Finite Group Databasebeta

← #394832

Subgroups — #394832

Order 512.

1692 conjugacy classes of subgroups.

The subgroup lattice has 629 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2628C211yesmaximal, characteristic
2C2×#2316C211yesmaximal, characteristic
3C2×#2058C211yesmaximal, characteristic, complemented
4C2×#2057C211yesmaximal, characteristic, complemented
5C2×#1854C212yesmaximal
6#11328C212yesmaximal, complemented
7#11327C212yesmaximal, complemented
8#11208C211yesmaximal, characteristic, complemented
9#11208C211yesmaximal, characteristic, complemented
10#10730C212yesmaximal
11#10713C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C22×#107C2212yes—
17C22×#106C2212yes—
18C22×C22⋊C8C2212yes—
19#2628—28no—
20C2×#400—21no—
21C2×#399—21no—
22C2×#395—21no—
23C2×#393—21no—
24#2316—24no—
25#2316—24no—
26C2×#389—21no—
27C2×#389—21no—
28C2×#389C2211yescharacteristic
29C2×#389C2211yescharacteristic
30C2×#389C2212yes—
31C2×#389C2211yescharacteristic
32C2×#389C2211yescharacteristic
33C2×#382C2211yescharacteristic
34C2×#380—21no—
35C2×#380—21no—
36C2×#380C2212yes—
37#2058—24no—
38#2057—24no—
39#1854—28no—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#340C2212yes—
45C2×#340C2212yes—
46C2×#336C2211yescharacteristic
47C2×#336C2211yescharacteristic
48C2×#336C2212yes—
49C23×Q8—21no—
50C24×C4—21no—
51C24×C4C2311yescharacteristic
52C23×C8C2312yes—
53C23×C8C2312yes—
54C2×C4⋊Q8—24no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—42no—
58C2×#117—24no—
59C2×#117—22no—
60C2×#117—22no—
61C2×#117—22no—
62C2×#117—22no—
63C2×#117—24no—
64C2×#117—42no—
65C2×#117—42no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4D411yescharacteristic
79C22×C4⋊C4D411yescharacteristic
80C22×C4⋊C4Q814yescomplemented
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2311yescharacteristic
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2311yescharacteristic
87C22×C22⋊C4—21no—
88C22×C22⋊C4—22no—
89C22×C22⋊C4D411yescharacteristic
90C22×C22⋊C4D411yescharacteristic
91C22×C42—21no—
92C22×C42C2311yescharacteristic
93C22×C42C2311yescharacteristic
94C2×#107—24no—
95C2×#107—24no—
96C2×#107—42no—
97C2×#106—24no—
98C2×#106—24no—
99C2×#106—42no—
100C2×C22⋊C8—24no—
101C2×C22⋊C8—24no—
102C2×C22⋊C8Q818yescomplemented
103#400—28no—
104#399—28no—
105#395—28no—
106#393—28no—
107#389—28no—
108#389—28no—
109#389—22no—
110#389—22no—
111#389—22no—
112#389—22no—
113#389—24no—
114#389—24no—
115#389—22no—
116#389—22no—
117#389D414yes—
118#389D414yes—
119#382—24no—
120#380—28no—
121#380—28no—
122#380—24no—
123#380D418yes—
124C2×#95—21no—
125C2×#95—22no—
126C2×#95—22no—
127C2×#95—21no—
128C2×#95—21no—
129C2×#95—21no—
130C2×#95—21no—
131C2×#95—21no—
132#340—24no—
133#340—24no—
134#340—24no—
135#340—24no—
136#340—24no—
137#340—24no—
138#336—22no—
139#336—22no—
140#336—42no—
141C25C2×D411yescharacteristic
142C22×Q8—24no—
143C22×Q8—42no—
144C22×Q8—42no—
145C22×Q8—82no—
146C22×Q8—82no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—24no—
156C23×C4—24no—
157C23×C4—24no—
158C23×C4—22no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—22no—
162C23×C4—82no—
163C23×C4—82no—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D414yes—
170C23×C4C2×Q814yes—
171C23×C4C2×Q812yes—
172C23×C4#4112yes—
173C23×C4C2411yescharacteristic
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×C8C2×Q814yes—
187C22×C8C2×Q814yes—
188C4⋊Q8—48no—
189C4⋊Q8—48no—
190#117—48no—
191#117—48no—
192#117—48no—
193#117—48no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—28no—
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—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
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—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—41no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—82no—
238C2×C4⋊C4—82no—
239C2×C4⋊C4—82no—
240C2×C4⋊C4—82no—
241C2×C4⋊C4—82no—
242C2×C4⋊C4—82no—
243C2×C4⋊C4C2×D412yes—
244C2×C4⋊C4C2×D412yes—
245C2×C4⋊C4C2×D414yes—
246C2×C4⋊C4C2×D414yes—
247C2×C4⋊C4C2×D412yes—
248C2×C4⋊C4C2×D412yes—
249C2×C4⋊C4C2×Q812yes—
250C2×C4⋊C4C2×Q812yes—
251C2×C4⋊C4#4112yes—
252C2×C4⋊C4#4112yes—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—41no—
260C2×C22⋊C4—41no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—82no—
264C2×C22⋊C4—82no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—24no—
268C2×C42—22no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—22no—
272C2×C42—42no—
273C2×C42—42no—
274C2×C42C2×D412yes—
275C2×C42C2×D412yes—
276#107—48no—
277#106—48no—
278C22⋊C8—216no—
279#95—24no—
280#95—24no—
281#95—24no—
282#95—24no—
283#95—28no—
284#95—24no—
285#95—24no—
286#95—28no—
287C24—21no—
288C24—21no—
289C24—21no—
290C24—21no—
291C24—22no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24#12012yes—
296C24C22⋊Q812yes—
297C24C4⋊Q812yes—
298C24C22×D411yescharacteristic
299C2×Q8—48no—
300C2×Q8—84no—
301C2×Q8—84no—
302C2×Q8—88no—
303C2×Q8—88no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
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—24no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—216no—
337C22×C4—21no—
338C22×C4—24no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—88no—
376C22×C4—88no—
377C22×C4—84no—
378C22×C4—84no—
379C22×C4#12012yes—
380C22×C4#12012yes—
381C22×C4#12012yes—
382C22×C4C22⋊Q818yes—
383C22×C4C22⋊Q812yes—
384C22×C4C4⋊Q818yes—
385C22×C4C4⋊Q812yes—
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4C22×D412yes—
389C22×C4C22×Q811yescharacteristic
390C22×C4C22×Q812yes—
391C22×C4C2×#4111yescharacteristic
392C22×C4C2×#4111yescharacteristic
393C22×C4C2×#4111yescharacteristic
394C22×C4H(5,𝔽2)12yes—
395C22×C42−1+412yes—
396C2×C8—24no—
397C2×C8—24no—
398C2×C8—24no—
399C2×C8—24no—
400C2×C8—24no—
401C2×C8—24no—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—42no—
409C2×C8—42no—
410C4⋊C4—216no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—44no—
424C4⋊C4—44no—
425C4⋊C4—48no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—84no—
433C4⋊C4—84no—
434C4⋊C4—84no—
435C4⋊C4—84no—
436C4⋊C4—88no—
437C4⋊C4C22⋊Q814yes—
438C4⋊C4C22⋊Q814yes—
439C22⋊C4—44no—
440C22⋊C4—44no—
441C22⋊C4—48no—
442C22⋊C4—88no—
443C42—24no—
444C42—24no—
445C42—48no—
446C42—42no—
447C42—42no—
448C23—24no—
449C23—24no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—24no—
454C23—24no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—24no—
459C23—24no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23#39514yes—
471C23C2×#12011yescharacteristic
472C23C2×#12011yescharacteristic
473C23C2×C22⋊Q811yescharacteristic
474C23#52511yescharacteristic
475C23C2×C4⋊Q811yescharacteristic
476C23#53411yescharacteristic
477C23#53611yescharacteristic
478C23#57612yes—
479C23#57712yes—
480C23#57712yes—
481C23#57812yes—
482Q8—816no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—28no—
490C2×C4—28no—
491C2×C4—28no—
492C2×C4—28no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—816no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4#39514yes—
549C2×C4C2×C22⋊Q811yescharacteristic
550C2×C4C2×C22⋊Q811yescharacteristic
551C2×C4C2×C22⋊Q812yes—
552C2×C4#52511yescharacteristic
553C2×C4#52511yescharacteristic
554C2×C4C2×C4⋊Q812yes—
555C2×C4#53912yes—
556C2×C4#54412yes—
557C2×C4D4212yes—
558C2×C4D4212yes—
559C2×C4#54712yes—
560C2×C4#54712yes—
561C2×C4D4×Q812yes—
562C2×C4D4×Q812yes—
563C2×C4#55412yes—
564C2×C4#55412yes—
565C8—44no—
566C8—44no—
567C22—28no—
568C22—24no—
569C22—44no—
570C22—44no—
571C22—42no—
572C22—42no—
573C22—44no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22C2×#39511yescharacteristic
579C22#240311yescharacteristic
580C22#240611yescharacteristic
581C22#243411yescharacteristic
582C22#258311yescharacteristic
583C22#262711yescharacteristic
584C22#284811yescharacteristic
585C22#300811yescharacteristic
586C22#300911yescharacteristic
587C22#301011yescharacteristic
588C22#301111yescharacteristic
589C22#301212yes—
590C22#301312yes—
591C22#308112yes—
592C22#308212yes—
593C22#311812yes—
594C22#312112yes—
595C22#316312yes—
596C22#316412yes—
597C22#319812yes—
598C22#320112yes—
599C22#324312yes—
600C22#324412yes—
601C4—24no—
602C4—42no—
603C4—42no—
604C4—44no—
605C4—44no—
606C4—84no—
607C4—81no—
608C4—82no—
609C4—84no—
610C4—81no—
611C4—82no—
612C4#244012yes—
613C4#244012yes—
614C4#244912yes—
615C4#244912yes—
616C2—44no—
617C2#1472011yescharacteristic
618C2#2355712yes—
619C2#2392411yescharacteristic
620C2#2392711yescharacteristic
621C2#2407512yes—
622C2#2582011yescharacteristic
623C2#2582111yescharacteristic
624C2#2930812yes—
625C2#2997512yes—
626C2#3296211yescharacteristic
627C2#3296411yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#2058C2semidirect
4. C2×#2057C2semidirect
6. #11328C2semidirect
7. #11327C2semidirect
8. #11208C2semidirect
9. #11208C2semidirect
80. C22×C4⋊C4Q8semidirect
102. C2×C22⋊C8Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#26282
C2×#23162
C2×#20582
C2×#20572
C2×#18542
#113282
#113272
#112082
#107302
#107132