Finite Group Databasebeta

← #369350

Subgroups — #369350

Order 512.

1728 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#436C211yesmaximal, characteristic
2C2×#2562C211yesmaximal, characteristic
3C2×#1991C211yesmaximal, characteristic, complemented
4C2×#1989C211yesmaximal, characteristic, complemented
5#13905C211yesmaximal, characteristic
6#11200C212yesmaximal, complemented
7#11199C212yesmaximal, complemented
8#10850C211yesmaximal, characteristic, complemented
9#10722C212yesmaximal, complemented
10#10722C212yesmaximal, complemented
11C4×#340C211yesmaximal, characteristic, complemented
12C23×M16C2212yes—
13C22×#125C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C2×#436—24no—
16C2×#436—24no—
17C2×#436—24no—
18C22×C4⋊C8C2211yescharacteristic
19C22×C22⋊C8C2212yes—
20C22×C4×C8C2211yescharacteristic
21#2562—24no—
22#2562C418yescomplemented
23C2×#400—21no—
24C2×#394—21no—
25C2×#388—22no—
26C2×#382C2211yescharacteristic
27C2×#380C2212yes—
28C2×#380C2212yes—
29C2×#380C2212yes—
30C2×#380C2212yes—
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33#1991—24no—
34#1989—24no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2211yescharacteristic
38C2×#340C2211yescharacteristic
39C2×#340C2211yescharacteristic
40C2×#340C2211yescharacteristic
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#340C2212yes—
45#1445—21no—
46#1438—21no—
47#1437—22no—
48C23×D4D411yescharacteristic
49C24×C4C2312yes—
50C22×M16—24no—
51C22×M16—24no—
52C22×M16—24no—
53C22×M16—24no—
54C22×M16—24no—
55C22×M16—42no—
56C22×M16—42no—
57C23×C8C2311yescharacteristic
58C23×C8C2312yes—
59C23×C8C2311yescharacteristic
60C2×#125—22no—
61C2×#125—22no—
62C2×#125—22no—
63C2×#125—22no—
64C2×#125—22no—
65C2×#125C2×C414yes—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×C4×D4C2×C414yes—
71C2×C4×D4D414yes—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4D411yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2312yes—
81C22×C22⋊C4—22no—
82C22×C22⋊C4—22no—
83C22×C22⋊C4—22no—
84C22×C22⋊C4—22no—
85C22×C22⋊C4D412yes—
86C22×C42—21no—
87C22×C42—21no—
88C22×C42C2311yescharacteristic
89#436—232no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8D414yes—
93C2×C22⋊C8—24no—
94C2×C22⋊C8—24no—
95C2×C22⋊C8—24no—
96C2×C4×C8—22no—
97C2×C4×C8—22no—
98C2×C4×C8D414yes—
99#400—28no—
100#394—28no—
101#388—216no—
102#382—22no—
103#382—22no—
104#380—24no—
105#380—24no—
106#380—24no—
107#380—24no—
108#380—42no—
109#380—42no—
110#380C2×C418yes—
111#380C2×C418yes—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4C2×C414yes—
116C2×#95—21no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—21no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95—21no—
123C2×#95—21no—
124#340—24no—
125#340—24no—
126#340—22no—
127#340—22no—
128#340—22no—
129#340—22no—
130#340—24no—
131#340—24no—
132#340—42no—
133#340—42no—
134C25C2×D412yes—
135C22×D4—22no—
136C22×D4—22no—
137C22×D4—42no—
138C22×D4—42no—
139C22×D4C22⋊C414yes—
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×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4C22×C414yes—
159C23×C4C2×D412yes—
160C23×C4C2×D414yes—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2411yescharacteristic
164C2×M16—216no—
165C2×M16—48no—
166C2×M16—48no—
167C2×M16—48no—
168C2×M16—48no—
169C2×M16—48no—
170C2×M16—48no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8C2×D414yes—
185C22×C8C2×D412yes—
186C22×C8C2×D412yes—
187C22×C8#4114yes—
188#125—28no—
189#125—28no—
190#125—44no—
191#125—44no—
192C4×D4—28no—
193C4×D4—28no—
194C4×D4—28no—
195C4×D4—28no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
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—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4C22⋊C414yes—
233C2×C4⋊C4C22×C412yes—
234C2×C4⋊C4C22×C412yes—
235C2×C4⋊C4C22×C414yes—
236C2×C4⋊C4#4114yes—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—82no—
248C2×C22⋊C4—82no—
249C2×C22⋊C4—82no—
250C2×C22⋊C4—82no—
251C2×C22⋊C4C22⋊C418yes—
252C2×C42—24no—
253C2×C42—42no—
254C2×C42—81no—
255C2×C42—81no—
256C2×C42—81no—
257C2×C42—81no—
258C2×C42C22×C412yes—
259C2×C42C2×D412yes—
260C2×C42C2×D412yes—
261C2×C42C2×D412yes—
262C2×C42#4112yes—
263C2×C42#4112yes—
264C4⋊C8—28no—
265C22⋊C8—216no—
266C4×C8—28no—
267#95—24no—
268#95—28no—
269#95—28no—
270#95—44no—
271#95—44no—
272#95—42no—
273#95—42no—
274#95—42no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24—22no—
280C24—22no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24—22no—
289C24C2×C22⋊C414yes—
290C24C22×D411yescharacteristic
291C2×D4—28no—
292C2×D4—48no—
293C2×D4—48no—
294C2×D4—44no—
295C2×D4—44no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—42no—
325C22×C4—42no—
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—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
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—41no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4C2×C22⋊C414yes—
373C22×C4C2×C22⋊C412yes—
374C22×C4C2×C22⋊C412yes—
375C22×C4#12112yes—
376C22×C4#12112yes—
377C22×C4#12112yes—
378C22×C4#12112yes—
379C22×C4C23×C411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C22×C4C22×D411yescharacteristic
382C22×C4C22×D411yescharacteristic
383C22×C4C2×#4111yescharacteristic
384C22×C4C2×#4111yescharacteristic
385C22×C4C2×#4112yes—
386C22×C4C2×#4112yes—
387C22×C4H(5,𝔽2)12yes—
388C22×C42−1+412yes—
389M16—432no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—24no—
393C2×C8—24no—
394C2×C8—22no—
395C2×C8—24no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—24no—
399C2×C8—24no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8#12414yes—
407C4⋊C4—28no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—48no—
425C4⋊C4—48no—
426C4⋊C4—41no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C22⋊C4—216no—
433C22⋊C4—416no—
434C22⋊C4—88no—
435C22⋊C4—88no—
436C22⋊C4—88no—
437C42—84no—
438C42—84no—
439C42C4×D414yes—
440C42C4×D414yes—
441C42#12014yes—
442C42#12314yes—
443C23—24no—
444C23—24no—
445C23—24no—
446C23—24no—
447C23—24no—
448C23—24no—
449C23—24no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
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—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23C22×C22⋊C411yescharacteristic
479C23C2×#12111yescharacteristic
480C23C2×#12111yescharacteristic
481C23#54612yes—
482C23#54712yes—
483C23#57512yes—
484C23#57512yes—
485C23#57611yescharacteristic
486C23#57711yescharacteristic
487C23#57711yescharacteristic
488C23#57811yescharacteristic
489D4—416no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4—81no—
536C2×C4—82no—
537C2×C4—81no—
538C2×C4—81no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—81no—
542C2×C4—81no—
543C2×C4—81no—
544C2×C4—81no—
545C2×C4—81no—
546C2×C4—81no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4#38614yes—
552C2×C4#38614yes—
553C2×C4C2×C4×D411yescharacteristic
554C2×C4C2×C4×D411yescharacteristic
555C2×C4C4×#4112yes—
556C2×C4#52012yes—
557C2×C4C2×#12011yescharacteristic
558C2×C4C2×#12311yescharacteristic
559C2×C4C2×#12411yescharacteristic
560C2×C4#52912yes—
561C2×C4#53211yescharacteristic
562C2×C4#53511yescharacteristic
563C2×C4#53611yescharacteristic
564C2×C4#54212yes—
565C2×C4#54612yes—
566C2×C4#54612yes—
567C2×C4#54712yes—
568C2×C4#54812yes—
569C2×C4#55612yes—
570C2×C4#55912yes—
571C8—24no—
572C8—42no—
573C8—44no—
574C22—44no—
575C22—44no—
576C22—44no—
577C22—44no—
578C22—44no—
579C22—44no—
580C22—44no—
581C22—82no—
582C22—82no—
583C22—82no—
584C22—82no—
585C22—82no—
586C22—82no—
587C22—82no—
588C22—82no—
589C22C2×#38611yescharacteristic
590C22#233912yes—
591C22#242712yes—
592C22#254411yescharacteristic
593C22#254711yescharacteristic
594C22#289612yes—
595C22#289711yescharacteristic
596C22#289811yescharacteristic
597C22#305412yes—
598C22#305411yescharacteristic
599C22#305411yescharacteristic
600C22#306011yescharacteristic
601C22#306111yescharacteristic
602C22#306211yescharacteristic
603C22#306311yescharacteristic
604C22#332212yes—
605C22#332312yes—
606C22#332612yes—
607C22#332712yes—
608C22#335112yes—
609C22#335112yes—
610C22#335212yes—
611C22#335512yes—
612C4—24no—
613C4—24no—
614C4—44no—
615C4—84no—
616C4—84no—
617C4—82no—
618C4—82no—
619C4—161no—
620C4—161no—
621C4C4×#12411yescharacteristic
622C4#232611yescharacteristic
623C4#236712yes—
624C4#243211yescharacteristic
625C4#243511yescharacteristic
626C4#244712yes—
627C2—84no—
628C2#1402711yescharacteristic
629C2#2097011yescharacteristic
630C2#2097611yescharacteristic
631C2#2249912yes—
632C2#2250112yes—
633C2#2534412yes—
634C2#2534512yes—
635C2#2860311yescharacteristic
636C2#2860411yescharacteristic
637C2#2870911yescharacteristic
638C2#2871011yescharacteristic
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
3. C2×#1991C2semidirect
4. C2×#1989C2semidirect
6. #11200C2semidirect
7. #11199C2semidirect
8. #10850C2semidirect
9. #10722C2semidirect
10. #10722C2semidirect
11. C4×#340C2semidirect
22. #2562C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4362
C2×#25622
C2×#19912
C2×#19892
#139052
#112002
#111992
#108502
#107222
C4×#3402