Finite Group Databasebeta

← #382832

Subgroups — #382832

Order 512.

1600 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2290C211yesmaximal, characteristic
2C2×#1840C211yesmaximal, characteristic, complemented
3C2×#1832C211yesmaximal, characteristic, complemented
4#11346C212yesmaximal, complemented
5#11042C212yesmaximal, complemented
6#11042C212yesmaximal, complemented
7#10964C212yesmaximal
8#10713C211yesmaximal, characteristic, complemented
9#10713C211yesmaximal, characteristic, complemented
10#10712C211yesmaximal, characteristic, complemented
11#10712C211yesmaximal, characteristic, complemented
12C22×C4×D4C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C42C2211yescharacteristic
15C22×C4⋊C8C2212yes—
16C22×C22⋊C8C2212yes—
17C22×C22⋊C8C2212yes—
18C22×C22⋊C8C2212yes—
19C22×C22⋊C8C2212yes—
20#2290—24no—
21#2290—24no—
22C2×#388C412yes—
23C2×#387C412yes—
24C2×#386C412yes—
25C2×#385C412yes—
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C22⋊C4C2211yescharacteristic
28C2×C4×C22⋊C4C2211yescharacteristic
29C2×#376C2211yescharacteristic
30#1840—24no—
31#1832—24no—
32C2×#336C2212yes—
33C2×#336C2211yescharacteristic
34C2×#336C2211yescharacteristic
35C2×#336C2212yes—
36C2×#336C2212yes—
37C2×#336C2211yescharacteristic
38C2×#336C2211yescharacteristic
39C2×#336C2212yes—
40C2×#336C2212yes—
41C2×#336C2212yes—
42C2×#336C2212yes—
43C23×D4C2×C411yescharacteristic
44C24×C4C2×C412yes—
45C24×C4C2311yescharacteristic
46C24×C4C2311yescharacteristic
47C23×C8C2312yes—
48C23×C8C2312yes—
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×C4×D4—24no—
52C2×C4×D4—24no—
53C2×C4×D4—22no—
54C2×C4×D4—22no—
55C2×#117—22no—
56C2×#117—22no—
57C2×#117—42no—
58C2×#117—42no—
59C22×C4⋊C4C2×C411yescharacteristic
60C22×C4⋊C4C2×C412yes—
61C22×C4⋊C4C2×C412yes—
62C22×C22⋊C4C2×C411yescharacteristic
63C22×C22⋊C4C2×C411yescharacteristic
64C22×C22⋊C4C2×C412yes—
65C22×C22⋊C4C2×C412yes—
66C22×C22⋊C4C2×C412yes—
67C22×C42—22no—
68C22×C42—22no—
69C22×C42—42no—
70C22×C42—42no—
71C22×C42C2×C412yes—
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C2×C4⋊C8—24no—
78C2×C4⋊C8—42no—
79C2×C4⋊C8—42no—
80C2×C22⋊C8—24no—
81C2×C22⋊C8—24no—
82C2×C22⋊C8—24no—
83C2×C22⋊C8—24no—
84C2×C22⋊C8—42no—
85C2×C22⋊C8—42no—
86C2×C22⋊C8—42no—
87C2×C22⋊C8—42no—
88C2×C22⋊C8—42no—
89C2×C22⋊C8—42no—
90C2×C22⋊C8—42no—
91C2×C22⋊C8—42no—
92#388—28no—
93#387—28no—
94#386—28no—
95#385—28no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4—22no—
98C4×C22⋊C4—22no—
99C4×C22⋊C4—22no—
100C4×C22⋊C4—22no—
101C4×C22⋊C4—22no—
102#376D414yes—
103#376D414yes—
104C2×#95C2×C412yes—
105C2×#95C2×C412yes—
106C2×#95C2×C412yes—
107C2×#95C2×C412yes—
108C2×#95C2×C412yes—
109#336—22no—
110#336—22no—
111#336—24no—
112#336—22no—
113#336—22no—
114#336—24no—
115#336—24no—
116#336—42no—
117#336—42no—
118#336—42no—
119#336—42no—
120C25C22×C411yescharacteristic
121C25C22×C411yescharacteristic
122C22×D4—22no—
123C22×D4—22no—
124C22×D4—24no—
125C22×D4—22no—
126C22×D4—42no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—24no—
146C23×C4—24no—
147C23×C4—24no—
148C23×C4—42no—
149C23×C4—42no—
150C23×C4—42no—
151C23×C4—42no—
152C23×C4C22×C412yes—
153C23×C4C22×C411yescharacteristic
154C23×C4C22×C412yes—
155C23×C4C22×C412yes—
156C23×C4C22×C411yescharacteristic
157C23×C4C22×C411yescharacteristic
158C23×C4C22×C411yescharacteristic
159C23×C4C22×C412yes—
160C23×C4#4114yes—
161C23×C4#4114yes—
162C23×C4C2411yescharacteristic
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—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C4×D4—216no—
192C4×D4—216no—
193#117—48no—
194#117—48no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4—42no—
217C2×C22⋊C4—42no—
218C2×C22⋊C4—42no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—24no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—48no—
237C2×C42—48no—
238C2×C42—42no—
239C2×C42—42no—
240C2×C42C2×D412yes—
241C2×C42C2×D412yes—
242C2×C42C2×D412yes—
243C2×C42C2×D412yes—
244C2×C42C2×D412yes—
245C2×C42C2×D412yes—
246C4⋊C8—48no—
247C22⋊C8—48no—
248C22⋊C8—48no—
249C22⋊C8—48no—
250C22⋊C8—48no—
251#95—24no—
252#95—24no—
253#95—24no—
254#95C22⋊C418yes—
255#95C22⋊C418yes—
256C24—21no—
257C24—21no—
258C24—21no—
259C24—21no—
260C24—21no—
261C24—21no—
262C24—22no—
263C24—22no—
264C24—22no—
265C24—21no—
266C24—21no—
267C24—21no—
268C24—21no—
269C24—21no—
270C24—21no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24#11712yes—
276C24#11712yes—
277C24#13114yes—
278C24C23×C411yescharacteristic
279C2×D4—28no—
280C2×D4—48no—
281C2×D4—48no—
282C2×D4—44no—
283C2×D4—44no—
284C22×C4—22no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—48no—
342C22×C4—48no—
343C22×C4—48no—
344C22×C4—48no—
345C22×C4—48no—
346C22×C4—48no—
347C22×C4—48no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
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×C4C2×C22⋊C414yes—
369C22×C4C2×C22⋊C412yes—
370C22×C4C2×C22⋊C414yes—
371C22×C4C2×C22⋊C412yes—
372C22×C4#11712yes—
373C22×C4#11712yes—
374C22×C4#13114yes—
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D411yescharacteristic
377C22×C4C2×#4112yes—
378C22×C4C2×#4112yes—
379C22×C4H(5,𝔽2)12yes—
380C22×C4H(5,𝔽2)12yes—
381C22×C4H(5,𝔽2)11yescharacteristic
382C22×C4H(5,𝔽2)11yescharacteristic
383C22×C42−1+411yescharacteristic
384C22×C42−1+411yescharacteristic
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—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8—42no—
413C4⋊C4—28no—
414C4⋊C4—48no—
415C4⋊C4—48no—
416C22⋊C4—28no—
417C22⋊C4—28no—
418C22⋊C4—48no—
419C22⋊C4—48no—
420C22⋊C4—48no—
421C42—24no—
422C42—24no—
423C42—22no—
424C42—24no—
425C42—24no—
426C42—22no—
427C42—22no—
428C42—22no—
429C42—48no—
430C42—42no—
431C42—42no—
432C42—42no—
433C42—42no—
434C23—22no—
435C23—22no—
436C23—24no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—24no—
441C23—24no—
442C23—24no—
443C23—24no—
444C23—24no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—24no—
449C23—24no—
450C23—22no—
451C23—41no—
452C23—41no—
453C23—41no—
454C23—41no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—42no—
466C23—42no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23#43314yes—
480C23C22×C22⋊C411yescharacteristic
481C23C2×#11711yescharacteristic
482C23C2×#11711yescharacteristic
483C23#51811yescharacteristic
484C23#51811yescharacteristic
485C23#52011yescharacteristic
486C23#52011yescharacteristic
487C23C2×#13112yes—
488C23C2×#13112yes—
489C23#56812yes—
490C23#56812yes—
491D4—416no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—432no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4#52512yes—
554C2×C4#52512yes—
555C2×C4#53511yescharacteristic
556C2×C4#53512yes—
557C2×C4#53512yes—
558C2×C4#53611yescharacteristic
559C2×C4#53711yescharacteristic
560C2×C4#53711yescharacteristic
561C2×C4#53812yes—
562C2×C4#53812yes—
563C2×C4#53812yes—
564C2×C4#53812yes—
565C2×C4#53912yes—
566C2×C4#53912yes—
567C2×C4#53912yes—
568C2×C4#53912yes—
569C8—82no—
570C8—82no—
571C8—82no—
572C8—82no—
573C22—24no—
574C22—24no—
575C22—24no—
576C22—42no—
577C22—44no—
578C22—44no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—44no—
583C22—44no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—44no—
588C22—44no—
589C22—82no—
590C22—82no—
591C22—81no—
592C22—81no—
593C22—81no—
594C22—81no—
595C22#228011yescharacteristic
596C22#230911yescharacteristic
597C22#231111yescharacteristic
598C22#231811yescharacteristic
599C22#231811yescharacteristic
600C22#231811yescharacteristic
601C22#231811yescharacteristic
602C22#288012yes—
603C22#288012yes—
604C22C2×#43311yescharacteristic
605C22#292111yescharacteristic
606C22#292211yescharacteristic
607C22#292311yescharacteristic
608C22#292412yes—
609C22#292512yes—
610C22#297612yes—
611C22#297612yes—
612C22#297712yes—
613C22#297812yes—
614C22#299212yes—
615C22#299312yes—
616C22#299412yes—
617C22#299512yes—
618C4—42no—
619C4—42no—
620C4—44no—
621C4—44no—
622C4—44no—
623C4—42no—
624C4—42no—
625C4—82no—
626C4#263112yes—
627C4#263312yes—
628C4#263612yes—
629C4#263612yes—
630C2—44no—
631C2—82no—
632C2#1434511yescharacteristic
633C2#2071611yescharacteristic
634C2#2072011yescharacteristic
635C2#2179212yes—
636C2#2185512yes—
637C2#2199711yescharacteristic
638C2#2199911yescharacteristic
639C2#2200111yescharacteristic
640C2#2200311yescharacteristic
641C2#2200412yes—
642C2#2200612yes—
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
2. C2×#1840C2semidirect
3. C2×#1832C2semidirect
4. #11346C2semidirect
5. #11042C2semidirect
6. #11042C2semidirect
8. #10713C2semidirect
9. #10713C2semidirect
10. #10712C2semidirect
11. #10712C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22902
C2×#18402
C2×#18322
#113462
#110422
#109642
#107132
#107122