Finite Group Databasebeta

← #404964

Subgroups — #404964

Order 512.

1558 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×#2354C211yesmaximal, characteristic
2C2×#2071C211yesmaximal, characteristic
3C2×#1947C211yesmaximal, characteristic, complemented
4C2×#1862C212yesmaximal
5#15151C211yesmaximal, characteristic, complemented
6#11207C212yesmaximal, complemented
7#11202C211yesmaximal, characteristic, complemented
8#11189C212yesmaximal
9#11076C211yesmaximal, characteristic
10#10907C212yesmaximal, complemented
11#10806C211yesmaximal, characteristic, complemented
12C22×C4×D4C2211yescharacteristic
13C22×#117—21no—
14C22×#107C2212yes—
15C22×C4⋊C8C2211yescharacteristic
16C22×#102C2212yes—
17C22×#102C2212yes—
18C2×#398—21no—
19C2×#398—21no—
20#2354—24no—
21#2354—24no—
22C2×#390C2211yescharacteristic
23C2×#390C2211yescharacteristic
24C2×#389C2212yes—
25C2×#388—21no—
26C2×#388—21no—
27C2×#384C2211yescharacteristic
28C2×#384C2211yescharacteristic
29C2×#380—21no—
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33#2071—24no—
34#1947—24no—
35#1862—28no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2211yescharacteristic
42C2×#336C2212yes—
43C2×#336C2212yes—
44#1445—21no—
45#1445—21no—
46C4⋊#95C2211yescharacteristic
47C4⋊#95C2212yes—
48C23×D4C2311yescharacteristic
49C24×C4—21no—
50C23×C8C2312yes—
51C23×C8C2312yes—
52C2×C4×D4—22no—
53C2×C4×D4—22no—
54C2×C4×D4—22no—
55C2×C4×D4—22no—
56C2×C4×D4—22no—
57C2×C4×D4D414yes—
58C2×#117—28no—
59C2×#117—44no—
60C2×#117—44no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4D411yescharacteristic
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—21no—
77C22×C22⋊C4—21no—
78C22×C42—21no—
79C22×C42—21no—
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C2×#107—24no—
84C2×#107—24no—
85C2×#107—24no—
86C2×C4⋊C8—24no—
87C2×C4⋊C8—22no—
88C2×#102—24no—
89C2×#102—24no—
90C2×#102—24no—
91C2×#102—24no—
92C2×#102—24no—
93C2×#102—24no—
94#398—44no—
95#398—44no—
96#390—22no—
97#390—22no—
98#390—22no—
99#390D414yes—
100#389—24no—
101#389—24no—
102#388—28no—
103#388—28no—
104#384—22no—
105#384—22no—
106#384—22no—
107#384—22no—
108#380—28no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4D414yes—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—22no—
119C2×#95—21no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95D411yescharacteristic
123C2×#95D411yescharacteristic
124#340—24no—
125#340—24no—
126#340—24no—
127#340—24no—
128#340—24no—
129#336—24no—
130#336—22no—
131#336—24no—
132C25—21no—
133C22×D4—21no—
134C22×D4—22no—
135C22×D4—22no—
136C22×D4—21no—
137C22×D4—21no—
138C22×D4—22no—
139C22×D4—22no—
140C22×D4C2×D412yes—
141C22×D4C2×D412yes—
142C22×D4#4112yes—
143C23×C4—24no—
144C23×C4—24no—
145C23×C4—21no—
146C23×C4—24no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
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#4114yes—
176C4×D4—28no—
177C4×D4—28no—
178C4×D4—28no—
179C4×D4—28no—
180#117—432no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—82no—
220C2×C4⋊C4D814yes—
221C2×C4⋊C4C2×D412yes—
222C2×C4⋊C4C2×D412yes—
223C2×C4⋊C4C2×D412yes—
224C2×C4⋊C4C2×D412yes—
225C2×C4⋊C4#4114yes—
226C2×C4⋊C4#4112yes—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—82no—
239C2×C22⋊C4—82no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—24no—
247C2×C42—24no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—42no—
253C2×C42—42no—
254C2×C42—42no—
255C2×C42—42no—
256C2×C42C2×D412yes—
257C2×C42C2×D412yes—
258C2×C42C2×D412yes—
259C2×C42#4112yes—
260#107—48no—
261C4⋊C8—28no—
262#102—216no—
263#102—48no—
264#95—24no—
265#95—24no—
266#95—22no—
267#95—22no—
268#95—42no—
269#95—44no—
270#95—44no—
271#95—42no—
272#95—44no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24—42no—
277C24—42no—
278C24—41no—
279C24—41no—
280C24—41no—
281C24—41no—
282C24—42no—
283C24—42no—
284C24C22×D411yescharacteristic
285C2×D4—22no—
286C2×D4—24no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—24no—
290C2×D4—24no—
291C2×D4—24no—
292C2×D4—24no—
293C2×D4—24no—
294C2×D4—41no—
295C2×D4—42no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4#12114yes—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—216no—
310C22×C4—22no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—48no—
334C22×C4—48no—
335C22×C4—48no—
336C22×C4—48no—
337C22×C4—48no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—48no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—42no—
351C22×C4—41no—
352C22×C4—41no—
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#12012yes—
365C22×C4#12012yes—
366C22×C4#12112yes—
367C22×C4#12112yes—
368C22×C4C2×D812yes—
369C22×C4C2×D812yes—
370C22×C4C2×D812yes—
371C22×C4AGL(1,ℤ/8)12yes—
372C22×C4AGL(1,ℤ/8)12yes—
373C22×C4C22×D411yescharacteristic
374C22×C4C22×D411yescharacteristic
375C22×C4C22×D411yescharacteristic
376C22×C4C2×#4111yescharacteristic
377C22×C4C2×#4111yescharacteristic
378C22×C4C2×#4112yes—
379C22×C4C2×#4111yescharacteristic
380C22×C4C2×#4112yes—
381C22×C42−1+412yes—
382C22×C42−1+412yes—
383C2×C8—24no—
384C2×C8—24no—
385C2×C8—24no—
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—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—28no—
401C4⋊C4—24no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—44no—
407C4⋊C4—48no—
408C4⋊C4—41no—
409C4⋊C4—42no—
410C4⋊C4—41no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—84no—
418C4⋊C4—84no—
419C4⋊C4—84no—
420C4⋊C4—88no—
421C4⋊C4#12114yes—
422C22⋊C4—216no—
423C22⋊C4—48no—
424C22⋊C4—48no—
425C22⋊C4—84no—
426C22⋊C4—84no—
427C42—22no—
428C42—22no—
429C42—22no—
430C42—22no—
431C42—22no—
432C42—22no—
433C42—24no—
434C42—48no—
435C42—41no—
436C42—41no—
437C42—42no—
438C42—48no—
439C23—24no—
440C23—42no—
441C23—42no—
442C23—44no—
443C23—42no—
444C23—44no—
445C23—42no—
446C23—44no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—44no—
454C23—82no—
455C23—82no—
456C23—82no—
457C23—82no—
458C23—82no—
459C23—82no—
460C23—82no—
461C23—82no—
462C23C2×#12011yescharacteristic
463C23C2×#12111yescharacteristic
464C23#52511yescharacteristic
465C23#54712yes—
466C23#54812yes—
467C23C22×D811yescharacteristic
468C23C2×AGL(1,ℤ/8)11yescharacteristic
469C23C2×AGL(1,ℤ/8)11yescharacteristic
470C23#57511yescharacteristic
471C23#57712yes—
472C23#57812yes—
473D4—28no—
474D4—42no—
475D4—42no—
476D4—42no—
477D4—42no—
478D4—44no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—432no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—41no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—42no—
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#44714yes—
542C2×C4#45914yes—
543C2×C4C2×#12111yescharacteristic
544C2×C4C2×#12111yescharacteristic
545C2×C4#52511yescharacteristic
546C2×C4#52511yescharacteristic
547C2×C4#52512yes—
548C2×C4#52511yescharacteristic
549C2×C4#52912yes—
550C2×C4#53211yescharacteristic
551C2×C4#53611yescharacteristic
552C2×C4#53611yescharacteristic
553C2×C4#53612yes—
554C2×C4#54112yes—
555C2×C4#54712yes—
556C2×C4#54712yes—
557C2×C4#55212yes—
558C2×C4#55212yes—
559C2×C4#55212yes—
560C2×C4#55612yes—
561C8—44no—
562C8—82no—
563C22—44no—
564C22—44no—
565C22—44no—
566C22—82no—
567C22—82no—
568C22—82no—
569C22—82no—
570C22—84no—
571C22—84no—
572C22—84no—
573C22—84no—
574C22#240611yescharacteristic
575C22#241512yes—
576C22#243211yescharacteristic
577C22#247512yes—
578C22#284711yescharacteristic
579C22C2×#44711yescharacteristic
580C22#300611yescharacteristic
581C22#300911yescharacteristic
582C22#301111yescharacteristic
583C22C2×#45911yescharacteristic
584C22#303611yescharacteristic
585C22#304612yes—
586C22#310011yescharacteristic
587C22#310911yescharacteristic
588C22#312012yes—
589C22#333412yes—
590C22#334312yes—
591C22#334812yes—
592C22#335412yes—
593C22#335812yes—
594C22#336712yes—
595C22#336812yes—
596C22#337712yes—
597C4—24no—
598C4—24no—
599C4—42no—
600C4—42no—
601C4—44no—
602C4—42no—
603C4—81no—
604C4—82no—
605C4—82no—
606C4—84no—
607C4—162no—
608C4#239211yescharacteristic
609C4#241312yes—
610C4#247812yes—
611C4#259911yescharacteristic
612C4#261111yescharacteristic
613C4#262111yescharacteristic
614C2—84no—
615C2#1490711yescharacteristic
616C2#2408811yescharacteristic
617C2#2410411yescharacteristic
618C2#2467612yes—
619C2#2469012yes—
620C2#2564511yescharacteristic
621C2#2566111yescharacteristic
622C2#2735812yes—
623C2#2737912yes—
624C2#3290211yescharacteristic
625C2#3291611yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1947C2semidirect
5. #15151C2semidirect
6. #11207C2semidirect
7. #11202C2semidirect
10. #10907C2semidirect
11. #10806C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23542
C2×#20712
C2×#19472
C2×#18622
#151512
#112072
#112022
#111892
#110762
#109072
#108062