Finite Group Databasebeta

← #214229

Subgroups — #214229

Order 512.

1090 conjugacy classes of subgroups.

The subgroup lattice has 651 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34952C211yesmaximal, characteristic
2#21373C211yesmaximal, characteristic
3#20629C211yesmaximal, characteristic
4#9075C212yesmaximal, complemented
5#9071C212yesmaximal, complemented
6#8942C212yesmaximal, complemented
7#8937C212yesmaximal, complemented
8#8710C212yesmaximal, complemented
9#8311C212yesmaximal, complemented
10C2×C4×#41C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#423C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C2×#408—21no—
16C22×C4×C8C2211yescharacteristic
17#2582C412yes—
18#2581—21no—
19#2580—21no—
20#2579—21no—
21#2579—21no—
22#2538C412yes—
23#2487C411yescharacteristic
24#2487C411yescharacteristic
25#2409C411yescharacteristic
26#2409C411yescharacteristic
27#2396—21no—
28#2396—21no—
29C4×C22⋊Q8C2212yes—
30C4×C22⋊Q8C2212yes—
31#2297—21no—
32#2296—21no—
33#2285C2211yescharacteristic
34#1984—21no—
35#1954—21no—
36#1869—21no—
37#1869—21no—
38#1798—21no—
39#1798—21no—
40#1797C2212yes—
41#1797C2212yes—
42C4×C22⋊C8C2212yes—
43C4×C22⋊C8C2212yes—
44#1729—21no—
45C2×Q8⋊C8C2212yes—
46C2×Q8⋊C8C2212yes—
47C2×Q8⋊C8C2212yes—
48C2×Q8⋊C8C2212yes—
49#1279C2212yes—
50#1279C2212yes—
51#1279C2212yes—
52#1279C2212yes—
53C22×#41D411yescharacteristic
54C23×Q8C2×C411yescharacteristic
55C22×M16—21no—
56C23×C8—21no—
57C23×C8—21no—
58C2×C22⋊Q8C2×C412yes—
59C2×C22⋊Q8C2×C412yes—
60C4×#41—41no—
61C4×#41—41no—
62C4×#41—41no—
63C4×#41—41no—
64C2×C4×Q8—21no—
65C2×C4×Q8—21no—
66C2×C4×Q8—41no—
67C2×C4×Q8—41no—
68C2×C4×Q8—41no—
69C2×C4×Q8—41no—
70C2×C4×Q8C2×C412yes—
71C2×C4×Q8C2×C411yescharacteristic
72C2×C4×Q8D411yescharacteristic
73C2×C4×Q8C2312yes—
74C2×C4×D4C2×C411yescharacteristic
75C2×C4×D4C2×C411yescharacteristic
76C2×C4×D4C2×C411yescharacteristic
77C2×C4×D4D411yescharacteristic
78C2×C4×D4D411yescharacteristic
79C2×C4×D4D411yescharacteristic
80C2×#117—21no—
81C2×#117D411yescharacteristic
82C2×#117D411yescharacteristic
83C2×#117D411yescharacteristic
84C22×C4⋊C4—21no—
85C22×C4⋊C4C2×C411yescharacteristic
86C22×C42—21no—
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90#431—41no—
91#431—41no—
92#423—41no—
93#423—41no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—41no—
98C2×C4⋊C8—41no—
99C2×C4⋊C8C2312yes—
100C2×C4⋊C8C2312yes—
101#408—44no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8—21no—
105C2×C22⋊C8—22no—
106C2×C22⋊C8—22no—
107C2×C4×C8—21no—
108C2×C4×C8—41no—
109C2×C4×C8—41no—
110C2×C4×C8C2312yes—
111C2×C4×C8C2312yes—
112#400—21no—
113#400—21no—
114#399—21no—
115#399—21no—
116#398—22no—
117#398—22no—
118#396—22no—
119#396—22no—
120#395—21no—
121#395—21no—
122#393—21no—
123#393—21no—
124#391—22no—
125#391C2×C412yes—
126#391C2×C412yes—
127#390—22no—
128#389—22no—
129#389—22no—
130#389C2×C412yes—
131#389C2×C412yes—
132#387—22no—
133#387—22no—
134#386—21no—
135#386—21no—
136#386—21no—
137#386—21no—
138#385—22no—
139#385—22no—
140#385—21no—
141#385—21no—
142#385—21no—
143#385—21no—
144#384C2×C412yes—
145#382—21no—
146#380—21no—
147#380C2×C411yescharacteristic
148#380C2×C411yescharacteristic
149#380C2×C412yes—
150#380C2×C412yes—
151C4×C4⋊C4—21no—
152C4×C4⋊C4—22no—
153C4×C4⋊C4—22no—
154C4×C4⋊C4C2312yes—
155C4×C22⋊C4—22no—
156C4×C22⋊C4—22no—
157#336—22no—
158#336—22no—
159#336—22no—
160#336—22no—
161Q8⋊C8—42no—
162Q8⋊C8—42no—
163Q8⋊C8—42no—
164Q8⋊C8—42no—
165C2×#41—41no—
166C2×#41—41no—
167C2×#41—41no—
168C2×#41—41no—
169C2×#41SD1614yes—
170C2×#41Q1614yes—
171C22×Q8—21no—
172C22×Q8—24no—
173C22×Q8—41no—
174C22×Q8—41no—
175C22×Q8—42no—
176C22×Q8C22⋊C411yescharacteristic
177C22×Q8C22×C412yes—
178C22×D4C22⋊C411yescharacteristic
179C22×D4C22⋊C411yescharacteristic
180C22×D4C22⋊C411yescharacteristic
181C23×C4—21no—
182C23×C4—21no—
183C23×C4C22×C411yescharacteristic
184C23×C4C22×C411yescharacteristic
185C23×C4C22×C411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C2×M16—42no—
193C2×M16—42no—
194C2×M16—42no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—44no—
200C22×C8—42no—
201C22×C8—42no—
202C22×C8—42no—
203C22×C8—42no—
204C22⋊Q8—24no—
205C22⋊Q8—24no—
206C22⋊Q8—42no—
207C22⋊Q8—42no—
208C4×Q8—22no—
209C4×Q8—22no—
210C4×Q8—22no—
211C4×Q8—22no—
212C4×Q8—41no—
213C4×Q8—41no—
214C4×Q8—41no—
215C4×Q8—41no—
216C4×Q8—42no—
217C4×Q8—42no—
218C4×Q8—42no—
219C4×Q8—42no—
220C4×Q8—42no—
221C4×Q8—42no—
222C4×D4—41no—
223C4×D4—41no—
224C4×D4—41no—
225C4×D4—41no—
226C4×D4—41no—
227C4×D4—41no—
228C4×D4—41no—
229C4×D4—41no—
230C4×D4—41no—
231C4×D4—41no—
232C4×D4—41no—
233C4×D4—41no—
234#117—41no—
235#117—41no—
236#117—41no—
237#117—41no—
238#117—41no—
239#117—41no—
240#117—42no—
241#117—42no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—41no—
252C2×C4⋊C4—41no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4C22⋊C411yescharacteristic
256C2×C4⋊C4C22⋊C411yescharacteristic
257C2×C4⋊C4C22⋊C411yescharacteristic
258C2×C4⋊C4C22⋊C411yescharacteristic
259C2×C4⋊C4C22⋊C411yescharacteristic
260C2×C4⋊C4C22⋊C411yescharacteristic
261C2×C4⋊C4C22×C412yes—
262C2×C4⋊C4C22×C412yes—
263C2×C4⋊C4C22×C412yes—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4C22⋊C411yescharacteristic
271C2×C22⋊C4C22⋊C411yescharacteristic
272C2×C22⋊C4C22⋊C411yescharacteristic
273C2×C22⋊C4C22⋊C411yescharacteristic
274C2×C22⋊C4C22⋊C411yescharacteristic
275C2×C22⋊C4C22⋊C411yescharacteristic
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—42no—
292C2×C42—42no—
293C2×C42C22×C411yescharacteristic
294C2×C42C22×C411yescharacteristic
295C2×C42C22×C411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D411yescharacteristic
301C2×C42C2×D411yescharacteristic
302C2×C42C2411yescharacteristic
303C4⋊C8—22no—
304C4⋊C8—22no—
305C4⋊C8—22no—
306C4⋊C8—22no—
307C4⋊C8—42no—
308C4⋊C8—42no—
309C4⋊C8—42no—
310C22⋊C8—44no—
311C22⋊C8—44no—
312C22⋊C8—42no—
313C22⋊C8—44no—
314C22⋊C8—44no—
315C4×C8—22no—
316C4×C8—22no—
317C4×C8—22no—
318C4×C8—22no—
319C4×C8—42no—
320#95—22no—
321#95—22no—
322#95—22no—
323#95—22no—
324#95—21no—
325#95—21no—
326#95—22no—
327#95—22no—
328#95—22no—
329#95—22no—
330#95—22no—
331C24C2×C22⋊C411yescharacteristic
332C24C2×C22⋊C411yescharacteristic
333C24C2×C22⋊C411yescharacteristic
334#41—44no—
335#41—44no—
336#41—44no—
337#41—44no—
338C2×Q8—22no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—22no—
343C2×Q8—41no—
344C2×Q8—41no—
345C2×Q8—44no—
346C2×Q8—42no—
347C2×Q8—44no—
348C2×Q8—48no—
349C2×Q8—44no—
350C2×Q8#10314yes—
351C2×Q8#13114yes—
352C2×D4—41no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4—41no—
356C2×D4—41no—
357C2×D4—41no—
358C2×D4#10314yes—
359C2×D4#10314yes—
360C2×D4#10314yes—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—44no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—82no—
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C2×C22⋊C411yescharacteristic
404C22×C4C2×C22⋊C411yescharacteristic
405C22×C4C2×C22⋊C411yescharacteristic
406C22×C4C2×C22⋊C411yescharacteristic
407C22×C4C2×C22⋊C411yescharacteristic
408C22×C4C2×C22⋊C411yescharacteristic
409C22×C4C2×C22⋊C411yescharacteristic
410C22×C4C2×C22⋊C411yescharacteristic
411C22×C4C2×SD1612yes—
412C22×C4C2×SD1612yes—
413C22×C4C2×SD1612yes—
414C22×C4C2×Q1612yes—
415C22×C4C2×Q1612yes—
416C22×C4C2×Q1612yes—
417C22×C4C23×C411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D411yescharacteristic
420M16—84no—
421C2×C8—24no—
422C2×C8—42no—
423C2×C8—48no—
424C2×C8—42no—
425C2×C8—42no—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8—42no—
432C2×C8—42no—
433C2×C8—42no—
434C2×C8—84no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—41no—
442C4⋊C4—41no—
443C4⋊C4—41no—
444C4⋊C4—41no—
445C4⋊C4—41no—
446C4⋊C4—41no—
447C4⋊C4—44no—
448C4⋊C4—44no—
449C4⋊C4—44no—
450C4⋊C4—42no—
451C4⋊C4—44no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—82no—
455C4⋊C4—82no—
456C4⋊C4—82no—
457C22⋊C4—41no—
458C22⋊C4—41no—
459C22⋊C4—41no—
460C22⋊C4—41no—
461C22⋊C4—41no—
462C22⋊C4—41no—
463C22⋊C4—44no—
464C22⋊C4—44no—
465C22⋊C4—42no—
466C22⋊C4—42no—
467C22⋊C4—82no—
468C22⋊C4—82no—
469C42—21no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42—21no—
474C42—21no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—44no—
488C42—42no—
489C42—42no—
490C42—42no—
491C42—42no—
492C42H(5,𝔽2)11yescharacteristic
493C42H(5,𝔽2)11yescharacteristic
494C42H(5,𝔽2)11yescharacteristic
495C42H(5,𝔽2)11yescharacteristic
496C23—21no—
497C23—21no—
498C23—21no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23C2×#10312yes—
509C23C2×#10312yes—
510C23C2×#10312yes—
511C23C22×C22⋊C411yescharacteristic
512Q8—24no—
513Q8—24no—
514Q8—44no—
515Q8—48no—
516Q8—42no—
517Q8—42no—
518D4—44no—
519D4—44no—
520D4—44no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—48no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—81no—
577C2×C4—81no—
578C2×C4—81no—
579C2×C4—81no—
580C2×C4—84no—
581C2×C4C2×#10312yes—
582C2×C4C2×#10312yes—
583C2×C4C2×#10312yes—
584C2×C4#51811yescharacteristic
585C2×C4#51811yescharacteristic
586C2×C4#53411yescharacteristic
587C2×C4#53411yescharacteristic
588C2×C4#53411yescharacteristic
589C2×C4#53411yescharacteristic
590C2×C4C2×#13112yes—
591C2×C4#56812yes—
592C2×C4C22×SD1611yescharacteristic
593C2×C4C22×Q1611yescharacteristic
594C2×C4#57511yescharacteristic
595C2×C4#57511yescharacteristic
596C8—44no—
597C8—82no—
598C8—82no—
599C8—82no—
600C22—24no—
601C22—24no—
602C22—24no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22#230811yescharacteristic
616C22#288012yes—
617C22C22×#10311yescharacteristic
618C22#289611yescharacteristic
619C22#290112yes—
620C4—21no—
621C4—21no—
622C4—22no—
623C4—22no—
624C4—22no—
625C4—41no—
626C4—41no—
627C4—41no—
628C4—41no—
629C4—41no—
630C4—41no—
631C4—44no—
632C4—82no—
633C4—82no—
634C4—82no—
635C4#297612yes—
636C4#297711yescharacteristic
637C4#297811yescharacteristic
638C4#318711yescharacteristic
639C4#318711yescharacteristic
640C4#318911yescharacteristic
641C4#318911yescharacteristic
642C2—42no—
643C2—42no—
644C2—42no—
645C2#1750912yes—
646C2#1751612yes—
647C2#2174411yescharacteristic
648C2#2175911yescharacteristic
649C2#2177511yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9075C2semidirect
5. #9071C2semidirect
6. #8942C2semidirect
7. #8937C2semidirect
8. #8710C2semidirect
9. #8311C2semidirect

Maximal subgroups

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

subgroupindex
#349522
#213732
#206292
#90752
#90712
#89422
#89372
#87102
#83112