Finite Group Databasebeta

← #386109

Subgroups — #386109

Order 512.

1554 conjugacy classes of subgroups.

The subgroup lattice has 721 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2440C211yesmaximal, characteristic
2C2×#1996C212yesmaximal
3#13882C212yesmaximal
4#11381C212yesmaximal
5#11353C212yesmaximal
6#11203C212yesmaximal
7#10912C212yesmaximal
8#10801C211yesmaximal, characteristic
9#10801C211yesmaximal, characteristic
10C22×C4⋊Q8C411yescharacteristic, complemented
11C22×C22⋊Q8C2212yes—
12C23×C4⋊C4C2211yescharacteristic
13#2440—22no—
14#2440—22no—
15#2440—22no—
16#2440—22no—
17C2×#398C412yescomplemented
18C2×#395C412yescomplemented
19C2×#389C411yescharacteristic, complemented
20C2×#389C411yescharacteristic, complemented
21C2×#387—22no—
22C2×#385C2212yes—
23C2×#384C411yescharacteristic, complemented
24C2×#380C2212yes—
25C2×#380C2211yescharacteristic
26C2×C4×C22⋊C4C2212yes—
27C2×C4×C22⋊C4C2211yescharacteristic
28#1996—28no—
29C2×#342C2212yes—
30C2×#342C2211yescharacteristic
31C2×#342C2212yes—
32C2×#342C2211yescharacteristic
33C2×#342C2212yes—
34C2×#342C2212yes—
35C2×#342C2212yes—
36C2×#342C2212yes—
37C2×#342C2211yescharacteristic
38C2×#342C2211yescharacteristic
39#1445—22no—
40#1441—22no—
41#1436—22no—
42C22⋊#95C2212yes—
43C22⋊#95C2212yes—
44C22⋊#95C2212yes—
45C22⋊#95C2212yes—
46C23×Q8C2×C412yes—
47C24×C4C2311yescharacteristic
48C24×C4C2312yes—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—44no—
53C2×C22⋊Q8—22no—
54C2×C22⋊Q8—22no—
55C2×C22⋊Q8—22no—
56C2×C22⋊Q8—22no—
57C2×C22⋊Q8—24no—
58C2×C22⋊Q8—24no—
59C2×C22⋊Q8—42no—
60C2×C22⋊Q8—42no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—44no—
63C22×C4⋊C4C2×C412yes—
64C22×C4⋊C4C2×C412yes—
65C22×C4⋊C4C2×C412yes—
66C22×C4⋊C4C2×C411yescharacteristic
67C22×C4⋊C4C2×C412yes—
68C22×C4⋊C4C2×C411yescharacteristic
69C22×C4⋊C4C2×C411yescharacteristic
70C22×C4⋊C4C2×C412yes—
71C22×C4⋊C4C2×C412yes—
72C22×C4⋊C4C2×C411yescharacteristic
73C22×C4⋊C4D412yes—
74C22×C4⋊C4D412yes—
75C22×C4⋊C4D412yes—
76C22×C4⋊C4D412yes—
77C22×C22⋊C4C2312yes—
78C22×C22⋊C4C2312yes—
79C22×C22⋊C4C2312yes—
80C22×C22⋊C4C2312yes—
81C22×C22⋊C4C2312yes—
82C22×C22⋊C4C2311yescharacteristic
83C22×C22⋊C4C2311yescharacteristic
84C22×C42—22no—
85C22×C42C2×C411yescharacteristic
86C22×C42C2×C411yescharacteristic
87#398—22no—
88#398—22no—
89#398—22no—
90#398—22no—
91#395—22no—
92#395—22no—
93#395—22no—
94#395—22no—
95#389—21no—
96#389—22no—
97#389—21no—
98#389—22no—
99#389—21no—
100#389—21no—
101#387—48no—
102#385—22no—
103#385—22no—
104#385—22no—
105#385—22no—
106#384—22no—
107#384—21no—
108#384—21no—
109#380—22no—
110#380—22no—
111#380—24no—
112#380—42no—
113C4×C22⋊C4—22no—
114C4×C22⋊C4—22no—
115C4×C22⋊C4—24no—
116C4×C22⋊C4—42no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95—22no—
127C2×#95C2×C411yescharacteristic
128C2×#95C2×C412yes—
129C2×#95C2×C412yes—
130C2×#95C2×C411yescharacteristic
131C2×#95C2×C412yes—
132#342—24no—
133#342—24no—
134#342—24no—
135#342—24no—
136#342—22no—
137#342—22no—
138#342—42no—
139#342—42no—
140#342D414yes—
141#342Q814yescomplemented
142C25C2411yescharacteristic
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—24no—
146C22×Q8—42no—
147C22×Q8—44no—
148C22×Q8—82no—
149C23×C4—22no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4—44no—
163C23×C4—82no—
164C23×C4C22×C411yescharacteristic
165C23×C4C22×C412yes—
166C23×C4C22×C412yes—
167C23×C4C22×C412yes—
168C23×C4C22×C412yes—
169C23×C4C22×C411yescharacteristic
170C23×C4C22×C411yescharacteristic
171C23×C4C22×C411yescharacteristic
172C23×C4C22×C411yescharacteristic
173C23×C4C22×C411yescharacteristic
174C23×C4C2×D412yes—
175C23×C4C2×D412yes—
176C23×C4C2×D412yes—
177C23×C4C2×D412yes—
178C23×C4C2×D412yes—
179C23×C4C2×D412yes—
180C4⋊Q8—48no—
181C4⋊Q8—44no—
182C4⋊Q8—44no—
183C22⋊Q8—44no—
184C22⋊Q8—44no—
185C22⋊Q8—44no—
186C22⋊Q8—44no—
187C22⋊Q8—44no—
188C22⋊Q8—44no—
189C22⋊Q8—44no—
190C22⋊Q8—44no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—82no—
224C2×C4⋊C4—82no—
225C2×C4⋊C4C22⋊C411yescharacteristic
226C2×C4⋊C4C22⋊C412yes—
227C2×C4⋊C4C22⋊C412yes—
228C2×C4⋊C4C22⋊C412yes—
229C2×C4⋊C4C22⋊C412yes—
230C2×C4⋊C4C22⋊C411yescharacteristic
231C2×C4⋊C4C22⋊C411yescharacteristic
232C2×C4⋊C4C22⋊C412yes—
233C2×C4⋊C4C22⋊C412yes—
234C2×C4⋊C4C22⋊C411yescharacteristic
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4C2×D414yes—
268C2×C22⋊C4C2×D412yes—
269C2×C22⋊C4C2×Q814yes—
270C2×C22⋊C4C2×Q812yes—
271C2×C22⋊C4#4112yes—
272C2×C22⋊C4#4112yes—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—42no—
278C2×C42—42no—
279C2×C42—44no—
280C2×C42—44no—
281C2×C42—82no—
282#95—22no—
283#95—22no—
284#95—22no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—44no—
291#95—44no—
292#95—44no—
293#95—44no—
294#95—44no—
295#95—44no—
296#95—44no—
297#95—44no—
298#95—82no—
299#95—82no—
300C24—41no—
301C24—41no—
302C24—41no—
303C24—41no—
304C24C23×C411yescharacteristic
305C24C22×D412yes—
306C24C22×D411yescharacteristic
307C24C22×Q811yescharacteristic
308C24C2×#4111yescharacteristic
309C24C2×#4111yescharacteristic
310C24H(5,𝔽2)12yes—
311C242−1+412yes—
312C242−1+411yescharacteristic
313C242−1+411yescharacteristic
314C242−1+412yes—
315C2×Q8—44no—
316C2×Q8—44no—
317C2×Q8—44no—
318C2×Q8—44no—
319C2×Q8—82no—
320C2×Q8—82no—
321C2×Q8—88no—
322C2×Q8—88no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—42no—
367C22×C4—41no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—41no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—48no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
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×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—84no—
421C22×C4—84no—
422C22×C4—84no—
423C22×C4—84no—
424C22×C4C2×C22⋊C412yes—
425C22×C4C2×C22⋊C412yes—
426C22×C4C2×C22⋊C412yes—
427C22×C4C2×C22⋊C412yes—
428C22×C4C2×C22⋊C412yes—
429C22×C4C2×C22⋊C411yescharacteristic
430C22×C4C2×C22⋊C411yescharacteristic
431C22×C4C4×D412yes—
432C22×C4C4×D412yes—
433C22×C4C4×Q812yes—
434C22×C4C4×Q812yes—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—22no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—44no—
445C4⋊C4—44no—
446C4⋊C4—44no—
447C4⋊C4—44no—
448C4⋊C4—44no—
449C4⋊C4—44no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—44no—
453C4⋊C4—44no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—82no—
457C4⋊C4—82no—
458C4⋊C4—88no—
459C4⋊C4#9914yes—
460C22⋊C4—24no—
461C22⋊C4—24no—
462C22⋊C4—24no—
463C22⋊C4—24no—
464C22⋊C4—24no—
465C22⋊C4—24no—
466C22⋊C4—24no—
467C22⋊C4—24no—
468C22⋊C4—42no—
469C22⋊C4—42no—
470C22⋊C4—42no—
471C22⋊C4—42no—
472C22⋊C4—42no—
473C22⋊C4—42no—
474C22⋊C4—42no—
475C22⋊C4—42no—
476C22⋊C4—42no—
477C22⋊C4—42no—
478C22⋊C4—42no—
479C22⋊C4—42no—
480C22⋊C4—42no—
481C22⋊C4—42no—
482C22⋊C4—42no—
483C22⋊C4—42no—
484C42—42no—
485C42—42no—
486C42—42no—
487C42—42no—
488C42—88no—
489C23—21no—
490C23—22no—
491C23—21no—
492C23—22no—
493C23—21no—
494C23—22no—
495C23—21no—
496C23—22no—
497C23—42no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—42no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—42no—
510C23—42no—
511C23—41no—
512C23—41no—
513C23—42no—
514C23—42no—
515C23C22×C22⋊C411yescharacteristic
516C23C2×C4×D411yescharacteristic
517C23C2×C4×Q811yescharacteristic
518C23#51811yescharacteristic
519C23#51911yescharacteristic
520C23#51911yescharacteristic
521C23#51911yescharacteristic
522C23#53612yes—
523C23#53612yes—
524C23#53612yes—
525C23#54112yes—
526C23#54212yes—
527C23#54412yes—
528C23D4212yes—
529C23#54612yes—
530C23#54711yescharacteristic
531C23#54812yes—
532C23D4×Q812yes—
533C23D4×Q811yescharacteristic
534C23#55611yescharacteristic
535C23#55711yescharacteristic
536C23#56312yes—
537C23#56312yes—
538Q8—88no—
539Q8—88no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
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—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—82no—
605C2×C4—82no—
606C2×C4—82no—
607C2×C4—82no—
608C2×C4—82no—
609C2×C4—82no—
610C2×C4—82no—
611C2×C4—82no—
612C2×C4—82no—
613C2×C4—82no—
614C2×C4—82no—
615C2×C4—82no—
616C2×C4—82no—
617C2×C4—82no—
618C2×C4—82no—
619C2×C4—82no—
620C2×C4—82no—
621C2×C4—82no—
622C2×C4—82no—
623C2×C4—82no—
624C2×C4—82no—
625C2×C4—82no—
626C2×C4—82no—
627C2×C4—82no—
628C2×C4—82no—
629C2×C4—82no—
630C2×C4—82no—
631C2×C4—82no—
632C2×C4—82no—
633C2×C4—82no—
634C2×C4—82no—
635C2×C4—82no—
636C2×C4C2×#9912yes—
637C2×C4C2×#9914yes—
638C2×C4#41012yes—
639C22—21no—
640C22—21no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—21no—
646C22—21no—
647C22—21no—
648C22—21no—
649C22—22no—
650C22—22no—
651C22—21no—
652C22—21no—
653C22—22no—
654C22—22no—
655C22—21no—
656C22—21no—
657C22—22no—
658C22—22no—
659C22—21no—
660C22—21no—
661C22—22no—
662C22—22no—
663C22—42no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22—41no—
668C22—41no—
669C22—41no—
670C22—41no—
671C22—42no—
672C22—42no—
673C22#231011yescharacteristic
674C22#232711yescharacteristic
675C22#233111yescharacteristic
676C22D4×C22⋊C411yescharacteristic
677C22Q8×C22⋊C411yescharacteristic
678C22#236511yescharacteristic
679C22#238111yescharacteristic
680C22#267412yes—
681C22#267712yes—
682C22#271312yes—
683C22#271612yes—
684C22C22×#9911yescharacteristic
685C22C2×#41011yescharacteristic
686C22#288311yescharacteristic
687C22#288411yescharacteristic
688C4—22no—
689C4—22no—
690C4—24no—
691C4—42no—
692C4—42no—
693C4—42no—
694C4—42no—
695C4—42no—
696C4—42no—
697C4—42no—
698C4—42no—
699C4—84no—
700C4—84no—
701C4—82no—
702C4—82no—
703C4—162no—
704C4—162no—
705C2—22no—
706C2—22no—
707C2—22no—
708C2—21no—
709C2—21no—
710C2—21no—
711C2—21no—
712C2—22no—
713C2#1442011yescharacteristic
714C2#2182511yescharacteristic
715C2#2182811yescharacteristic
716C2D4×#9911yescharacteristic
717C2#2240611yescharacteristic
718C2Q8×#9911yescharacteristic
719C2#2251311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, prime radical, cosocleC25C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
10. C22×C4⋊Q8C4semidirect
17. C2×#398C4semidirect
18. C2×#395C4semidirect
19. C2×#389C4semidirect
20. C2×#389C4semidirect
23. C2×#384C4semidirect
141. #342Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#24402
C2×#19962
#138822
#113812
#113532
#112032
#109122
#108012