Finite Group Databasebeta

← #387124

Subgroups — #387124

Order 512.

1128 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34953C211yesmaximal, characteristic
2#21303C211yesmaximal, characteristic
3#13850C211yesmaximal, characteristic
4#11386C212yesmaximal
5#11226C212yesmaximal
6#11209C212yesmaximal
7#10886C212yesmaximal
8#10879C212yesmaximal
9#10808C212yesmaximal
10C2×#533C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14#2584C412yescomplemented
15#2584C412yescomplemented
16#2583—21no—
17#2583—21no—
18#2567C411yescharacteristic, complemented
19#2567C411yescharacteristic, complemented
20#2546C411yescharacteristic, complemented
21#2546C411yescharacteristic, complemented
22#2502—21no—
23#2434—21no—
24#2399—21no—
25#2399—21no—
26C4×C4⋊Q8—21no—
27C4×C4⋊Q8—21no—
28#2296C2212yes—
29#2296C2212yes—
30C4×#117C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×#376C2211yescharacteristic
33#1985—21no—
34#1985—21no—
35#1984—21no—
36#1984—21no—
37C2×#339C2212yes—
38C2×#339C2212yes—
39C2×#339C2212yes—
40C2×#339C2212yes—
41C2×#339C2212yes—
42C2×#339C2212yes—
43C2×#339C2212yes—
44C2×#339C2212yes—
45#1728C2212yes—
46#1728C2212yes—
47#1446—21no—
48#1445—21no—
49#1441—21no—
50#1441—21no—
51C4×#95C2212yes—
52C4×#95C2212yes—
53C23×Q8D411yescharacteristic
54C22×M16C2312yes—
55C22×M16C2312yes—
56#533—41no—
57#533—41no—
58#533—41no—
59#533—41no—
60C2×C4⋊Q8—21no—
61C2×C4⋊Q8—21no—
62C2×C4⋊Q8C2×C411yescharacteristic
63C2×C4⋊Q8C2×C411yescharacteristic
64C2×#125C2×C412yes—
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8—21no—
67C2×C4×Q8—21no—
68C2×C4×Q8—21no—
69C2×C4×Q8—41no—
70C2×C4×Q8—41no—
71C2×C4×Q8—41no—
72C2×C4×Q8—41no—
73C2×C4×Q8C2×C411yescharacteristic
74C2×C4×Q8C2×C411yescharacteristic
75C2×C4×Q8C2×C411yescharacteristic
76C2×C4×Q8C2×C411yescharacteristic
77C2×C4×Q8D411yescharacteristic
78C2×C4×Q8D411yescharacteristic
79C2×C4×Q8D411yescharacteristic
80C2×C4×Q8D411yescharacteristic
81C2×#117C2312yes—
82C2×#117C2312yes—
83C22×C4⋊C4—21no—
84C22×C4⋊C4D411yescharacteristic
85C22×C4⋊C4D411yescharacteristic
86C22×C4⋊C4D411yescharacteristic
87C22×C42C2312yes—
88C22×C42C2312yes—
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91C22×C42C2311yescharacteristic
92#423—41no—
93#423—41no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C4×M16—41no—
97C4×M16—41no—
98C2×#97—21no—
99C2×C4×C8—21no—
100#398—21no—
101#398—21no—
102#398—21no—
103#398—21no—
104#398—22no—
105#398—22no—
106#395—22no—
107#395—22no—
108#395—21no—
109#395—21no—
110#395—21no—
111#395—21no—
112#391—21no—
113#391—21no—
114#391—21no—
115#391—21no—
116#389—22no—
117#389—22no—
118#389—22no—
119#389—22no—
120#389C2×C412yes—
121#389C2×C412yes—
122#389C2×C412yes—
123#389C2×C412yes—
124#389C2×C412yes—
125#389C2×C412yes—
126#389C2×C412yes—
127#389C2×C412yes—
128#385—22no—
129#385—22no—
130#385—22no—
131#385—22no—
132#384—21no—
133#384—21no—
134C4×C4⋊C4—22no—
135C4×C4⋊C4—22no—
136C4×C4⋊C4—22no—
137C4×C4⋊C4—22no—
138C4×C4⋊C4—41no—
139C4×C4⋊C4—41no—
140C4×C4⋊C4C2×C411yescharacteristic
141C4×C4⋊C4C2×C411yescharacteristic
142C4×C4⋊C4C2×C412yes—
143C4×C22⋊C4—21no—
144C4×C22⋊C4—21no—
145#376—21no—
146#376—41no—
147#376—41no—
148C2×#95—22no—
149C2×#95—22no—
150C2×#95—22no—
151C2×#95—22no—
152C2×#95—21no—
153C2×#95—21no—
154C2×#95—21no—
155C43—21no—
156#339—24no—
157#339—24no—
158#339—24no—
159#339—42no—
160#339—42no—
161#339—42no—
162#339—42no—
163#339Q818yescomplemented
164#336—22no—
165#336—22no—
166#336—22no—
167#336—22no—
168C22×Q8—21no—
169C22×Q8—21no—
170C22×Q8—41no—
171C22×Q8—41no—
172C22×Q8—42no—
173C22×Q8—42no—
174C22×Q8C22⋊C411yescharacteristic
175C22×Q8C22⋊C411yescharacteristic
176C22×Q8C22⋊C411yescharacteristic
177C22×Q8C22⋊C411yescharacteristic
178C23×C4—22no—
179C23×C4—22no—
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2411yescharacteristic
187C2×M16—22no—
188C2×M16—22no—
189C2×M16—22no—
190C2×M16—22no—
191C2×M16—22no—
192C2×M16—22no—
193C2×M16—22no—
194C2×M16—22no—
195C2×M16—22no—
196C2×M16—22no—
197C2×M16—22no—
198C2×M16C2×Q814yes—
199C22×C8—22no—
200C22×C8—22no—
201C4⋊Q8—41no—
202C4⋊Q8—41no—
203C4⋊Q8—41no—
204C4⋊Q8—41no—
205C4⋊Q8—44no—
206C4⋊Q8—44no—
207#125—42no—
208#125—42no—
209C22⋊Q8—42no—
210C22⋊Q8—42no—
211C22⋊Q8—42no—
212C22⋊Q8—42no—
213C4×Q8—41no—
214C4×Q8—41no—
215C4×Q8—41no—
216C4×Q8—41no—
217C4×Q8—41no—
218C4×Q8—41no—
219C4×Q8—41no—
220C4×Q8—41no—
221C4×Q8—41no—
222C4×Q8—41no—
223C4×Q8—41no—
224C4×Q8—41no—
225C4×Q8—41no—
226C4×Q8—41no—
227C4×Q8—41no—
228C4×Q8—41no—
229C4×Q8—42no—
230C4×Q8—42no—
231C4×Q8—42no—
232C4×Q8—42no—
233#117—22no—
234#117—22no—
235#117—22no—
236#117—22no—
237#117—22no—
238#117—22no—
239#117C2×Q814yes—
240#117#4114yes—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
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—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—41no—
254C2×C4⋊C4—41no—
255C2×C4⋊C4—41no—
256C2×C4⋊C4—41no—
257C2×C4⋊C4—41no—
258C2×C4⋊C4—41no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4C22⋊C411yescharacteristic
262C2×C4⋊C4C22⋊C411yescharacteristic
263C2×C4⋊C4C22⋊C411yescharacteristic
264C2×C4⋊C4C22⋊C411yescharacteristic
265C2×C4⋊C4C22⋊C411yescharacteristic
266C2×C4⋊C4C22⋊C411yescharacteristic
267C2×C4⋊C4C22⋊C411yescharacteristic
268C2×C4⋊C4C22⋊C411yescharacteristic
269C2×C4⋊C4C22⋊C411yescharacteristic
270C2×C4⋊C4C22⋊C411yescharacteristic
271C2×C4⋊C4C22⋊C411yescharacteristic
272C2×C4⋊C4C22⋊C411yescharacteristic
273C2×C4⋊C4C22×C412yes—
274C2×C4⋊C4C22×C412yes—
275C2×C4⋊C4C22×C412yes—
276C2×C4⋊C4C22×C412yes—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C42—22no—
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—21no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42—22no—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42C22×C411yescharacteristic
308C2×C42C22×C411yescharacteristic
309C2×C42C22×C411yescharacteristic
310C2×C42C22×C411yescharacteristic
311C2×C42C22×C411yescharacteristic
312C2×C42C22×C411yescharacteristic
313C2×C42C2×D411yescharacteristic
314C2×C42C2×D411yescharacteristic
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D411yescharacteristic
319C2×C42C2×Q814yes—
320C4⋊C8—42no—
321C4⋊C8—42no—
322#97—42no—
323C4×C8—42no—
324#95—22no—
325#95—21no—
326#95—21no—
327#95—22no—
328#95—22no—
329#95—22no—
330#95—22no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—42no—
340#95—42no—
341#95—42no—
342C24C22×D411yescharacteristic
343C2×Q8—22no—
344C2×Q8—22no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—41no—
348C2×Q8—41no—
349C2×Q8—41no—
350C2×Q8—41no—
351C2×Q8—41no—
352C2×Q8—41no—
353C2×Q8—41no—
354C2×Q8—41no—
355C2×Q8—42no—
356C2×Q8—42no—
357C2×Q8—42no—
358C2×Q8—48no—
359C2×Q8—48no—
360C2×Q8—42no—
361C2×Q8—82no—
362C22×C4—22no—
363C22×C4—22no—
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—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—42no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4C2×C22⋊C411yescharacteristic
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×C22⋊C411yescharacteristic
412C22×C4C2×C22⋊C411yescharacteristic
413C22×C4C23×C411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C22×Q812yes—
416C22×C4C2×#4112yes—
417C22×C4H(5,𝔽2)11yescharacteristic
418C22×C4H(5,𝔽2)11yescharacteristic
419C22×C4H(5,𝔽2)11yescharacteristic
420C22×C4H(5,𝔽2)11yescharacteristic
421C22×C42−1+412yes—
422C22×C42−1+412yes—
423M16—24no—
424M16—24no—
425M16—42no—
426M16—42no—
427M16—42no—
428M16—42no—
429M16—42no—
430M16—42no—
431C2×C8—24no—
432C2×C8—42no—
433C2×C8—42no—
434C2×C8—42no—
435C2×C8—42no—
436C2×C8—42no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—41no—
444C4⋊C4—41no—
445C4⋊C4—41no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C4⋊C4—41no—
449C4⋊C4—41no—
450C4⋊C4—41no—
451C4⋊C4—41no—
452C4⋊C4—41no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C4⋊C4—44no—
456C4⋊C4—44no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—44no—
460C4⋊C4—44no—
461C4⋊C4—44no—
462C4⋊C4—48no—
463C4⋊C4—44no—
464C4⋊C4—82no—
465C4⋊C4—82no—
466C4⋊C4—82no—
467C4⋊C4—82no—
468C4⋊C4C4×Q814yes—
469C4⋊C4C4×Q814yes—
470C22⋊C4—24no—
471C22⋊C4—24no—
472C22⋊C4—42no—
473C22⋊C4—42no—
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—21no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—21no—
495C42—21no—
496C42—21no—
497C42—21no—
498C42—24no—
499C42—24no—
500C42—24no—
501C42—24no—
502C42—42no—
503C42—42no—
504C42—42no—
505C42—42no—
506C42—42no—
507C42—42no—
508C42—42no—
509C42—42no—
510C42—42no—
511C42—42no—
512C42—82no—
513C23—21no—
514C23—21no—
515C23—22no—
516C23C22×C22⋊C411yescharacteristic
517C23#53411yescharacteristic
518C23#53411yescharacteristic
519C23#54812yes—
520C23D4×Q812yes—
521Q8—44no—
522Q8—44no—
523Q8—44no—
524Q8—44no—
525Q8—84no—
526Q8—84no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—81no—
582C2×C4—81no—
583C2×C4—81no—
584C2×C4—81no—
585C2×C4—82no—
586C2×C4—82no—
587C2×C4—82no—
588C2×C4—82no—
589C2×C4—82no—
590C2×C4—82no—
591C2×C4—82no—
592C2×C4—82no—
593C2×C4#42112yes—
594C2×C4#42112yes—
595C2×C4C2×C4×Q812yes—
596C2×C4#51811yescharacteristic
597C2×C4#51811yescharacteristic
598C2×C4#51912yes—
599C2×C4#53411yescharacteristic
600C2×C4#53411yescharacteristic
601C2×C4#54212yes—
602C2×C4#54212yes—
603C2×C4#54412yes—
604C2×C4#54412yes—
605C2×C4#54812yes—
606C2×C4D4×Q812yes—
607C2×C4#55412yes—
608C2×C4#55511yescharacteristic
609C2×C4Q8211yescharacteristic
610C2×C4#56312yes—
611C2×C4#56311yescharacteristic
612C2×C4#56311yescharacteristic
613C8—44no—
614C8—82no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—21no—
620C22—21no—
621C22—21no—
622C22—21no—
623C22#230811yescharacteristic
624C22Q8×C22⋊C412yes—
625C22#272912yes—
626C22#273111yescharacteristic
627C22#273411yescharacteristic
628C22C2×#42111yescharacteristic
629C22C2×#42111yescharacteristic
630C22#290112yes—
631C4—22no—
632C4—22no—
633C4—44no—
634C4—44no—
635C4—41no—
636C4—41no—
637C4—41no—
638C4—41no—
639C4—41no—
640C4—41no—
641C4—41no—
642C4—41no—
643C4—41no—
644C4—41no—
645C4—84no—
646C4—84no—
647C4—82no—
648C4—82no—
649C4#233112yes—
650C4#236911yescharacteristic
651C4#238111yescharacteristic
652C4#272912yes—
653C4#273111yescharacteristic
654C4#273411yescharacteristic
655C2—21no—
656C2—22no—
657C2—21no—
658C2#1445011yescharacteristic
659C2#2177411yescharacteristic
660C2#2177511yescharacteristic
661C2#2252912yes—
662C2#2252912yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC24C22×D4
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
14. #2584C4semidirect
15. #2584C4semidirect
18. #2567C4semidirect
19. #2567C4semidirect
20. #2546C4semidirect
21. #2546C4semidirect
163. #339Q8semidirect

Maximal subgroups

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

subgroupindex
#349532
#213032
#138502
#113862
#112262
#112092
#108862
#108792
#108082