Finite Group Databasebeta

← #119702

Subgroups — #119702

Order 512.

992 conjugacy classes of subgroups.

The subgroup lattice has 734 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1954C211yesmaximal, characteristic
2C2×#1802C211yesmaximal, characteristic
3C2×#1802C211yesmaximal, characteristic
4#10841C211yesmaximal, characteristic, complemented
5C2×#1296C211yesmaximal, characteristic, complemented
6C2×#1296C211yesmaximal, characteristic, complemented
7C2×#1277C211yesmaximal, characteristic, complemented
8C22×C4×D4C2211yescharacteristic
9C22×C4⋊C8C2211yescharacteristic
10C22×C4⋊C8C2211yescharacteristic
11C22×C4⋊C8C2211yescharacteristic
12C22×#102—21no—
13C22×#102—21no—
14C22×C22⋊C8—21no—
15C22×C22⋊C8—21no—
16C22×C4×C8C2211yescharacteristic
17C2×#384C2211yescharacteristic
18C2×#384C2211yescharacteristic
19#1954—22no—
20#1954—22no—
21#1802—22no—
22#1802—22no—
23#1802—22no—
24#1802C414yescomplemented
25C2×#340—21no—
26C2×#340—21no—
27C2×#340—21no—
28C2×#340—21no—
29#1296—22no—
30#1296—22no—
31#1277—22no—
32C23×D4D411yescharacteristic
33C24×C4—21no—
34C23×C8—21no—
35C23×C8—21no—
36C23×C8—21no—
37C23×C8D411yescharacteristic
38C23×C8D411yescharacteristic
39C2×C4×D4—21no—
40C2×C4×D4—21no—
41C2×C4×D4—21no—
42C2×C4×D4—21no—
43C2×C4×D4—21no—
44C2×C4×D4—21no—
45C2×C4×D4—21no—
46C2×C4×D4—21no—
47C2×C4×D4—21no—
48C2×C4×D4—21no—
49C2×C4×D4C2×C412yes—
50C2×C4×D4D412yes—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4D411yescharacteristic
58C22×C22⋊C4—21no—
59C22×C42C2311yescharacteristic
60C2×C4⋊C8—21no—
61C2×C4⋊C8—21no—
62C2×C4⋊C8—21no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8—21no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8C2×C412yes—
76C2×C4⋊C8D412yes—
77C2×C4⋊C8D412yes—
78C2×#102—24no—
79C2×#102—24no—
80C2×#102—42no—
81C2×#102—42no—
82C2×#102—42no—
83C2×#102—42no—
84C2×C22⋊C8—24no—
85C2×C22⋊C8—24no—
86C2×C22⋊C8—42no—
87C2×C22⋊C8—42no—
88C2×C22⋊C8—42no—
89C2×C22⋊C8—42no—
90C2×C4×C8—21no—
91C2×C4×C8—21no—
92C2×C4×C8—21no—
93C2×C4×C8—21no—
94C2×C4×C8—21no—
95C2×C4×C8—21no—
96#384—21no—
97#384—21no—
98#384—21no—
99#384—21no—
100#384—21no—
101#384—21no—
102#384C2×C412yes—
103#384D412yes—
104#340—24no—
105#340—24no—
106#340—42no—
107#340—42no—
108C25—21no—
109C22×D4—21no—
110C22×D4—21no—
111C22×D4—21no—
112C22×D4—21no—
113C22×D4—21no—
114C22×D4—21no—
115C22×D4—21no—
116C22×D4—21no—
117C22×D4—21no—
118C22×D4—21no—
119C22×D4—21no—
120C22×D4C22⋊C412yes—
121C22×D4SD1612yes—
122C22×D4Q1612yes—
123C23×C4—21no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—41no—
127C23×C4—41no—
128C23×C4—41no—
129C23×C4—41no—
130C23×C4—41no—
131C23×C4—41no—
132C23×C4—41no—
133C23×C4—41no—
134C23×C4—41no—
135C23×C4—41no—
136C23×C4—41no—
137C23×C4—41no—
138C23×C4—41no—
139C23×C4—41no—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4#4111yescharacteristic
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—41no—
159C22×C8—41no—
160C22×C8—41no—
161C22×C8—41no—
162C22×C8—41no—
163C22×C8—41no—
164C22×C8—41no—
165C22×C8—41no—
166C22×C8—41no—
167C22×C8—41no—
168C22×C8—41no—
169C22×C8—41no—
170C22×C8—41no—
171C22×C8—41no—
172C22×C8—41no—
173C22×C8—41no—
174C22×C8—41no—
175C22×C8—41no—
176C22×C8D812yes—
177C22×C8SD1612yes—
178C4×D4—22no—
179C4×D4—22no—
180C4×D4—22no—
181C4×D4—22no—
182C4×D4—22no—
183C4×D4—22no—
184C4×D4—22no—
185C4×D4—22no—
186C4×D4—41no—
187C4×D4—41no—
188C4×D4—41no—
189C4×D4—41no—
190C4×D4—41no—
191C4×D4—41no—
192C4×D4—41no—
193C4×D4—41no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—81no—
228C2×C4⋊C4—81no—
229C2×C4⋊C4—81no—
230C2×C4⋊C4—81no—
231C2×C4⋊C4C22⋊C412yes—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—41no—
235C2×C22⋊C4—41no—
236C2×C22⋊C4—41no—
237C2×C22⋊C4—41no—
238C2×C42C22×C411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42#4111yescharacteristic
244C2×C42#4111yescharacteristic
245C2×C42#4111yescharacteristic
246C2×C42#4111yescharacteristic
247C2×C42#4111yescharacteristic
248C2×C42#4111yescharacteristic
249C2×C42#4111yescharacteristic
250C4⋊C8—22no—
251C4⋊C8—22no—
252C4⋊C8—22no—
253C4⋊C8—22no—
254C4⋊C8—22no—
255C4⋊C8—22no—
256C4⋊C8—41no—
257C4⋊C8—41no—
258C4⋊C8—41no—
259C4⋊C8—41no—
260C4⋊C8—41no—
261C4⋊C8—41no—
262#102—48no—
263#102—48no—
264C22⋊C8—48no—
265C22⋊C8—48no—
266C4×C8—41no—
267C4×C8—41no—
268C4×C8—41no—
269C4×C8—41no—
270C24—22no—
271C24—22no—
272C24—22no—
273C24—41no—
274C24—41no—
275C24—41no—
276C24—41no—
277C24—41no—
278C24—41no—
279C24—41no—
280C24—41no—
281C24—41no—
282C24—41no—
283C24—41no—
284C24—41no—
285C24#12111yescharacteristic
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—22no—
299C2×D4—22no—
300C2×D4—22no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4#10314yes—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—81no—
340C22×C4—81no—
341C22×C4—81no—
342C22×C4—81no—
343C22×C4—81no—
344C22×C4—81no—
345C22×C4—81no—
346C22×C4—81no—
347C22×C4—81no—
348C22×C4—81no—
349C22×C4—81no—
350C22×C4—81no—
351C22×C4—81no—
352C22×C4—81no—
353C22×C4—81no—
354C22×C4—81no—
355C22×C4—81no—
356C22×C4—81no—
357C22×C4—81no—
358C22×C4—81no—
359C22×C4—81no—
360C22×C4—81no—
361C22×C4—81no—
362C22×C4—81no—
363C22×C4—81no—
364C22×C4—81no—
365C22×C4—81no—
366C22×C4—81no—
367C22×C4C2×C22⋊C411yescharacteristic
368C22×C4C4×D411yescharacteristic
369C22×C4C4×D411yescharacteristic
370C22×C4#12011yescharacteristic
371C22×C4#12011yescharacteristic
372C22×C4#12111yescharacteristic
373C22×C4#12111yescharacteristic
374C22×C4#12111yescharacteristic
375C22×C4#12311yescharacteristic
376C22×C4#12311yescharacteristic
377C22×C4#12311yescharacteristic
378C22×C4#12311yescharacteristic
379C22×C4#12311yescharacteristic
380C22×C4#12311yescharacteristic
381C22×C4#12411yescharacteristic
382C22×C4#12411yescharacteristic
383C22×C4#12611yescharacteristic
384C22×C4#12611yescharacteristic
385C22×C4C2×D811yescharacteristic
386C22×C4C2×SD1611yescharacteristic
387C22×C4C2×SD1611yescharacteristic
388C22×C4C2×Q1611yescharacteristic
389C22×C4#13511yescharacteristic
390C22×C4#13511yescharacteristic
391C22×C4#13511yescharacteristic
392C22×C4#13511yescharacteristic
393C22×C4AGL(1,ℤ/8)11yescharacteristic
394C22×C4AGL(1,ℤ/8)11yescharacteristic
395C22×C4AGL(1,ℤ/8)11yescharacteristic
396C22×C4AGL(1,ℤ/8)11yescharacteristic
397C22×C4AGL(1,ℤ/8)11yescharacteristic
398C22×C4AGL(1,ℤ/8)11yescharacteristic
399C22×C4#13711yescharacteristic
400C22×C4#13711yescharacteristic
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8—42no—
419C2×C8—42no—
420C2×C8—42no—
421C2×C8—42no—
422C2×C8—42no—
423C2×C8—42no—
424C2×C8—81no—
425C2×C8—81no—
426C2×C8—81no—
427C2×C8—81no—
428C2×C8—81no—
429C2×C8—81no—
430C2×C8—81no—
431C2×C8—81no—
432C2×C8—81no—
433C2×C8—81no—
434C2×C8—81no—
435C2×C8—81no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—41no—
439C4⋊C4—41no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—82no—
449C4⋊C4—82no—
450C4⋊C4—82no—
451C4⋊C4—82no—
452C4⋊C4—81no—
453C4⋊C4—81no—
454C4⋊C4—81no—
455C4⋊C4—81no—
456C4⋊C4—81no—
457C4⋊C4—81no—
458C4⋊C4—81no—
459C4⋊C4—81no—
460C22⋊C4—42no—
461C22⋊C4—42no—
462C22⋊C4—42no—
463C22⋊C4—42no—
464C42#11711yescharacteristic
465C42#11711yescharacteristic
466C42C4×D411yescharacteristic
467C42C4×D411yescharacteristic
468C42#12111yescharacteristic
469C42#12111yescharacteristic
470C42#12111yescharacteristic
471C42#12111yescharacteristic
472C42#12311yescharacteristic
473C42#12311yescharacteristic
474C42#12311yescharacteristic
475C42#12311yescharacteristic
476C42#12411yescharacteristic
477C42#12411yescharacteristic
478C42#12611yescharacteristic
479C42#12611yescharacteristic
480C23—24no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—81no—
500C23—81no—
501C23—81no—
502C23—81no—
503C23—81no—
504C23—81no—
505C23—81no—
506C23—81no—
507C23—81no—
508C23—81no—
509C23—81no—
510C23—81no—
511C23—81no—
512C23—81no—
513C23—81no—
514C23—81no—
515C23#38611yescharacteristic
516C23#39411yescharacteristic
517C23#39711yescharacteristic
518C23#45911yescharacteristic
519C23#46111yescharacteristic
520C23#46311yescharacteristic
521C23#46311yescharacteristic
522C23#46511yescharacteristic
523C23#46711yescharacteristic
524C23#46811yescharacteristic
525C23#46911yescharacteristic
526C23#47111yescharacteristic
527C23#47111yescharacteristic
528C23#47211yescharacteristic
529C23#47311yescharacteristic
530D4—24no—
531D4—24no—
532D4—42no—
533D4—42no—
534D4—42no—
535D4—42no—
536D4—42no—
537D4—42no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—81no—
554C2×C4—81no—
555C2×C4—81no—
556C2×C4—81no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—161no—
572C2×C4—161no—
573C2×C4—161no—
574C2×C4—161no—
575C2×C4—161no—
576C2×C4—161no—
577C2×C4—161no—
578C2×C4—161no—
579C2×C4#38111yescharacteristic
580C2×C4#38611yescharacteristic
581C2×C4#38811yescharacteristic
582C2×C4#38811yescharacteristic
583C2×C4#38811yescharacteristic
584C2×C4#38811yescharacteristic
585C2×C4#39411yescharacteristic
586C2×C4#39411yescharacteristic
587C2×C4#39411yescharacteristic
588C2×C4#39711yescharacteristic
589C2×C4#39711yescharacteristic
590C2×C4#39711yescharacteristic
591C2×C4C2×#10311yescharacteristic
592C2×C4#41611yescharacteristic
593C2×C4#41711yescharacteristic
594C2×C4#41811yescharacteristic
595C2×C4C4×D811yescharacteristic
596C2×C4C4×SD1611yescharacteristic
597C2×C4#44011yescharacteristic
598C2×C4#44211yescharacteristic
599C2×C4#44711yescharacteristic
600C2×C4#44911yescharacteristic
601C2×C4#44911yescharacteristic
602C2×C4#44911yescharacteristic
603C2×C4#45011yescharacteristic
604C2×C4#45011yescharacteristic
605C2×C4#45111yescharacteristic
606C2×C4#45211yescharacteristic
607C2×C4#45911yescharacteristic
608C2×C4#46011yescharacteristic
609C2×C4#46311yescharacteristic
610C2×C4#46411yescharacteristic
611C2×C4#46511yescharacteristic
612C2×C4#46511yescharacteristic
613C2×C4#46611yescharacteristic
614C2×C4#46711yescharacteristic
615C2×C4#46811yescharacteristic
616C2×C4#46811yescharacteristic
617C2×C4#46911yescharacteristic
618C2×C4#46911yescharacteristic
619C2×C4#48011yescharacteristic
620C2×C4#48111yescharacteristic
621C2×C4#48111yescharacteristic
622C2×C4#48111yescharacteristic
623C2×C4#48211yescharacteristic
624C2×C4#48211yescharacteristic
625C2×C4#48211yescharacteristic
626C2×C4#48211yescharacteristic
627C2×C4#48211yescharacteristic
628C2×C4#48211yescharacteristic
629C2×C4#48311yescharacteristic
630C2×C4#48411yescharacteristic
631C2×C4#48411yescharacteristic
632C2×C4#48411yescharacteristic
633C2×C4#48511yescharacteristic
634C2×C4#48511yescharacteristic
635C2×C4#48611yescharacteristic
636C2×C4#48811yescharacteristic
637C2×C4#48911yescharacteristic
638C2×C4#49011yescharacteristic
639C8—42no—
640C8—42no—
641C8—82no—
642C8—82no—
643C8—82no—
644C22—44no—
645C22—44no—
646C22—44no—
647C22—82no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22—82no—
657C22—82no—
658C22—82no—
659C22#144011yescharacteristic
660C22#166511yescharacteristic
661C22#166911yescharacteristic
662C22#167111yescharacteristic
663C22#167511yescharacteristic
664C22#167711yescharacteristic
665C22#167911yescharacteristic
666C22#168311yescharacteristic
667C22#168311yescharacteristic
668C22#168511yescharacteristic
669C22#168711yescharacteristic
670C22#169111yescharacteristic
671C22#169311yescharacteristic
672C22#169711yescharacteristic
673C22#169911yescharacteristic
674C22#170111yescharacteristic
675C22#170111yescharacteristic
676C22#187711yescharacteristic
677C22#187911yescharacteristic
678C22#189211yescharacteristic
679C22#189511yescharacteristic
680C22#190711yescharacteristic
681C22#190811yescharacteristic
682C22#203411yescharacteristic
683C22#203611yescharacteristic
684C22#205111yescharacteristic
685C22#205211yescharacteristic
686C22#205411yescharacteristic
687C22#205511yescharacteristic
688C22#207111yescharacteristic
689C22#207311yescharacteristic
690C22#207611yescharacteristic
691C22#207711yescharacteristic
692C22#207811yescharacteristic
693C22#207911yescharacteristic
694C4—21no—
695C4—21no—
696C4—21no—
697C4—21no—
698C4—82no—
699C4—162no—
700C4—162no—
701C4#182411yescharacteristic
702C4#182511yescharacteristic
703C4#187211yescharacteristic
704C4#187511yescharacteristic
705C4#192511yescharacteristic
706C4#192611yescharacteristic
707C4#193411yescharacteristic
708C4#193611yescharacteristic
709C4#203311yescharacteristic
710C4#203411yescharacteristic
711C4#203411yescharacteristic
712C4#203611yescharacteristic
713C4#204111yescharacteristic
714C4#204211yescharacteristic
715C4#207611yescharacteristic
716C4#207711yescharacteristic
717C2—84no—
718C2#921511yescharacteristic
719C2#921911yescharacteristic
720C2#922511yescharacteristic
721C2#922911yescharacteristic
722C2#1047311yescharacteristic
723C2#1047711yescharacteristic
724C2#1048111yescharacteristic
725C2#1048511yescharacteristic
726C2#1061611yescharacteristic
727C2#1061911yescharacteristic
728C2#1062211yescharacteristic
729C2#1062511yescharacteristic
730C2#1118911yescharacteristic
731C2#1119211yescharacteristic
732C2#1120011yescharacteristic
subgroupG/H
center, socleC24#121
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC23×D4D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #10841C2semidirect
5. C2×#1296C2semidirect
6. C2×#1296C2semidirect
7. C2×#1277C2semidirect
24. #1802C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19542
C2×#18022
#108412
C2×#12962
C2×#12772