C4×D4—48no—
112C2×C4×D4D418yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2318yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2318yes—
131C22×C22⋊C4C2314yes—
132C22×C22⋊C4C2318yes—
133C22×C22⋊C4C2312yes—
134C22×C42C2312yes—
135C22×C42C2312yes—
136C22×C42C2312yes—
137C22×C42C2311yescharacteristic
138#399—28no—
139#399—216no—
140#399—28no—
141#398—28no—
142#395—232no—
143#394—28no—
144#393—232no—
145#392—28no—
146#392—28no—
147#391—216no—
148#391D4116yescomplemented
149#390—28no—
150#388—216no—
151#386—232no—
152#385—232no—
153#384—28no—
154#382—24no—
155#380—216no—
156#380—28no—
157C4×C4⋊C4—24no—
158#376—24no—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2318yes—
162C2×#95C2318yes—
163C2×#95C2312yes—
164C2×#95C2318yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2318yes—
169C2×#95C2318yes—
170C25C2412yes—
171C25C2412yes—
172C22×Q8—264no—
173C22×Q8—22no—
174C22×Q8—22no—
175C22×Q8—28no—
176C22×Q8—28no—
177C22×Q8—28no—
178C22×Q8—24no—
179C22×Q8—24no—
180C22×Q8—28no—
181C22×Q8—216no—
182C22×Q8—24no—
183C22×Q8—28no—
184C22×Q8—48no—
185C22×Q8—464no—
186C22×Q8—464no—
187C22×Q8—432no—
188C22×Q8—432no—
189C22×Q8C2×D418yes—
190C22×Q8C2×D414yes—
191C22×Q8C2×D418yes—
192C22×D4—28no—
193C22×D4—24no—
194C22×D4—28no—
195C22×D4—28no—
196C22×D4—28no—
197C22×D4—28no—
198C22×D4—22no—
199C22×D4—22no—
200C22×D4—28no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—44no—
205C22×D4—42no—
206C22×D4—44no—
207C22×D4C2×D414yes—
208C23×C4—28no—
209C23×C4—28no—
210C23×C4—28no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—28no—
214C23×C4—28no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—28no—
218C23×C4—24no—
219C23×C4—216no—
220C23×C4—28no—
221C23×C4—48no—
222C23×C4C2×D418yes—
223C23×C4C2414yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2414yes—
227C23×C4C2412yes—
228C23×C4C2414yes—
229C23×C4C2414yes—
230C23×C4C2411yescharacteristic
231C23×C4C2412yes—
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C4⋊Q8—264no—
235C4⋊Q8—232no—
236C22⋊Q8—264no—
237C22⋊Q8—2128no—
238C22⋊Q8—464no—
239C22⋊Q8—432no—
240C4×Q8—232no—
241C4×D4—264no—
242C4×D4—232no—
243C4×D4—464no—
244C4×D4—416no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—216no—
268C2×C4⋊C4—216no—
269C2×C4⋊C4—216no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—48no—
273C2×C4⋊C4C2×D418yes—
274C2×C4⋊C4C2×D414yes—
275C2×C4⋊C4C2×D414yes—
276C2×C4⋊C4C2×D418yes—
277C2×C4⋊C4C2×D418yes—
278C2×C4⋊C4C2×D418yes—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—216no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—216no—
294C2×C22⋊C4—216no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—216no—
297C2×C22⋊C4—216no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—48no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—48no—
304C2×C22⋊C4C2×D418yes—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—44no—
311C2×C42C2×D418yes—
312C2×C42C2×D418yes—
313C2×C42C2×D414yes—
314C2×C42C2×D414yes—
315C2×C42C2×D414yes—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—28no—
320#95—216no—
321#95—24no—
322#95—24no—
323#95—216no—
324#95—216no—
325#95—48no—
326#95C2×D4132yes—
327C24—22no—
328C24—22no—
329C24—24no—
330C24—24no—
331C24—24no—
332C24—24no—
333C24—24no—
334C24—22no—
335C24—24no—
336C24—22no—
337C24—22no—
338C24—24no—
339C24—24no—
340C24—22no—
341C24—24no—
342C24—22no—
343C24—22no—
344C24C22×D414yes—
345C242−1+414yes—
346C242−1+418yes—
347C24C2511yescharacteristic
348C2×Q8—232no—
349C2×Q8—28no—
350C2×Q8—216no—
351C2×Q8—28no—
352C2×Q8—216no—
353C2×Q8—216no—
354C2×Q8—232no—
355C2×Q8—432no—
356C2×Q8—432no—
357C2×Q8—416no—
358C2×Q8—4256no—
359C2×Q8—416no—
360C2×Q8—48no—
361C2×Q8—48no—
362C2×Q8—416no—
363C2×Q8#127116yes—
364C2×D4—216no—
365C2×D4—216no—
366C2×D4—216no—
367C2×D4—28no—
368C2×D4—232no—
369C2×D4—28no—
370C2×D4—416no—
371C2×D4—48no—
372C2×D4—48no—
373C2×D4—416no—
374C2×D4—44no—
375C2×D4—416no—
376C2×D4—416no—
377C2×D4—416no—
378C2×D4—416no—
379C2×D4—44no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—232no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—28no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—28no—
401C22×C4—24no—
402C22×C4—28no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—28no—
409C22×C4—216no—
410C22×C4—232no—
411C22×C4—24no—
412C22×C4—28no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—28no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—432no—
422C22×C4—432no—
423C22×C4—416no—
424C22×C4—416no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—416no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4—416no—
432C22×C4—416no—
433C22×C4C22×D418yes—
434C22×C4C22×D418yes—
435C22×C4C22×D414yes—
436C22×C4C22×D418yes—
437C22×C4C22×D414yes—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D418yes—
445C22×C4C22×D414yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D414yes—
450C22×C4H(5,𝔽2)14yes—
451C4⋊C4—232no—
452C4⋊C4—216no—
453C4⋊C4—28no—
454C4⋊C4—216no—
455C4⋊C4—28no—
456C4⋊C4—216no—
457C4⋊C4—216no—
458C4⋊C4—416no—
459C4⋊C4—432no—
460C4⋊C4—44no—
461C4⋊C4—48no—
462C4⋊C4—416no—
463C22⋊C4—232no—
464C22⋊C4—216no—
465C22⋊C4—416no—
466C22⋊C4—432no—
467C22⋊C4—432no—
468C22⋊C4—416no—
469C22⋊C4—416no—
470C22⋊C4—48no—
471C22⋊C4—416no—
472C42—216no—
473C42—216no—
474C42—48no—
475C42#120116yes—
476C23—24no—
477C23—24no—
478C23—28no—
479C23—28no—
480C23—28no—
481C23—28no—
482C23—28no—
483C23—28no—
484C23—28no—
485C23—28no—
486C23—28no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—28no—
491C23—42no—
492C23—42no—
493C23—44no—
494C23—44no—
495C23—42no—
496C23—42no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—42no—
508C23—42no—
509C23—44no—
510C23—44no—
511C23—42no—
512C23—44no—
513C23—42no—
514C23—42no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23#53618yes—
520C23C23×D411yescharacteristic
521C23C23×D411yescharacteristic
522C23C23×D411yescharacteristic
523C23C23×D412yes—
524C23C2×H(5,𝔽2)12yes—
525C23C2×2−1+414yes—
526C23C2×2−1+412yes—
527C23#58512yes—
528Q8—232no—
529Q8—464no—
530Q8—432no—
531D4—232no—
532D4—416no—
533D4—464no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—28no—
538C2×C4—28no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—28no—
547C2×C4—24no—
548C2×C4—28no—
549C2×C4—28no—
550C2×C4—28no—
551C2×C4—216no—
552C2×C4—28no—
553C2×C4—28no—
554C2×C4—216no—
555C2×C4—28no—
556C2×C4—464no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—48no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—48no—
572C2×C4—44no—
573C2×C4—432no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—48no—
581C2×C4—44no—
582C2×C4—48no—
583C2×C4—44no—
584C2×C4—816no—
585C2×C4C2×#12014yes—
586C2×C4C2×#12014yes—
587C2×C4C2×#12014yes—
588C2×C4C2×#12718yes—
589C2×C4C2×#12714yes—
590C2×C4#53514yes—
591C2×C4D4218yes—
592C2×C4D4218yes—
593C22—216no—
594C22—28no—
595C22—48no—
596C22—48no—
597C22—44no—
598C22—48no—
599C22—48no—
600C22—44no—
601C22—48no—
602C22—48no—
603C22—44no—
604C22—44no—
605C22—44no—
606C22—48no—
607C22—84no—
608C22—82no—
609C22—82no—
610C22—82no—
611C22—82no—
612C22—84no—
613C22C22×#12011yescharacteristic
614C22C22×#12711yescharacteristic
615C22C2×#53511yescharacteristic
616C22C2×#53612yes—
617C22C2×D4212yes—
618C22C2×D4212yes—
619C22#348614yes—
620C22#348612yes—
621C22#348611yescharacteristic
622C22#348612yes—
623C22#348711yescharacteristic
624C22#348912yes—
625C22#348914yes—
626C22#348912yes—
627C22#353814yes—
628C22#354114yes—
629C4—24no—
630C4—28no—
631C4—24no—
632C4—48no—
633C4—44no—
634C4—44no—
635C4—48no—
636C4—48no—
637C4—84no—
638C4—82no—
639C4#243318yes—
640C2—48no—
641C2—84no—
642C2C2×#243311yescharacteristic
643C2#3524211yescharacteristic
644C2#3565911yescharacteristic
645C2#3592712yes—
646C2#3765114yes—
647C2#4280812yes—
648C2#4526712yes—
649C2#4541812yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2397C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×D4×Q8C2semidirect
2. C2×#2542C2semidirect
3. C2×#2536C2semidirect
4. C2×#2409C2semidirect
6. #16511C2semidirect
7. #15453C2semidirect
8. #15430C2semidirect
9. #15177C2semidirect
10. #14796C2semidirect
11. #14274C2semidirect
12. #13905C2semidirect
13. #13786C2semidirect
86. C2×C4⋊Q8D4semidirect
100. C2×C4×Q8D4semidirect
148. #391D4semidirect

Maximal subgroups

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

subgroupindex
C22×D4×Q82
C2×#25422
C2×#25362
C2×#24092
C2×#23972
#165112
#154532
#154302
#151772
#147962
#142742
#139052
#137862