Finite Group Databasebeta

← #364784

Subgroups — #364784

Order 512.

1514 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34952C211yesmaximal, characteristic
2C2×#2295C211yesmaximal, characteristic, complemented
3#21303C211yesmaximal, characteristic, complemented
4C2×#1806C211yesmaximal, characteristic
5C2×C4×#104C211yesmaximal, characteristic
6#11226C212yesmaximal, complemented
7#11075C212yesmaximal
8#10924C212yesmaximal
9#10879C212yesmaximal, complemented
10#10851C211yesmaximal, characteristic, complemented
11#10839C211yesmaximal, characteristic, complemented
12C2×C4×#41C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C2×#423C2211yescharacteristic
15C22×#104C2211yescharacteristic
16C22×#104C2212yes—
17C22×#104C2211yescharacteristic
18C2×C4×M16C2211yescharacteristic
19#2582C412yescomplemented
20#2581C412yescomplemented
21#2580C412yescomplemented
22#2579—22no—
23#2538C412yescomplemented
24#2487—21no—
25#2409—21no—
26#2396—22no—
27C4×C22⋊Q8—22no—
28#2297C2212yes—
29#2296C2212yes—
30#2295—24no—
31#2295—24no—
32C2×#391—21no—
33C2×#389—21no—
34C2×#387—22no—
35#2285C2211yescharacteristic
36C2×C4×C4⋊C4C2212yes—
37C2×C4×C4⋊C4C2211yescharacteristic
38C2×#376C2212yes—
39C2×C43C2211yescharacteristic
40#1985—22no—
41#1984—22no—
42#1806—24no—
43C4×#104—24no—
44C2×#339C2212yes—
45C2×#339C2212yes—
46C2×#339C2211yescharacteristic
47C2×#339C2212yes—
48C2×#339C2211yescharacteristic
49C2×#339C2212yes—
50C2×#339C2212yes—
51#1728C2212yes—
52#1728C2212yes—
53C22×#41C2311yescharacteristic
54C23×Q8D411yescharacteristic
55C22×M16C2312yes—
56C22×M16C2311yescharacteristic
57C22×M16C2311yescharacteristic
58C2×C22⋊Q8—22no—
59C4×#41—22no—
60C4×#41—22no—
61C4×#41—22no—
62C4×#41—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8—22no—
65C2×C4×Q8—22no—
66C2×C4×Q8—22no—
67C2×C4×Q8C2×C411yescharacteristic
68C2×C4×Q8C2×C411yescharacteristic
69C2×C4×Q8C2×C412yes—
70C2×C4×Q8C2×C412yes—
71C2×C4×Q8D412yes—
72C2×C4×Q8D412yes—
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4C2×C411yescharacteristic
76C2×C4×D4C2×C411yescharacteristic
77C2×#117—21no—
78C2×#117C2311yescharacteristic
79C2×#117C2312yes—
80C22×C4⋊C4—22no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4D412yes—
83C22×C4⋊C4D411yescharacteristic
84C22×C42—21no—
85C22×C42—21no—
86C22×C42—21no—
87C22×C42C2311yescharacteristic
88C22×C42C2312yes—
89C22×C42C2312yes—
90C22×C42C2311yescharacteristic
91C22×C42C2311yescharacteristic
92#423—22no—
93#423—22no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×#104—22no—
97C2×#104—22no—
98C2×#104—22no—
99C2×#104—22no—
100C2×#104—22no—
101C2×#104—24no—
102C2×#104—24no—
103C2×#104—42no—
104C2×#104Q814yescomplemented
105C4×M16—22no—
106C4×M16—22no—
107C2×#97—21no—
108C2×C4×C8—21no—
109#400—22no—
110#399—22no—
111#398—24no—
112#396—24no—
113#395—22no—
114#393—22no—
115#391—28no—
116#391—22no—
117#391C2×C414yes—
118#390C2×C414yes—
119#389—28no—
120#389—22no—
121#389C2×C414yes—
122#389C2×C414yes—
123#387—216no—
124#387—24no—
125#386—22no—
126#386—22no—
127#385—22no—
128#385—22no—
129#385—24no—
130#384C2×C412yes—
131#382C2×C412yes—
132#380—21no—
133#380—21no—
134#380—22no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4—24no—
137C4×C4⋊C4—24no—
138C4×C4⋊C4—24no—
139C4×C4⋊C4—22no—
140C4×C4⋊C4—21no—
141C4×C4⋊C4D414yes—
142C4×C22⋊C4—24no—
143#376—24no—
144#376—24no—
145C2×#95—22no—
146C2×#95—22no—
147C2×#95—21no—
148C2×#95—22no—
149C43—22no—
150C43D414yes—
151#339—24no—
152#339—24no—
153#339—22no—
154#339—22no—
155#339—24no—
156#339—42no—
157#339—42no—
158#336—24no—
159#336—24no—
160C2×#41—21no—
161C2×#41—21no—
162C2×#41—21no—
163C2×#41—21no—
164C2×#41—21no—
165C2×#41—21no—
166C2×#41—21no—
167C2×#41—21no—
168C2×#41—21no—
169C2×#41—21no—
170C2×#41C2×Q812yes—
171C2×#41#4112yes—
172C22×Q8—22no—
173C22×Q8—22no—
174C22×Q8—44no—
175C22×Q8C22⋊C412yes—
176C22×Q8C22⋊C412yes—
177C22×Q8C22×C411yescharacteristic
178C22×D4—21no—
179C22×D4C22×C411yescharacteristic
180C23×C4—21no—
181C23×C4—21no—
182C23×C4—21no—
183C23×C4—22no—
184C23×C4—21no—
185C23×C4—21no—
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D412yes—
188C23×C4C2×D412yes—
189C23×C4C2×D411yescharacteristic
190C23×C4C2411yescharacteristic
191C2×M16—21no—
192C2×M16—21no—
193C2×M16—21no—
194C2×M16—21no—
195C2×M16—21no—
196C2×M16—21no—
197C2×M16—21no—
198C2×M16—21no—
199C2×M16—21no—
200C2×M16—21no—
201C2×M16—22no—
202C2×M16—22no—
203C2×M16—22no—
204C2×M16—22no—
205C2×M16—22no—
206C2×M16—22no—
207C2×M16C2×Q812yes—
208C2×M16#4112yes—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—22no—
212C22⋊Q8—48no—
213C4×Q8—24no—
214C4×Q8—24no—
215C4×Q8—24no—
216C4×Q8—24no—
217C4×Q8—24no—
218C4×Q8—24no—
219C4×Q8—24no—
220C4×Q8—24no—
221C4×Q8—22no—
222C4×Q8—22no—
223C4×Q8—22no—
224C4×Q8—22no—
225C4×D4—22no—
226C4×D4—22no—
227C4×D4—22no—
228C4×D4—22no—
229C4×D4—42no—
230C4×D4—42no—
231C4×D4—42no—
232C4×D4—42no—
233#117—21no—
234#117—21no—
235#117—21no—
236#117—21no—
237#117—22no—
238#117—22no—
239#117—22no—
240#117—22no—
241#117—42no—
242#117—42no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—48no—
257C2×C4⋊C4C22⋊C412yes—
258C2×C4⋊C4C22⋊C414yes—
259C2×C4⋊C4C22⋊C414yes—
260C2×C4⋊C4C22⋊C412yes—
261C2×C4⋊C4C22×C411yescharacteristic
262C2×C4⋊C4C22×C411yescharacteristic
263C2×C4⋊C4C22×C414yes—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—21no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—22no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—22no—
281C2×C42—21no—
282C2×C42—22no—
283C2×C42—21no—
284C2×C42—22no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—21no—
290C2×C42—22no—
291C2×C42—21no—
292C2×C42—22no—
293C2×C42—44no—
294C2×C42—44no—
295C2×C42—42no—
296C2×C42—42no—
297C2×C42C22×C414yes—
298C2×C42C22×C411yescharacteristic
299C2×C42C22×C411yescharacteristic
300C2×C42C2×D412yes—
301C2×C42C2×D414yes—
302C2×C42C2×D414yes—
303C2×C42C2×D412yes—
304C2×C42C2×D412yes—
305C2×C42C2×D412yes—
306C2×C42C2×D411yescharacteristic
307C2×C42C2×D411yescharacteristic
308C2×C42C2×Q812yes—
309C2×C42#4112yes—
310C2×C42#4112yes—
311C2×C42#4112yes—
312C4⋊C8—42no—
313C4⋊C8—42no—
314#104—28no—
315#104—28no—
316#104—48no—
317#97—42no—
318C4×C8—42no—
319#95—28no—
320#95—28no—
321#95—28no—
322#95—24no—
323#95—24no—
324#95—22no—
325#95—22no—
326#95—24no—
327#95—24no—
328#95—24no—
329C24—21no—
330C24C22×D411yescharacteristic
331#41—22no—
332#41—22no—
333#41—22no—
334#41—22no—
335#41—22no—
336#41—22no—
337#41—22no—
338#41—22no—
339#41—41no—
340#41—41no—
341#41—41no—
342#41—41no—
343#41—41no—
344#41—41no—
345#41—41no—
346#41—41no—
347C2×Q8—21no—
348C2×Q8—24no—
349C2×Q8—24no—
350C2×Q8—24no—
351C2×Q8—24no—
352C2×Q8—21no—
353C2×Q8—21no—
354C2×Q8—21no—
355C2×Q8—21no—
356C2×Q8—44no—
357C2×Q8—48no—
358C2×Q8—48no—
359C2×Q8C4×Q812yes—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—21no—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—21no—
366C2×D4—21no—
367C2×D4—41no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4C4×Q812yes—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—24no—
389C22×C4—42no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—44no—
408C22×C4—82no—
409C22×C4C2×C22⋊C414yes—
410C22×C4C2×C22⋊C414yes—
411C22×C4C2×C22⋊C411yescharacteristic
412C22×C4C2×C22⋊C411yescharacteristic
413C22×C4C2×C22⋊C411yescharacteristic
414C22×C4C2×C22⋊C411yescharacteristic
415C22×C4#12012yes—
416C22×C4#12012yes—
417C22×C4#12312yes—
418C22×C4#12312yes—
419C22×C4C23×C411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×Q811yescharacteristic
424C22×C4C2×#4111yescharacteristic
425C22×C4C2×#4111yescharacteristic
426C22×C4C2×#4111yescharacteristic
427C22×C4C2×#4111yescharacteristic
428C22×C4C2×#4111yescharacteristic
429C22×C4H(5,𝔽2)11yescharacteristic
430C22×C42−1+411yescharacteristic
431C22×C42−1+411yescharacteristic
432C22×C42−1+411yescharacteristic
433M16—22no—
434M16—22no—
435M16—22no—
436M16—22no—
437M16—41no—
438M16—41no—
439M16—41no—
440M16—41no—
441M16—42no—
442M16—42no—
443M16—42no—
444M16—42no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—42no—
452C2×C8—42no—
453C2×C8—42no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—28no—
457C4⋊C4—28no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—48no—
465C4⋊C4—416no—
466C4⋊C4—48no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—44no—
470C4⋊C4—82no—
471C4⋊C4—84no—
472C22⋊C4—48no—
473C22⋊C4—41no—
474C22⋊C4—41no—
475C22⋊C4—42no—
476C22⋊C4—42no—
477C22⋊C4—42no—
478C22⋊C4—42no—
479C22⋊C4—84no—
480C42—24no—
481C42—24no—
482C42—24no—
483C42—24no—
484C42—22no—
485C42—22no—
486C42—22no—
487C42—22no—
488C42—22no—
489C42—24no—
490C42—24no—
491C42—48no—
492C42—48no—
493C42—41no—
494C42—41no—
495C42—42no—
496C42—42no—
497C42—41no—
498C42—41no—
499C42—42no—
500C42—42no—
501C42—42no—
502C42—42no—
503C42C4×D412yes—
504C42C4×D412yes—
505C42C4×D412yes—
506C42C4×D412yes—
507C42#12112yes—
508C42#12112yes—
509C42#12112yes—
510C42#12112yes—
511C42C22⋊Q814yescomplemented
512C23—22no—
513C23—22no—
514C23—22no—
515C23—41no—
516C23—42no—
517C23—42no—
518C23—41no—
519C23C22×C22⋊C411yescharacteristic
520C23C2×#12011yescharacteristic
521C23C2×#12311yescharacteristic
522C23#54811yescharacteristic
523C23#54811yescharacteristic
524C23D4×Q811yescharacteristic
525C23#55011yescharacteristic
526Q8—22no—
527Q8—22no—
528Q8—48no—
529Q8—41no—
530Q8—41no—
531Q8—48no—
532D4—22no—
533D4—22no—
534D4—42no—
535D4—42no—
536D4—42no—
537D4—42no—
538D4—41no—
539D4—41no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—84no—
585C2×C4—84no—
586C2×C4—84no—
587C2×C4—84no—
588C2×C4#38612yes—
589C2×C4#38612yes—
590C2×C4#38612yes—
591C2×C4#38612yes—
592C2×C4#42114yes—
593C2×C4C2×C4×D411yescharacteristic
594C2×C4C2×C4×D411yescharacteristic
595C2×C4C2×C4×Q811yescharacteristic
596C2×C4C4×#4111yescharacteristic
597C2×C4#51911yescharacteristic
598C2×C4#52011yescharacteristic
599C2×C4C2×#12111yescharacteristic
600C2×C4C2×#12111yescharacteristic
601C2×C4C2×C22⋊Q811yescharacteristic
602C2×C4#52511yescharacteristic
603C2×C4#52911yescharacteristic
604C2×C4#53311yescharacteristic
605C2×C4#53611yescharacteristic
606C2×C4#53711yescharacteristic
607C2×C4#54111yescharacteristic
608C2×C4#54211yescharacteristic
609C2×C4#54611yescharacteristic
610C2×C4#54711yescharacteristic
611C2×C4#54811yescharacteristic
612C2×C4#54811yescharacteristic
613C2×C4#54811yescharacteristic
614C2×C4D4×Q811yescharacteristic
615C2×C4D4×Q811yescharacteristic
616C2×C4#55011yescharacteristic
617C2×C4#55211yescharacteristic
618C2×C4#55211yescharacteristic
619C2×C4#55411yescharacteristic
620C2×C4#55611yescharacteristic
621C8—42no—
622C8—42no—
623C8—82no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—44no—
633C22—42no—
634C22—82no—
635C22—81no—
636C22—81no—
637C22C2×#38611yescharacteristic
638C22Q8×C22⋊C411yescharacteristic
639C22#234111yescharacteristic
640C22#240711yescharacteristic
641C22#240911yescharacteristic
642C22#242111yescharacteristic
643C22#242311yescharacteristic
644C22C2×#42112yes—
645C22#290112yes—
646C4—22no—
647C4—22no—
648C4—22no—
649C4—44no—
650C4—44no—
651C4—42no—
652C4—44no—
653C4—44no—
654C4—84no—
655C4—84no—
656C4—81no—
657C4—81no—
658C4#189714yes—
659C4C4×C22⋊Q811yescharacteristic
660C4#232311yescharacteristic
661C4#236011yescharacteristic
662C4#236211yescharacteristic
663C4#242611yescharacteristic
664C4#242911yescharacteristic
665C4#253611yescharacteristic
666C4#253811yescharacteristic
667C2—21no—
668C2—21no—
669C2—21no—
670C2—21no—
671C2—82no—
672C2#1396911yescharacteristic
673C2C2×#189711yescharacteristic
674C2#2098011yescharacteristic
675C2#2252912yes—
676C2#2255112yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2295C2semidirect
3. #21303C2semidirect
6. #11226C2semidirect
9. #10879C2semidirect
10. #10851C2semidirect
11. #10839C2semidirect
19. #2582C4semidirect
20. #2581C4semidirect
21. #2580C4semidirect
23. #2538C4semidirect
104. C2×#104Q8semidirect
511. C42C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
#349522
C2×#22952
#213032
C2×#18062
C2×C4×#1042
#112262
#110752
#109242
#108792
#108512
#108392