Finite Group Databasebeta

← #391658

Subgroups — #391658

Order 512.

1454 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
1C4×#392C211yesmaximal, characteristic
2#11298C212yesmaximal, complemented
3#11147C212yesmaximal, complemented
4#11147C211yesmaximal, characteristic, complemented
5#10964C212yesmaximal, complemented
6#10964C212yesmaximal, complemented
7#10964C212yesmaximal, complemented
8#10922C212yesmaximal, complemented
9#10922C211yesmaximal, characteristic, complemented
10C22×C4×D4C2212yes—
11C22×C4×D4C2211yescharacteristic
12C22×C22⋊C8C2212yes—
13C22×C22⋊C8C2212yes—
14C22×C22⋊C8C2212yes—
15C22×C22⋊C8C2212yes—
16C22×C22⋊C8C2212yes—
17C22×C22⋊C8C2211yescharacteristic
18C22×C22⋊C8C2211yescharacteristic
19C22×#97C2211yescharacteristic
20C2×#392C411yescharacteristic
21C2×C4×C22⋊C4C2212yes—
22C2×C4×C22⋊C4C2211yescharacteristic
23C2×#336C2212yes—
24C2×#336C2212yes—
25C2×#336C2212yes—
26C2×#336C2212yes—
27C2×#336C2212yes—
28C2×#336C2212yes—
29C2×#336C2211yescharacteristic
30C2×#336C2212yes—
31#1444C411yescharacteristic
32#1439C412yes—
33#1439C411yescharacteristic
34#1438C412yes—
35#1438C411yescharacteristic
36C4×#95C2211yescharacteristic
37C23×D4C2×C412yes—
38C23×D4C2×C411yescharacteristic
39C24×C4C2312yes—
40C24×C4C2311yescharacteristic
41C23×C8C2312yes—
42C23×C8C2312yes—
43C23×C8C2312yes—
44C23×C8C2312yes—
45C2×C4×D4—42no—
46C2×C4×D4—42no—
47C2×C4×D4—42no—
48C2×C4×D4—42no—
49C2×C4×D4—42no—
50C2×C4×D4—42no—
51C2×C4×D4—41no—
52C2×C4×D4—42no—
53C2×C4×D4—41no—
54C2×C4×D4—42no—
55C22×C4⋊C4C2×C411yescharacteristic
56C22×C4⋊C4C2×C412yes—
57C22×C22⋊C4C2×C412yes—
58C22×C22⋊C4C2×C411yescharacteristic
59C22×C22⋊C4C2×C411yescharacteristic
60C22×C22⋊C4C2×C412yes—
61C22×C22⋊C4C2×C412yes—
62C22×C22⋊C4C2×C412yes—
63C22×C22⋊C4C2×C412yes—
64C22×C42C2311yescharacteristic
65C22×C42C2311yescharacteristic
66C22×C42C2312yes—
67C2×C22⋊C8—24no—
68C2×C22⋊C8—24no—
69C2×C22⋊C8—24no—
70C2×C22⋊C8—42no—
71C2×C22⋊C8—42no—
72C2×C22⋊C8—42no—
73C2×C22⋊C8—42no—
74C2×C22⋊C8—42no—
75C2×C22⋊C8—42no—
76C2×C22⋊C8—42no—
77C2×C22⋊C8—42no—
78C2×C22⋊C8—42no—
79C2×C22⋊C8—42no—
80C2×C22⋊C8—42no—
81C2×C22⋊C8—42no—
82C2×C22⋊C8—41no—
83C2×C22⋊C8—42no—
84C2×C22⋊C8—41no—
85C2×C22⋊C8—42no—
86C2×#97—41no—
87C2×#97—42no—
88#392—24no—
89C4×C22⋊C4—42no—
90C4×C22⋊C4—41no—
91C4×C22⋊C4—42no—
92C4×C22⋊C4—41no—
93C2×#95C2×C411yescharacteristic
94C2×#95C2×C412yes—
95C2×#95C2×C411yescharacteristic
96C2×#95C2×C411yescharacteristic
97C2×#95C2×C412yes—
98C2×#95C2×C411yescharacteristic
99C2×#95C2×C412yes—
100#336—42no—
101#336—42no—
102#336—42no—
103#336—42no—
104#336—42no—
105#336—42no—
106#336—41no—
107#336—42no—
108C25C22×C412yes—
109C25C22×C411yescharacteristic
110C22×D4—24no—
111C22×D4—24no—
112C22×D4—24no—
113C22×D4—41no—
114C22×D4—42no—
115C22×D4—42no—
116C22×D4—42no—
117C22×D4—42no—
118C22×D4—41no—
119C22×D4—42no—
120C22×D4—42no—
121C22×D4M1614yes—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—21no—
141C23×C4—22no—
142C23×C4—21no—
143C23×C4—22no—
144C23×C4C22×C411yescharacteristic
145C23×C4C22×C412yes—
146C23×C4C22×C411yescharacteristic
147C23×C4C22×C411yescharacteristic
148C23×C4C22×C411yescharacteristic
149C23×C4C22×C412yes—
150C23×C4C22×C411yescharacteristic
151C23×C4C22×C412yes—
152C23×C4C2411yescharacteristic
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
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#4114yes—
181C4×D4—48no—
182C4×D4—48no—
183C4×D4—48no—
184C4×D4—48no—
185C4×D4—44no—
186C4×D4—48no—
187C4×D4—44no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—41no—
198C2×C22⋊C4—41no—
199C2×C22⋊C4—42no—
200C2×C22⋊C4—42no—
201C2×C22⋊C4—42no—
202C2×C22⋊C4—42no—
203C2×C22⋊C4—42no—
204C2×C22⋊C4—42no—
205C2×C22⋊C4—42no—
206C2×C22⋊C4—42no—
207C2×C22⋊C4—42no—
208C2×C22⋊C4—42no—
209C2×C22⋊C4—42no—
210C2×C22⋊C4—42no—
211C2×C22⋊C4—41no—
212C2×C22⋊C4—41no—
213C2×C22⋊C4—42no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—21no—
221C2×C42—22no—
222C2×C42—21no—
223C2×C42—22no—
224C2×C42—21no—
225C2×C42—22no—
226C2×C42—21no—
227C2×C42—22no—
228C22⋊C8—48no—
229C22⋊C8—48no—
230C22⋊C8—48no—
231C22⋊C8—48no—
232C22⋊C8—44no—
233C22⋊C8—48no—
234C22⋊C8—44no—
235#97—44no—
236#95—22no—
237#95—24no—
238#95—41no—
239#95—41no—
240#95—42no—
241#95—41no—
242#95—42no—
243C24—21no—
244C24—22no—
245C24—22no—
246C24—21no—
247C24—22no—
248C24—21no—
249C24—22no—
250C24—22no—
251C24—22no—
252C24—22no—
253C24—22no—
254C24—22no—
255C24—21no—
256C24—22no—
257C24—22no—
258C24—22no—
259C24—22no—
260C24—22no—
261C24—21no—
262C24—22no—
263C24—21no—
264C24—22no—
265C24—22no—
266C24—22no—
267C24C2×M1614yes—
268C24#13112yes—
269C24C23×C411yescharacteristic
270C2×D4—44no—
271C2×D4—48no—
272C2×D4—48no—
273C2×D4—48no—
274C2×D4—48no—
275C2×D4—48no—
276C2×D4—48no—
277C2×D4—48no—
278C2×D4—48no—
279C2×D4—44no—
280C2×D4—48no—
281C2×D4—44no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
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—41no—
360C22×C4—42no—
361C22×C4—41no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—41no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4C2×M1612yes—
371C22×C4#13114yes—
372C22×C4C2×#4112yes—
373C22×C4H(5,𝔽2)11yescharacteristic
374C22×C4H(5,𝔽2)12yes—
375C22×C4H(5,𝔽2)12yes—
376C22×C4H(5,𝔽2)12yes—
377C22×C4H(5,𝔽2)12yes—
378C22×C4H(5,𝔽2)11yescharacteristic
379C22×C4H(5,𝔽2)12yes—
380C2×C8—24no—
381C2×C8—24no—
382C2×C8—24no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
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—
408C4⋊C4—48no—
409C4⋊C4—44no—
410C22⋊C4—48no—
411C22⋊C4—44no—
412C22⋊C4—44no—
413C22⋊C4—48no—
414C22⋊C4—48no—
415C22⋊C4—48no—
416C22⋊C4—48no—
417C42—41no—
418C42—41no—
419C42—42no—
420C42—42no—
421C42—42no—
422C42—42no—
423C42—42no—
424C42—41no—
425C42—42no—
426C42—41no—
427C23—22no—
428C23—24no—
429C23—24no—
430C23—24no—
431C23—24no—
432C23—24no—
433C23—22no—
434C23—24no—
435C23—24no—
436C23—22no—
437C23—24no—
438C23—24no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—41no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—41no—
449C23—42no—
450C23—41no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—41no—
456C23—41no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—41no—
473C23—42no—
474C23—42no—
475C23—41no—
476C23—42no—
477C23—41no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23C4×#4111yescharacteristic
486C23#51812yes—
487C23#51811yescharacteristic
488C23#51811yescharacteristic
489C23#51812yes—
490C23C22×M1611yescharacteristic
491C23C2×#13111yescharacteristic
492C23C2×#13112yes—
493C23#56811yescharacteristic
494C23#56812yes—
495C23#56811yescharacteristic
496D4—432no—
497D4—416no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4#53812yes—
557C2×C4#53812yes—
558C2×C4#53812yes—
559C2×C4#54012yes—
560C2×C4#54012yes—
561C2×C4#54012yes—
562C2×C4#56011yescharacteristic
563C2×C4#56012yes—
564C2×C4#56012yes—
565C2×C4#56012yes—
566C2×C4#56012yes—
567C2×C4#56011yescharacteristic
568C2×C4#56011yescharacteristic
569C2×C4#56012yes—
570C2×C4H(3,𝔽4)12yes—
571C2×C4H(3,𝔽4)11yescharacteristic
572C8—44no—
573C8—82no—
574C8—82no—
575C8—82no—
576C22—44no—
577C22—44no—
578C22—44no—
579C22—44no—
580C22—42no—
581C22—44no—
582C22—44no—
583C22—44no—
584C22—42no—
585C22—44no—
586C22—42no—
587C22—44no—
588C22—81no—
589C22—82no—
590C22—82no—
591C22—82no—
592C22—82no—
593C22—81no—
594C22—82no—
595C22—81no—
596C22—82no—
597C22—81no—
598C22—82no—
599C22—82no—
600C22—82no—
601C22—82no—
602C22#233311yescharacteristic
603C22#233312yes—
604C22#237512yes—
605C22#237511yescharacteristic
606C22#238011yescharacteristic
607C22#296811yescharacteristic
608C22#296912yes—
609C22#297011yescharacteristic
610C22#297312yes—
611C22#297311yescharacteristic
612C22#297411yescharacteristic
613C22#297412yes—
614C22#297612yes—
615C22#297612yes—
616C22#297611yescharacteristic
617C22#297712yes—
618C22#297812yes—
619C22#297811yescharacteristic
620C22#297812yes—
621C22#297911yescharacteristic
622C22#298011yescharacteristic
623C22#298112yes—
624C22#298112yes—
625C4—44no—
626C4—42no—
627C4—81no—
628C4—82no—
629C4—81no—
630C4—82no—
631C4—81no—
632C4—81no—
633C4#282512yes—
634C4#282812yes—
635C4#282812yes—
636C4#282812yes—
637C2—82no—
638C2—84no—
639C2#1465111yescharacteristic
640C2#2233211yescharacteristic
641C2#2233512yes—
642C2#2233612yes—
643C2#2233911yescharacteristic
644C2#2330712yes—
645C2#2330911yescharacteristic
646C2#2331011yescharacteristic
647C2#2331312yes—
648C2#2335611yescharacteristic
649C2#2335911yescharacteristic
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutator, socleC24C23×C4
prime radicalC2×#392C4
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #11298C2semidirect
3. #11147C2semidirect
4. #11147C2semidirect
5. #10964C2semidirect
6. #10964C2semidirect
7. #10964C2semidirect
8. #10922C2semidirect
9. #10922C2semidirect

Maximal subgroups

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

subgroupindex
C4×#3922
#112982
#111472
#109642
#109222