Finite Group Databasebeta

← #364112

Subgroups — #364112

Order 512.

1760 conjugacy classes of subgroups.

The subgroup lattice has 657 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2293C211yesmaximal, characteristic
2C2×#1870C211yesmaximal, characteristic
3C2×#1869C211yesmaximal, characteristic
4C2×#1798C211yesmaximal, characteristic, complemented
5C2×#1729C211yesmaximal, characteristic, complemented
6#11202C212yesmaximal, complemented
7#11042C212yesmaximal, complemented
8#10795C212yesmaximal, complemented
9#10715C211yesmaximal, characteristic, complemented
10#10714C211yesmaximal, characteristic, complemented
11#10713C212yesmaximal, complemented
12C23×M16C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C22×#117C2211yescharacteristic
17C22×C4⋊C8C2212yes—
18C22×C22⋊C8C2211yescharacteristic
19C22×C22⋊C8C2211yescharacteristic
20C22×C22⋊C8C2212yes—
21C22×C22⋊C8C2212yes—
22#2293—24no—
23#2293—24no—
24C2×#390C412yes—
25C2×#389C412yes—
26C2×#388C412yes—
27C2×#388C412yes—
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C22⋊C4C2211yescharacteristic
30C2×C4×C22⋊C4C2211yescharacteristic
31C2×C43C2211yescharacteristic
32#1870—24no—
33#1869—24no—
34#1798—24no—
35#1729—24no—
36C2×#336C2212yes—
37C2×#336C2212yes—
38C2×#336C2212yes—
39C2×#336C2212yes—
40C2×#336C2212yes—
41C2×#336C2212yes—
42C2×#336C2212yes—
43C2×#336C2212yes—
44C2×#336C2212yes—
45C23×D4C2×C411yescharacteristic
46C24×C4C2311yescharacteristic
47C24×C4C2311yescharacteristic
48C22×M16—22no—
49C22×M16—22no—
50C22×M16—22no—
51C22×M16—24no—
52C22×M16—42no—
53C23×C8—22no—
54C23×C8—24no—
55C23×C8—22no—
56C23×C8—22no—
57C23×C8—42no—
58C23×C8C2312yes—
59C23×C8C2312yes—
60C23×C8C2312yes—
61C23×C8C2312yes—
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4—22no—
66C2×C4×D4D414yes—
67C2×C4×D4Q814yes—
68C2×#117—22no—
69C2×#117—22no—
70C2×#117—42no—
71C2×#117—42no—
72C2×#117—42no—
73C2×#117—42no—
74C2×#117D414yes—
75C2×#117D414yes—
76C22×C4⋊C4C2×C412yes—
77C22×C4⋊C4C2×C411yescharacteristic
78C22×C4⋊C4C2×C412yes—
79C22×C4⋊C4C2×C412yes—
80C22×C22⋊C4C2×C412yes—
81C22×C22⋊C4C2×C412yes—
82C22×C22⋊C4C2×C411yescharacteristic
83C22×C22⋊C4C2×C411yescharacteristic
84C22×C22⋊C4C2×C412yes—
85C22×C22⋊C4C2×C412yes—
86C22×C42C2×C412yes—
87C22×C42C2×C412yes—
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91C22×C42C2311yescharacteristic
92C22×C42C2311yescharacteristic
93C2×C4⋊C8—24no—
94C2×C4⋊C8—42no—
95C2×C4⋊C8—42no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—22no—
98C2×C22⋊C8—24no—
99C2×C22⋊C8—22no—
100C2×C22⋊C8—24no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8—24no—
105C2×C22⋊C8—24no—
106C2×C22⋊C8—42no—
107C2×C22⋊C8—42no—
108#390—28no—
109#389—28no—
110#388—28no—
111#388—28no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C4×C22⋊C4—22no—
115C4×C22⋊C4—22no—
116C4×C22⋊C4—22no—
117C4×C22⋊C4—22no—
118C2×#95C2×C412yes—
119C2×#95C2×C412yes—
120C2×#95C2×C412yes—
121C43—22no—
122C43—22no—
123#336—24no—
124#336—24no—
125#336—24no—
126#336—24no—
127#336—24no—
128#336—24no—
129#336—24no—
130#336—42no—
131#336—42no—
132C25C22×C411yescharacteristic
133C25C22×C411yescharacteristic
134C22×D4—22no—
135C22×D4—22no—
136C22×D4—24no—
137C22×D4—42no—
138C22×D4C4⋊C414yes—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4C22×C412yes—
152C23×C4C22×C412yes—
153C23×C4C22×C412yes—
154C23×C4C22×C411yescharacteristic
155C23×C4C22×C411yescharacteristic
156C23×C4C22×C412yes—
157C23×C4C22×C411yescharacteristic
158C23×C4C22×C411yescharacteristic
159C23×C4C2×D412yes—
160C23×C4C2×D412yes—
161C23×C4C2×D412yes—
162C23×C4C2×D412yes—
163C23×C4C2×Q812yes—
164C23×C4C2×Q812yes—
165C23×C4#4112yes—
166C23×C4#4112yes—
167C23×C4C2411yescharacteristic
168C2×M16—28no—
169C2×M16—48no—
170C2×M16—48no—
171C2×M16—44no—
172C2×M16—44no—
173C22×C8—28no—
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—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—48no—
201C22×C8—48no—
202C22×C8—44no—
203C22×C8—44no—
204C22×C8#4114yes—
205C22×C8#4114yes—
206C4×D4—28no—
207C4×D4—28no—
208C4×D4—28no—
209C4×D4—28no—
210#117—48no—
211#117—44no—
212#117—44no—
213#117—48no—
214#117—44no—
215#117—44no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4C22⋊C418yes—
227C2×C4⋊C4C4⋊C414yes—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4C22⋊C418yes—
244C2×C22⋊C4C4⋊C414yes—
245C2×C22⋊C4C4⋊C414yes—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—24no—
255C2×C42—24no—
256C2×C42—24no—
257C2×C42—24no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—42no—
267C2×C42—42no—
268C2×C42C2×D412yes—
269C2×C42C2×D412yes—
270C2×C42C2×D412yes—
271C2×C42C2×D412yes—
272C2×C42C2×D412yes—
273C2×C42C2×Q812yes—
274C4⋊C8—48no—
275C22⋊C8—48no—
276C22⋊C8—48no—
277C22⋊C8—44no—
278C22⋊C8—44no—
279#95—24no—
280#95—24no—
281#95—24no—
282C24—22no—
283C24—22no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24—21no—
289C24—21no—
290C24—22no—
291C24—22no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24C2×C22⋊C412yes—
297C24C2×C4⋊C412yes—
298C24C2×C4⋊C412yes—
299C24#11712yes—
300C24C23×C411yescharacteristic
301C2×D4—28no—
302C2×D4—28no—
303C2×D4—44no—
304C2×D4—48no—
305C2×D4—48no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4C2×C22⋊C414yes—
377C22×C4C2×C22⋊C414yes—
378C22×C4C2×C22⋊C412yes—
379C22×C4C2×C4⋊C412yes—
380C22×C4C2×C4⋊C412yes—
381C22×C4C2×C4⋊C412yes—
382C22×C4C2×C4⋊C412yes—
383C22×C4#11712yes—
384C22×C4#12114yes—
385C22×C4#12114yes—
386C22×C4C22⋊Q814yes—
387C22×C4C22⋊Q814yes—
388C22×C4#13114yes—
389C22×C4#13114yes—
390C22×C4C22×D411yescharacteristic
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C22×Q811yescharacteristic
394C22×C4C2×#4111yescharacteristic
395C22×C4C2×#4111yescharacteristic
396C22×C4C2×#4112yes—
397C22×C4C2×#4112yes—
398C22×C4H(5,𝔽2)12yes—
399C22×C4H(5,𝔽2)12yes—
400M16—416no—
401C2×C8—24no—
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—24no—
405C2×C8—24no—
406C2×C8—24no—
407C2×C8—416no—
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—
424C2×C8—42no—
425C2×C8—42no—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—42no—
430C4⋊C4—28no—
431C4⋊C4—48no—
432C4⋊C4—48no—
433C4⋊C4—48no—
434C22⋊C4—28no—
435C22⋊C4—28no—
436C22⋊C4—48no—
437C22⋊C4—48no—
438C22⋊C4—48no—
439C22⋊C4—48no—
440C42—22no—
441C42—22no—
442C42—22no—
443C42—22no—
444C42—24no—
445C42—22no—
446C42—22no—
447C42—24no—
448C42—22no—
449C42—22no—
450C42—48no—
451C42—48no—
452C42—41no—
453C42—41no—
454C42—42no—
455C42—42no—
456C42—41no—
457C42—41no—
458C23—22no—
459C23—24no—
460C23—24no—
461C23—24no—
462C23—24no—
463C23—22no—
464C23—22no—
465C23—24no—
466C23—22no—
467C23—22no—
468C23—24no—
469C23—22no—
470C23—24no—
471C23—22no—
472C23—22no—
473C23—24no—
474C23—22no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—41no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—41no—
488C23—41no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—41no—
499C23#38014yes—
500C23C22×C22⋊C411yescharacteristic
501C23C22×C4⋊C411yescharacteristic
502C23C2×#11711yescharacteristic
503C23C4×#4112yes—
504C23#51812yes—
505C23C2×#13112yes—
506C23C2×#13112yes—
507C23#56812yes—
508C23#56812yes—
509D4—416no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—28no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—28no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4#38014yes—
569C2×C4#42414yes—
570C2×C4C2×#12111yescharacteristic
571C2×C4C2×#12111yescharacteristic
572C2×C4C2×C22⋊Q811yescharacteristic
573C2×C4C2×C22⋊Q811yescharacteristic
574C2×C4#52511yescharacteristic
575C2×C4#52511yescharacteristic
576C2×C4#53212yes—
577C2×C4#53412yes—
578C2×C4#53411yescharacteristic
579C2×C4#53411yescharacteristic
580C2×C4#54612yes—
581C2×C4#54612yes—
582C2×C4#55112yes—
583C2×C4#55112yes—
584C2×C4#55312yes—
585C2×C4#55312yes—
586C2×C4#55412yes—
587C2×C4#55412yes—
588C8—44no—
589C8—44no—
590C8—82no—
591C8—82no—
592C22—24no—
593C22—24no—
594C22—24no—
595C22—44no—
596C22—44no—
597C22—44no—
598C22—44no—
599C22—44no—
600C22—44no—
601C22—42no—
602C22—42no—
603C22—42no—
604C22—44no—
605C22—44no—
606C22—82no—
607C22—82no—
608C22—82no—
609C22—82no—
610C22C2×#38011yescharacteristic
611C22#229711yescharacteristic
612C22#230811yescharacteristic
613C22#234912yes—
614C22#235312yes—
615C22#288012yes—
616C22#288012yes—
617C22C2×#42411yescharacteristic
618C22#290511yescharacteristic
619C22#290612yes—
620C22#292211yescharacteristic
621C22#292311yescharacteristic
622C22#292412yes—
623C22#296612yes—
624C22#296712yes—
625C22#296812yes—
626C22#296912yes—
627C22#297912yes—
628C22#298012yes—
629C22#298112yes—
630C22#298112yes—
631C4—24no—
632C4—24no—
633C4—44no—
634C4—42no—
635C4—42no—
636C4—44no—
637C4—44no—
638C4—82no—
639C4#253411yescharacteristic
640C4#253411yescharacteristic
641C4#254312yes—
642C4#256412yes—
643C4#257911yescharacteristic
644C4#257911yescharacteristic
645C2—44no—
646C2—82no—
647C2#1396011yescharacteristic
648C2#2062811yescharacteristic
649C2#2063011yescharacteristic
650C2#2138112yes—
651C2#2174512yes—
652C2#2270112yes—
653C2#2270212yes—
654C2#2278912yes—
655C2#2279012yes—
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutator, socleC24C23×C4
prime radicalC23×D4C2×C4
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
4. C2×#1798C2semidirect
5. C2×#1729C2semidirect
6. #11202C2semidirect
7. #11042C2semidirect
8. #10795C2semidirect
9. #10715C2semidirect
10. #10714C2semidirect
11. #10713C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22932
C2×#18702
C2×#18692
C2×#17982
C2×#17292
#112022
#110422
#107952
#107152
#107142
#107132