Finite Group Databasebeta

← #404928

Subgroups — #404928

Order 512.

1526 conjugacy classes of subgroups.

The subgroup lattice has 621 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2344C211yesmaximal, characteristic
2C2×#2083C211yesmaximal, characteristic, complemented
3C2×#1947C211yesmaximal, characteristic, complemented
4C2×#1852C212yesmaximal
5#15150C211yesmaximal, characteristic
6#11227C211yesmaximal, characteristic, complemented
7#11219C212yesmaximal, complemented
8#11199C212yesmaximal
9#11042C211yesmaximal, characteristic
10#10907C212yesmaximal, complemented
11#10806C211yesmaximal, characteristic, complemented
12C22×#117—21no—
13C23×C4⋊C4C2211yescharacteristic
14C22×#107C2212yes—
15C22×#107C2212yes—
16C22×C4⋊C8C2211yescharacteristic
17C22×C22⋊C8C2212yes—
18C2×#397—21no—
19C2×#397—21no—
20#2344—24no—
21#2344—24no—
22C2×#389C2211yescharacteristic
23C2×#389C2211yescharacteristic
24C2×#387—21no—
25C2×#387—21no—
26C2×#384C2211yescharacteristic
27C2×#381—21no—
28C2×#380C2212yes—
29C2×#380C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C22⋊C4C2211yescharacteristic
33#2083—24no—
34#1947—24no—
35#1852—28no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2211yescharacteristic
43C2×#336C2212yes—
44#1445—21no—
45#1445—21no—
46C4⋊#95C2211yescharacteristic
47C4⋊#95C2212yes—
48C24×C4—21no—
49C24×C4C2311yescharacteristic
50C23×C8C2312yes—
51C23×C8C2312yes—
52C2×#117—28no—
53C2×#117—44no—
54C2×#117—44no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4D414yes—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C22⋊C4—21no—
73C22×C22⋊C4—22no—
74C22×C22⋊C4D411yescharacteristic
75C22×C22⋊C4D411yescharacteristic
76C22×C42—21no—
77C22×C42—21no—
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C2×#107—24no—
81C2×#107—24no—
82C2×#107—24no—
83C2×#107—24no—
84C2×#107—24no—
85C2×#107—24no—
86C2×C4⋊C8—24no—
87C2×C4⋊C8—22no—
88C2×C22⋊C8—24no—
89C2×C22⋊C8—24no—
90C2×C22⋊C8—24no—
91#397—44no—
92#397—44no—
93#389—22no—
94#389—22no—
95#389—22no—
96#389D414yes—
97#387—28no—
98#387—28no—
99#384—22no—
100#384—22no—
101#381—28no—
102#380—22no—
103#380—22no—
104#380—24no—
105#380—24no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4D414yes—
110C4×C22⋊C4—22no—
111C4×C22⋊C4—22no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—22no—
118C2×#95—21no—
119C2×#95—22no—
120C2×#95—21no—
121C2×#95—22no—
122C2×#95D411yescharacteristic
123C2×#95D411yescharacteristic
124#340—24no—
125#340—24no—
126#340—24no—
127#340—24no—
128#340—24no—
129#340—24no—
130#336—24no—
131#336—22no—
132C25C2×D411yescharacteristic
133C23×C4—24no—
134C23×C4—24no—
135C23×C4—24no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—42no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4—42no—
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D412yes—
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4#4112yes—
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#4114yes—
175C22×C8#4114yes—
176#117—432no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—28no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
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—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4C2×D412yes—
221C2×C4⋊C4C2×D412yes—
222C2×C4⋊C4C2×D412yes—
223C2×C4⋊C4C2×D412yes—
224C2×C4⋊C4C2×D412yes—
225C2×C4⋊C4C2×D412yes—
226C2×C4⋊C4#4112yes—
227C2×C4⋊C4#4112yes—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—41no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4D814yes—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—24no—
243C2×C42—24no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—42no—
249C2×C42—42no—
250C2×C42—42no—
251C2×C42—42no—
252C2×C42C2×D412yes—
253C2×C42C2×D412yes—
254#107—48no—
255#107—48no—
256C4⋊C8—28no—
257C22⋊C8—216no—
258#95—24no—
259#95—24no—
260#95—24no—
261#95—24no—
262#95—24no—
263#95—22no—
264#95—22no—
265#95—42no—
266#95—44no—
267#95—44no—
268#95—42no—
269#95—44no—
270C24—21no—
271C24—21no—
272C24—22no—
273C24—22no—
274C24—21no—
275C24—21no—
276C24—22no—
277C24—22no—
278C24#12112yes—
279C24C2×D812yes—
280C24AGL(1,ℤ/8)12yes—
281C24C22×D411yescharacteristic
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—216no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—48no—
319C22×C4—48no—
320C22×C4—48no—
321C22×C4—48no—
322C22×C4—48no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—48no—
332C22×C4—41no—
333C22×C4—42no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
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#12012yes—
365C22×C4#12012yes—
366C22×C4#12112yes—
367C22×C4C2×D812yes—
368C22×C4C2×D812yes—
369C22×C4AGL(1,ℤ/8)12yes—
370C22×C4C22×D411yescharacteristic
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C2×#4111yescharacteristic
374C22×C4C2×#4111yescharacteristic
375C22×C4C2×#4112yes—
376C22×C4C2×#4111yescharacteristic
377C22×C4C2×#4112yes—
378C22×C42−1+412yes—
379C22×C42−1+412yes—
380C2×C8—24no—
381C2×C8—24no—
382C2×C8—24no—
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—
394C4⋊C4—216no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—24no—
404C4⋊C4—48no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—42no—
408C4⋊C4—48no—
409C4⋊C4—42no—
410C4⋊C4—42no—
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#12114yes—
420C4⋊C4#12114yes—
421C22⋊C4—28no—
422C22⋊C4—44no—
423C22⋊C4—48no—
424C22⋊C4—88no—
425C42—22no—
426C42—22no—
427C42—24no—
428C42—48no—
429C42—41no—
430C42—41no—
431C42—42no—
432C42—48no—
433C23—24no—
434C23—24no—
435C23—24no—
436C23—24no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—24no—
441C23—22no—
442C23—22no—
443C23—22no—
444C23—24no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—41no—
451C23—41no—
452C23—42no—
453C23—41no—
454C23—41no—
455C23#45914yes—
456C23C2×#12011yescharacteristic
457C23C2×#12111yescharacteristic
458C23#52511yescharacteristic
459C23#54712yes—
460C23#54712yes—
461C23C22×D811yescharacteristic
462C23C2×AGL(1,ℤ/8)11yescharacteristic
463C23C2×AGL(1,ℤ/8)11yescharacteristic
464C23#57511yescharacteristic
465C23#57812yes—
466C23#57812yes—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—22no—
476C2×C4—24no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—432no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
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#44714yes—
537C2×C4C2×#12111yescharacteristic
538C2×C4C2×#12111yescharacteristic
539C2×C4#52511yescharacteristic
540C2×C4#52511yescharacteristic
541C2×C4#52512yes—
542C2×C4#52511yescharacteristic
543C2×C4#52912yes—
544C2×C4#53211yescharacteristic
545C2×C4#53611yescharacteristic
546C2×C4#53611yescharacteristic
547C2×C4#53612yes—
548C2×C4#54112yes—
549C2×C4#54712yes—
550C2×C4#54812yes—
551C2×C4#55212yes—
552C2×C4#55212yes—
553C2×C4#55212yes—
554C2×C4#55612yes—
555C8—44no—
556C8—44no—
557C22—28no—
558C22—24no—
559C22—44no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22—44no—
564C22—44no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22#240611yescharacteristic
569C22#241312yes—
570C22#243211yescharacteristic
571C22#247812yes—
572C22#284711yescharacteristic
573C22C2×#44711yescharacteristic
574C22#300611yescharacteristic
575C22#301111yescharacteristic
576C22#301111yescharacteristic
577C22C2×#45911yescharacteristic
578C22#303611yescharacteristic
579C22#304412yes—
580C22#310011yescharacteristic
581C22#310911yescharacteristic
582C22#312712yes—
583C22#333412yes—
584C22#333412yes—
585C22#334312yes—
586C22#334312yes—
587C22#334312yes—
588C22#334912yes—
589C22#335212yes—
590C22#335312yes—
591C4—24no—
592C4—42no—
593C4—42no—
594C4—42no—
595C4—44no—
596C4—44no—
597C4—82no—
598C4—82no—
599C4—81no—
600C4—162no—
601C4—162no—
602C4#239211yescharacteristic
603C4#241512yes—
604C4#247512yes—
605C4#259911yescharacteristic
606C4#261111yescharacteristic
607C4#262111yescharacteristic
608C2—44no—
609C2#1490711yescharacteristic
610C2#2408811yescharacteristic
611C2#2411011yescharacteristic
612C2#2457612yes—
613C2#2458812yes—
614C2#2570311yescharacteristic
615C2#2571611yescharacteristic
616C2#2745412yes—
617C2#2747512yes—
618C2#3290211yescharacteristic
619C2#3291511yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2083C2semidirect
3. C2×#1947C2semidirect
6. #11227C2semidirect
7. #11219C2semidirect
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×#23442
C2×#20832
C2×#19472
C2×#18522
#151502
#112272
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