Finite Group Databasebeta

← #391184

Subgroups — #391184

Order 512.

1608 conjugacy classes of subgroups.

The subgroup lattice has 673 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1869C212yesmaximal, complemented
2C4×#392C211yesmaximal, characteristic
3#11298C212yesmaximal, complemented
4#11147C212yesmaximal, complemented
5#11042C211yesmaximal, characteristic, complemented
6#11042C212yesmaximal, complemented
7#10964C212yesmaximal, complemented
8#10964C212yesmaximal, complemented
9#10714C211yesmaximal, characteristic, complemented
10C24×C8C2211yescharacteristic
11C22×C4×D4C2212yes—
12C22×C4×D4C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×C22⋊C8C2212yes—
15C22×C22⋊C8C2212yes—
16C22×C22⋊C8C2212yes—
17C22×C22⋊C8C2211yescharacteristic
18C22×C22⋊C8C2212yes—
19C22×C22⋊C8C2212yes—
20C2×#392C411yescharacteristic
21C2×C4×C22⋊C4C2212yes—
22C2×C4×C22⋊C4C2211yescharacteristic
23#1869—28no—
24C2×#336C2212yes—
25C2×#336C2212yes—
26C2×#336C2212yes—
27C2×#336C2212yes—
28C2×#336C2211yescharacteristic
29C2×#336C2212yes—
30C2×#336C2212yes—
31C2×#336C2212yes—
32#1444C411yescharacteristic
33#1439C411yescharacteristic
34#1439C412yes—
35#1438C411yescharacteristic
36#1438C412yes—
37C4×#95C2211yescharacteristic
38C23×D4C2×C412yes—
39C23×D4C2×C411yescharacteristic
40C24×C4C2312yes—
41C24×C4C2311yescharacteristic
42C23×C8—42no—
43C23×C8—42no—
44C23×C8—41no—
45C23×C8—41no—
46C23×C8D414yescomplemented
47C23×C8C2312yes—
48C23×C8C2312yes—
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×C4×D4—24no—
52C2×C4×D4—24no—
53C2×C4×D4—42no—
54C2×C4×D4—42no—
55C2×C4×D4—42no—
56C2×C4×D4—42no—
57C2×C4×D4—42no—
58C2×C4×D4—42no—
59C2×C4×D4—41no—
60C2×C4×D4—41no—
61C22×C4⋊C4C2×C412yes—
62C22×C4⋊C4C2×C411yescharacteristic
63C22×C22⋊C4C2×C412yes—
64C22×C22⋊C4C2×C412yes—
65C22×C22⋊C4C2×C411yescharacteristic
66C22×C22⋊C4C2×C412yes—
67C22×C22⋊C4C2×C412yes—
68C22×C22⋊C4C2×C411yescharacteristic
69C22×C22⋊C4C2×C412yes—
70C22×C42C2311yescharacteristic
71C22×C42C2312yes—
72C22×C42C2311yescharacteristic
73C2×C4⋊C8—41no—
74C2×C4⋊C8—42no—
75C2×C22⋊C8—24no—
76C2×C22⋊C8—22no—
77C2×C22⋊C8—24no—
78C2×C22⋊C8—24no—
79C2×C22⋊C8—42no—
80C2×C22⋊C8—42no—
81C2×C22⋊C8—42no—
82C2×C22⋊C8—42no—
83C2×C22⋊C8—42no—
84C2×C22⋊C8—41no—
85C2×C22⋊C8—41no—
86C2×C22⋊C8—42no—
87C2×C22⋊C8—42no—
88C2×C22⋊C8—42no—
89C2×C22⋊C8—42no—
90C2×C22⋊C8—42no—
91C2×C22⋊C8—42no—
92C2×C22⋊C8—42no—
93#392C818yescomplemented
94C4×C22⋊C4—41no—
95C4×C22⋊C4—42no—
96C4×C22⋊C4—41no—
97C4×C22⋊C4—42no—
98C2×#95C2×C411yescharacteristic
99C2×#95C2×C412yes—
100C2×#95C2×C412yes—
101C2×#95C2×C411yescharacteristic
102C2×#95C2×C412yes—
103C2×#95C2×C411yescharacteristic
104C2×#95C2×C411yescharacteristic
105#336—24no—
106#336—24no—
107#336—42no—
108#336—42no—
109#336—42no—
110#336—42no—
111#336—42no—
112#336—41no—
113C25C22×C412yes—
114C25C22×C411yescharacteristic
115C22×D4—24no—
116C22×D4—24no—
117C22×D4—42no—
118C22×D4—41no—
119C22×D4—42no—
120C22×D4—42no—
121C22×D4—42no—
122C22×D4—41no—
123C22×D4—42no—
124C22×D4—42no—
125C22×D4C2×C818yes—
126C22×D4C2×C814yescomplemented
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4C22×C412yes—
150C23×C4C22×C411yescharacteristic
151C23×C4C22×C411yescharacteristic
152C23×C4C22×C412yes—
153C23×C4C22×C411yescharacteristic
154C23×C4C22×C411yescharacteristic
155C23×C4C22×C412yes—
156C23×C4C22×C411yescharacteristic
157C23×C4C2×D412yes—
158C23×C4#4112yes—
159C23×C4C2411yescharacteristic
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—44no—
188C22×C8—48no—
189C22×C8—48no—
190C22×C8—44no—
191C22×C8—44no—
192C22×C8C2×D414yes—
193C4×D4—48no—
194C4×D4—48no—
195C4×D4—48no—
196C4×D4—48no—
197C4×D4—48no—
198C4×D4—44no—
199C4×D4—44no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—42no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—41no—
209C2×C22⋊C4—42no—
210C2×C22⋊C4—41no—
211C2×C22⋊C4—42no—
212C2×C22⋊C4—42no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—42no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4—42no—
217C2×C22⋊C4—42no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—41no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—41no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4C2×C814yes—
225C2×C22⋊C4C2×C818yescomplemented
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—22no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—22no—
240C4⋊C8—44no—
241C22⋊C8—48no—
242C22⋊C8—48no—
243C22⋊C8—48no—
244C22⋊C8—44no—
245C22⋊C8—48no—
246C22⋊C8—48no—
247#95—42no—
248#95—41no—
249#95—41no—
250#95—41no—
251#95—42no—
252#95—42no—
253#95C2×C814yescomplemented
254C24—21no—
255C24—22no—
256C24—22no—
257C24—21no—
258C24—21no—
259C24—21no—
260C24—22no—
261C24—22no—
262C24—22no—
263C24—22no—
264C24—22no—
265C24—22no—
266C24—22no—
267C24—22no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24C4×D412yes—
277C24C22×C814yes—
278C24C22×C812yes—
279C24#13112yes—
280C24C23×C411yescharacteristic
281C2×D4—216no—
282C2×D4—48no—
283C2×D4—48no—
284C2×D4—44no—
285C2×D4—48no—
286C2×D4—48no—
287C2×D4—48no—
288C2×D4—48no—
289C2×D4—48no—
290C2×D4—44no—
291C2×D4—48no—
292C2×D4—44no—
293C22×C4—22no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—42no—
344C22×C4—41no—
345C22×C4—42no—
346C22×C4—41no—
347C22×C4—42no—
348C22×C4—41no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4C4×D412yes—
382C22×C4C22×C812yes—
383C22×C4C22×C814yes—
384C22×C4C22×C812yes—
385C22×C4C22×D411yescharacteristic
386C22×C4C2×#4111yescharacteristic
387C22×C4H(5,𝔽2)11yescharacteristic
388C22×C4H(5,𝔽2)12yes—
389C22×C4H(5,𝔽2)12yes—
390C22×C4H(5,𝔽2)11yescharacteristic
391C22×C4H(5,𝔽2)12yes—
392C22×C4H(5,𝔽2)12yes—
393C22×C4H(5,𝔽2)11yescharacteristic
394C22×C42−1+411yescharacteristic
395C2×C8—24no—
396C2×C8—24no—
397C2×C8—24no—
398C2×C8—416no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8—42no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8—42no—
419C2×C8—42no—
420C2×C8—42no—
421C2×C8—42no—
422C2×C8—42no—
423C2×C8—42no—
424C4⋊C4—48no—
425C4⋊C4—44no—
426C22⋊C4—44no—
427C22⋊C4—48no—
428C22⋊C4—48no—
429C22⋊C4—44no—
430C22⋊C4—48no—
431C22⋊C4—48no—
432C22⋊C4—48no—
433C42—42no—
434C42—42no—
435C42—42no—
436C42—42no—
437C42—41no—
438C42—42no—
439C42—41no—
440C42—42no—
441C42—41no—
442C42—41no—
443C23—24no—
444C23—22no—
445C23—24no—
446C23—22no—
447C23—22no—
448C23—24no—
449C23—24no—
450C23—22no—
451C23—24no—
452C23—24no—
453C23—22no—
454C23—22no—
455C23—24no—
456C23—24no—
457C23—24no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—24no—
462C23—41no—
463C23—42no—
464C23—41no—
465C23—42no—
466C23—41no—
467C23—42no—
468C23—42no—
469C23—41no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23C8×D414yes—
501C23C2×C4×D411yescharacteristic
502C23#51812yes—
503C23#51812yes—
504C23#51811yescharacteristic
505C23#52011yescharacteristic
506C23C23×C811yescharacteristic
507C23C2×#13111yescharacteristic
508C23#56812yes—
509C23#56812yes—
510C23#56811yescharacteristic
511C23#56811yescharacteristic
512D4—432no—
513D4—416no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4#53412yes—
575C2×C4#53411yescharacteristic
576C2×C4#53512yes—
577C2×C4#53711yescharacteristic
578C2×C4#53812yes—
579C2×C4#53811yescharacteristic
580C2×C4#53812yes—
581C2×C4#53911yescharacteristic
582C2×C4#56011yescharacteristic
583C2×C4#56012yes—
584C2×C4#56012yes—
585C2×C4#56012yes—
586C2×C4#56012yes—
587C2×C4#56011yescharacteristic
588C2×C4#56011yescharacteristic
589C2×C4H(3,𝔽4)11yescharacteristic
590C2×C4#56212yes—
591C2×C4#56212yes—
592C8—44no—
593C8—82no—
594C8—82no—
595C8—82no—
596C22—28no—
597C22—24no—
598C22—42no—
599C22—42no—
600C22—44no—
601C22—42no—
602C22—44no—
603C22—44no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—44no—
608C22—44no—
609C22—44no—
610C22—44no—
611C22—44no—
612C22—44no—
613C22—44no—
614C22—82no—
615C22—82no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22#232112yes—
623C22#232511yescharacteristic
624C22#237511yescharacteristic
625C22#237712yes—
626C22#238011yescharacteristic
627C22C2×C8×D411yescharacteristic
628C22#293012yes—
629C22#293211yescharacteristic
630C22#297212yes—
631C22#297211yescharacteristic
632C22#297212yes—
633C22#297612yes—
634C22#297612yes—
635C22#297611yescharacteristic
636C22#297812yes—
637C22#297811yescharacteristic
638C22#297812yes—
639C22#298012yes—
640C22#298011yescharacteristic
641C22#298012yes—
642C22#298811yescharacteristic
643C22#299211yescharacteristic
644C22#299512yes—
645C4—42no—
646C4—44no—
647C4—42no—
648C4—42no—
649C4—81no—
650C4—82no—
651C4—82no—
652C4—81no—
653C4#281211yescharacteristic
654C4#281312yes—
655C4#281511yescharacteristic
656C4#282411yescharacteristic
657C4#282512yes—
658C4#282611yescharacteristic
659C2—44no—
660C2—84no—
661C2#1462111yescharacteristic
662C2#2202812yes—
663C2#2203512yes—
664C2#2216811yescharacteristic
665C2#2217711yescharacteristic
666C2#2330611yescharacteristic
667C2#2331111yescharacteristic
668C2#2332612yes—
669C2#2333512yes—
670C2#2335511yescharacteristic
671C2#2335911yescharacteristic
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutatorC23C23×C8
socleC24C23×C4
prime radicalC2×#392C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#1869C2semidirect
3. #11298C2semidirect
4. #11147C2semidirect
5. #11042C2semidirect
6. #11042C2semidirect
7. #10964C2semidirect
8. #10964C2semidirect
9. #10714C2semidirect
46. C23×C8D4semidirect
93. #392C8semidirect
126. C22×D4C2×C8semidirect
225. C2×C22⋊C4C2×C8semidirect
253. #95C2×C8semidirect

Maximal subgroups

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

subgroupindex
C2×#18692
C4×#3922
#112982
#111472
#110422
#109642
#107142