Finite Group Databasebeta

← #387422

Subgroups — #387422

Order 512.

1580 conjugacy classes of subgroups.

The subgroup lattice has 627 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2563C211yesmaximal, characteristic
2C2×#1869C211yesmaximal, characteristic
3#13871C211yesmaximal, characteristic
4#11384C212yesmaximal, complemented
5#11200C212yesmaximal, complemented
6#11199C212yesmaximal, complemented
7#10806C212yesmaximal, complemented
8#10731C212yesmaximal, complemented
9#10721C212yesmaximal, complemented
10C24×C8C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12C22×C4×D4C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×C22⋊C8C2212yes—
15C22×C22⋊C8C2211yescharacteristic
16#2563—24no—
17#2563C418yescomplemented
18C2×#393—21no—
19C2×#393—21no—
20C2×#390—21no—
21C2×#382—21no—
22C2×#380C2212yes—
23C2×#380C2212yes—
24C2×#380C2212yes—
25C2×#380C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27#1869—24no—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2212yes—
37#1440—21no—
38#1438—21no—
39#1437—21no—
40#1436—21no—
41C4⋊#95C2212yes—
42C23×Q8—21no—
43C23×D4D411yescharacteristic
44C24×C4C2311yescharacteristic
45C24×C4C2311yescharacteristic
46C23×C8—22no—
47C23×C8—24no—
48C23×C8—41no—
49C23×C8—41no—
50C23×C8—42no—
51C23×C8C2312yes—
52C23×C8C2312yes—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—41no—
55C2×C4⋊Q8—41no—
56C2×C4⋊Q8—41no—
57C2×C4⋊Q8—41no—
58C2×C4⋊Q8C2×C414yes—
59C2×C4×D4—41no—
60C2×C4×D4—41no—
61C2×C4×D4—41no—
62C2×C4×D4—41no—
63C2×C4×D4C2×C414yes—
64C2×C4×D4D414yes—
65C2×#117—24no—
66C2×#117—42no—
67C2×#117—41no—
68C2×#117—41no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C22⋊C4—22no—
76C22×C22⋊C4—22no—
77C22×C22⋊C4—21no—
78C22×C22⋊C4—22no—
79C22×C22⋊C4—21no—
80C22×C22⋊C4D411yescharacteristic
81C22×C22⋊C4D411yescharacteristic
82C22×C42—21no—
83C22×C42—21no—
84C22×C42C2311yescharacteristic
85C2×C22⋊C8—24no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—24no—
88C2×C22⋊C8—42no—
89C2×C22⋊C8—41no—
90C2×C22⋊C8—41no—
91#393—28no—
92#393—28no—
93#390—28no—
94#382—28no—
95#380—24no—
96#380—24no—
97#380—24no—
98#380—42no—
99#380—41no—
100#380—41no—
101#380C2×C418yes—
102#380C2×C418yes—
103C4×C4⋊C4—22no—
104C4×C4⋊C4C2×C414yes—
105C2×#95—21no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—21no—
110C2×#95—22no—
111C2×#95—22no—
112#340—24no—
113#340—24no—
114#340—42no—
115#340—42no—
116#340—42no—
117#340—42no—
118#340—42no—
119#340—42no—
120#336—24no—
121C25C2×D411yescharacteristic
122C25C2×D411yescharacteristic
123C22×Q8—24no—
124C22×Q8—42no—
125C22×Q8—42no—
126C22×Q8—82no—
127C22×Q8—82no—
128C22×D4—24no—
129C22×D4—24no—
130C22×D4—41no—
131C22×D4—41no—
132C22×D4C22⋊C414yes—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—22no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4C22×C412yes—
154C23×C4C22×C412yes—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D412yes—
159C23×C4C2×D412yes—
160C23×C4C2×D411yescharacteristic
161C23×C4C2411yescharacteristic
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—44no—
176C22×C8—48no—
177C22×C8—48no—
178C22×C8—44no—
179C22×C8—44no—
180C22×C8#4114yes—
181C4⋊Q8—44no—
182C4⋊Q8—44no—
183C4⋊Q8—44no—
184C4⋊Q8—44no—
185C4×D4—44no—
186C4×D4—44no—
187C4×D4—44no—
188C4×D4—44no—
189#117—44no—
190#117—48no—
191#117—44no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4C22⋊C414yes—
219C2×C4⋊C4C22×C414yes—
220C2×C4⋊C4C22×C414yes—
221C2×C4⋊C4#4114yes—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—41no—
225C2×C22⋊C4—41no—
226C2×C22⋊C4—41no—
227C2×C22⋊C4—41no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—82no—
234C2×C22⋊C4—82no—
235C2×C22⋊C4—82no—
236C2×C22⋊C4—82no—
237C2×C22⋊C4—82no—
238C2×C22⋊C4—82no—
239C2×C22⋊C4C22⋊C414yes—
240C2×C22⋊C4C22⋊C414yes—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—24no—
246C2×C42—44no—
247C2×C42—42no—
248C2×C42—44no—
249C2×C42C22×C412yes—
250C2×C42C2×D412yes—
251C22⋊C8—48no—
252C22⋊C8—44no—
253#95—24no—
254#95—28no—
255#95—28no—
256#95—44no—
257#95—42no—
258#95—44no—
259#95—82no—
260C24—21no—
261C24—21no—
262C24—22no—
263C24—22no—
264C24—22no—
265C24—22no—
266C24—22no—
267C24—21no—
268C24—21no—
269C24—22no—
270C24—22no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24C2×C22⋊C412yes—
276C24C2×C22⋊C412yes—
277C24#13714yes—
278C24C22×D411yescharacteristic
279C2×Q8—48no—
280C2×Q8—88no—
281C2×Q8—88no—
282C2×Q8—84no—
283C2×Q8—84no—
284C2×D4—28no—
285C2×D4—28no—
286C2×D4—44no—
287C2×D4—48no—
288C2×D4—48no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—22no—
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—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
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—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×C4C2×C22⋊C412yes—
369C22×C4C2×C22⋊C412yes—
370C22×C4C2×C22⋊C412yes—
371C22×C4C2×C22⋊C412yes—
372C22×C4#13514yes—
373C22×C4C23×C411yescharacteristic
374C22×C4C22×D411yescharacteristic
375C22×C4C2×#4112yes—
376C22×C4C2×#4112yes—
377C22×C4H(5,𝔽2)11yescharacteristic
378C22×C4H(5,𝔽2)11yescharacteristic
379C22×C4H(5,𝔽2)11yescharacteristic
380C22×C4H(5,𝔽2)11yescharacteristic
381C22×C4H(5,𝔽2)12yes—
382C22×C42−1+412yes—
383C2×C8—24no—
384C2×C8—24no—
385C2×C8—24no—
386C2×C8—416no—
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—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—44no—
405C4⋊C4—48no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C22⋊C4—44no—
417C22⋊C4—44no—
418C22⋊C4—48no—
419C22⋊C4—88no—
420C22⋊C4—88no—
421C22⋊C4—88no—
422C22⋊C4—84no—
423C42—22no—
424C42—22no—
425C42—22no—
426C42—22no—
427C42—48no—
428C42—48no—
429C23—22no—
430C23—24no—
431C23—24no—
432C23—24no—
433C23—22no—
434C23—24no—
435C23—24no—
436C23—24no—
437C23—24no—
438C23—24no—
439C23—22no—
440C23—24no—
441C23—22no—
442C23—22no—
443C23—24no—
444C23—42no—
445C23—42no—
446C23—41no—
447C23—42no—
448C23—42no—
449C23—41no—
450C23—41no—
451C23—41no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—41no—
461C23—42no—
462C23—42no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23#41914yes—
471C23C22×C22⋊C411yescharacteristic
472C23#53411yescharacteristic
473C23#53411yescharacteristic
474C23#54612yes—
475C23#54712yes—
476C23C2×#13512yes—
477C23C2×#13712yes—
478C23#57612yes—
479C23#57712yes—
480Q8—816no—
481D4—416no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4#41614yes—
542C2×C4C4×#4112yes—
543C2×C4#51811yescharacteristic
544C2×C4#51811yescharacteristic
545C2×C4#52012yes—
546C2×C4#53411yescharacteristic
547C2×C4#53411yescharacteristic
548C2×C4#53812yes—
549C2×C4#53812yes—
550C2×C4#54212yes—
551C2×C4#54212yes—
552C2×C4#54612yes—
553C2×C4#54712yes—
554C2×C4#55111yescharacteristic
555C2×C4#55111yescharacteristic
556C2×C4#55512yes—
557C2×C4H(3,𝔽4)11yescharacteristic
558C2×C4#56212yes—
559C2×C4#56311yescharacteristic
560C8—44no—
561C8—82no—
562C22—24no—
563C22—28no—
564C22—42no—
565C22—44no—
566C22—44no—
567C22—44no—
568C22—44no—
569C22—42no—
570C22—44no—
571C22—42no—
572C22—42no—
573C22—42no—
574C22—44no—
575C22—44no—
576C22—82no—
577C22—82no—
578C22—82no—
579C22—82no—
580C22#230811yescharacteristic
581C22#233912yes—
582C22#268411yescharacteristic
583C22#268612yes—
584C22#271711yescharacteristic
585C22C2×#41611yescharacteristic
586C22C2×#41911yescharacteristic
587C22#289712yes—
588C22#319012yes—
589C22#319012yes—
590C22#319812yes—
591C22#319911yescharacteristic
592C22#320011yescharacteristic
593C22#330312yes—
594C22#330512yes—
595C22#331412yes—
596C22#331612yes—
597C22#334012yes—
598C22#334112yes—
599C22#334412yes—
600C22#334612yes—
601C4—24no—
602C4—42no—
603C4—42no—
604C4—42no—
605C4—84no—
606C4—84no—
607C4—84no—
608C4—162no—
609C4#233712yes—
610C4#237211yescharacteristic
611C4#237911yescharacteristic
612C4#268411yescharacteristic
613C4#268612yes—
614C4#271711yescharacteristic
615C2—44no—
616C2—82no—
617C2#1447311yescharacteristic
618C2#2176211yescharacteristic
619C2#2177011yescharacteristic
620C2#2248912yes—
621C2#2249712yes—
622C2#3126812yes—
623C2#3132712yes—
624C2#3132912yes—
625C2#3226612yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11384C2semidirect
5. #11200C2semidirect
6. #11199C2semidirect
7. #10806C2semidirect
8. #10731C2semidirect
9. #10721C2semidirect
17. #2563C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25632
C2×#18692
#138712
#113842
#112002
#111992
#108062
#107312
#107212