Finite Group Databasebeta

← #368625

Subgroups — #368625

Order 512.

1366 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1981C211yesmaximal, characteristic
2C2×#1981C211yesmaximal, characteristic
3C2×#1933C212yesmaximal
4C2×C4×C4⋊C8C211yesmaximal, characteristic
5#14714C211yesmaximal, characteristic
6C4×#389C211yesmaximal, characteristic
7#11190C212yesmaximal
8#11067C212yesmaximal
9#10926C212yesmaximal
10#10849C211yesmaximal, characteristic
11#10849C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×C4×C8C2212yes—
20C2×#391C2212yes—
21C2×#391C2212yes—
22C2×#389C2211yescharacteristic
23C2×#389C2211yescharacteristic
24C2×#384C2211yescharacteristic
25C2×C4×C4⋊C4—21no—
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C43C2211yescharacteristic
29#1981—24no—
30#1981C418yescomplemented
31#1933—28no—
32C4×C4⋊C8—24no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2212yes—
38#1436—21no—
39#1436—21no—
40C4⋊#95—21no—
41C4⋊#95—21no—
42C4⋊#95—21no—
43C4⋊#95—21no—
44C4⋊#95C2211yescharacteristic
45C4⋊#95C2211yescharacteristic
46C4×#95—21no—
47C23×Q8C2312yes—
48C23×C8C2312yes—
49C23×C8C2312yes—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8C2×C414yes—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C42—21no—
68C22×C42—21no—
69C22×C42D411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42D411yescharacteristic
72C22×C42D411yescharacteristic
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75C22×C42C2311yescharacteristic
76C2×C4⋊C8—22no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8C2×C414yes—
81C2×C4⋊C8D414yes—
82C2×#103—24no—
83C2×#103—24no—
84C2×#103—24no—
85C2×#103—24no—
86C2×#103—24no—
87C2×#103—24no—
88C2×#103—42no—
89C2×#103—42no—
90C2×#103—42no—
91C2×#103—42no—
92C2×#103C2×C418yes—
93C2×#103C2×C418yes—
94C2×C4×C8—24no—
95C2×C4×C8—24no—
96C2×C4×C8—42no—
97#391—24no—
98#391—24no—
99#391—42no—
100#391—42no—
101#389—22no—
102#389—22no—
103#389—22no—
104#389D414yes—
105#384—22no—
106#384—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—44no—
111C4×C4⋊C4C2×C414yes—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—21no—
124C43—22no—
125C43—22no—
126#340—24no—
127#340—24no—
128#340—42no—
129#340—42no—
130#336—24no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8C22×C414yes—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2411yescharacteristic
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8C22×C414yes—
171C22×C8C2×D414yes—
172C4⋊Q8—28no—
173C4⋊Q8—28no—
174C4⋊Q8—44no—
175C4⋊Q8—44no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4C22×C412yes—
219C2×C4⋊C4C22×C412yes—
220C2×C4⋊C4C2×D412yes—
221C2×C4⋊C4C2×D412yes—
222C2×C4⋊C4#4112yes—
223C2×C4⋊C4#4112yes—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—24no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—41no—
244C2×C42—41no—
245C2×C42—42no—
246C2×C42—42no—
247C2×C42—42no—
248C2×C42—42no—
249C2×C42—42no—
250C2×C42SD1614yes—
251C2×C42Q1614yes—
252C2×C42C22×C412yes—
253C2×C42C2×D412yes—
254C2×C42C2×D412yes—
255C2×C42#4112yes—
256C2×C42#4112yes—
257C2×C42#4112yes—
258C2×C42#4112yes—
259C2×C42#4112yes—
260C4⋊C8—28no—
261C4⋊C8—28no—
262#103—216no—
263#103—48no—
264#103—48no—
265#103—48no—
266C4×C8—48no—
267#95—24no—
268#95—24no—
269#95—24no—
270#95—42no—
271#95—44no—
272#95—44no—
273#95—42no—
274#95—44no—
275#95—44no—
276#95—42no—
277#95—42no—
278#95—42no—
279C24C22×D411yescharacteristic
280C2×Q8—24no—
281C2×Q8—24no—
282C2×Q8—24no—
283C2×Q8—24no—
284C2×Q8—24no—
285C2×Q8—24no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—42no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×Q8—42no—
306C2×Q8—42no—
307C2×Q8—42no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—42no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—42no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4C4×D412yes—
375C22×C4C4×D412yes—
376C22×C4#12112yes—
377C22×C4#12112yes—
378C22×C4#12112yes—
379C22×C4#12112yes—
380C22×C4#12412yes—
381C22×C4#12412yes—
382C22×C4C2×SD1612yes—
383C22×C4C2×SD1612yes—
384C22×C4C2×SD1612yes—
385C22×C4C2×Q1612yes—
386C22×C4C2×Q1612yes—
387C22×C4C2×Q1612yes—
388C22×C4#13512yes—
389C22×C4#13512yes—
390C22×C4#13712yes—
391C22×C4#13712yes—
392C22×C4C23×C411yescharacteristic
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C2×#4111yescharacteristic
396C22×C4C2×#4111yescharacteristic
397C22×C4C2×#4111yescharacteristic
398C22×C4C2×#4111yescharacteristic
399C22×C4C2×#4111yescharacteristic
400C22×C4C2×#4111yescharacteristic
401C22×C4C2×#4111yescharacteristic
402C22×C4H(5,𝔽2)11yescharacteristic
403C22×C4H(5,𝔽2)11yescharacteristic
404C22×C4H(5,𝔽2)11yescharacteristic
405C22×C42−1+411yescharacteristic
406C2×C8—24no—
407C2×C8—24no—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—24no—
411C2×C8—24no—
412C2×C8—42no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8—42no—
419C2×C8—42no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—48no—
430C4⋊C4—41no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C4⋊C4—41no—
434C4⋊C4—41no—
435C4⋊C4—41no—
436C4⋊C4—48no—
437C4⋊C4—48no—
438C4⋊C4—48no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—84no—
444C4⋊C4—84no—
445C4⋊C4C4×D414yes—
446C42—22no—
447C42—22no—
448C42—22no—
449C42—22no—
450C42—28no—
451C42—28no—
452C42—44no—
453C42—44no—
454C42—41no—
455C42—41no—
456C42—41no—
457C42—41no—
458C42—48no—
459C42—84no—
460C42#11714yes—
461C42C4×D414yes—
462C42#12314yes—
463C42#12614yes—
464C23C2×C4×D411yescharacteristic
465C23C2×#12111yescharacteristic
466C23C2×#12111yescharacteristic
467C23#52511yescharacteristic
468C23C2×#12411yescharacteristic
469C23#53511yescharacteristic
470C23#53711yescharacteristic
471C23C22×SD1611yescharacteristic
472C23C22×Q1611yescharacteristic
473C23C2×#13511yescharacteristic
474C23C2×#13511yescharacteristic
475C23C2×#13711yescharacteristic
476C23C2×#13711yescharacteristic
477C23#57511yescharacteristic
478C23#57511yescharacteristic
479Q8—44no—
480Q8—44no—
481Q8—44no—
482Q8—44no—
483Q8—82no—
484Q8—82no—
485Q8—82no—
486Q8—82no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—81no—
542C2×C4—81no—
543C2×C4—81no—
544C2×C4—81no—
545C2×C4—81no—
546C2×C4—81no—
547C2×C4—81no—
548C2×C4—81no—
549C2×C4—81no—
550C2×C4—81no—
551C2×C4—81no—
552C2×C4—81no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4#38814yes—
562C2×C4#38814yes—
563C2×C4C4×SD1614yes—
564C2×C4C4×Q1614yes—
565C2×C4#46114yes—
566C2×C4#46214yes—
567C2×C4#46514yes—
568C2×C4#46714yes—
569C2×C4#48714yes—
570C2×C4#48814yes—
571C2×C4C2×#11711yescharacteristic
572C2×C4C2×C4×D411yescharacteristic
573C2×C4C2×C4×D411yescharacteristic
574C2×C4C4×#4111yescharacteristic
575C2×C4#51811yescharacteristic
576C2×C4#52011yescharacteristic
577C2×C4C2×#12311yescharacteristic
578C2×C4C2×#12611yescharacteristic
579C2×C4#52911yescharacteristic
580C2×C4#53211yescharacteristic
581C2×C4#54011yescharacteristic
582C2×C4#54211yescharacteristic
583C2×C4D4211yescharacteristic
584C2×C4#54611yescharacteristic
585C2×C4#54711yescharacteristic
586C2×C4#54811yescharacteristic
587C2×C4#55011yescharacteristic
588C2×C4#55011yescharacteristic
589C2×C4#55011yescharacteristic
590C2×C4#55311yescharacteristic
591C2×C4#55311yescharacteristic
592C2×C4#55511yescharacteristic
593C2×C4#55611yescharacteristic
594C2×C4#55611yescharacteristic
595C2×C4#55911yescharacteristic
596C2×C4#55911yescharacteristic
597C2×C4#55911yescharacteristic
598C2×C4#55911yescharacteristic
599C8—44no—
600C8—44no—
601C22C2×#38811yescharacteristic
602C22C4×#12111yescharacteristic
603C22#232411yescharacteristic
604C22#241411yescharacteristic
605C22#241611yescharacteristic
606C22#254811yescharacteristic
607C22#259211yescharacteristic
608C22C2×C4×SD1611yescharacteristic
609C22C2×C4×Q1611yescharacteristic
610C22C4×#13711yescharacteristic
611C22C4×#13711yescharacteristic
612C22C2×#46111yescharacteristic
613C22C2×#46211yescharacteristic
614C22#303511yescharacteristic
615C22#303511yescharacteristic
616C22C2×#46511yescharacteristic
617C22C2×#46711yescharacteristic
618C22#305611yescharacteristic
619C22#305611yescharacteristic
620C22#310111yescharacteristic
621C22#310411yescharacteristic
622C22#310711yescharacteristic
623C22#310711yescharacteristic
624C22C2×#48711yescharacteristic
625C22C2×#48811yescharacteristic
626C22#313911yescharacteristic
627C22#314011yescharacteristic
628C22#320611yescharacteristic
629C22#320911yescharacteristic
630C22#321211yescharacteristic
631C22#321211yescharacteristic
632C22#324611yescharacteristic
633C22#324811yescharacteristic
634C22#324811yescharacteristic
635C22#324911yescharacteristic
636C4—24no—
637C4—24no—
638C4—24no—
639C4—42no—
640C4—42no—
641C4—44no—
642C4—82no—
643C4—82no—
644C4—82no—
645C4—84no—
646C4—162no—
647C4#193318yescomplemented
648C4#235411yescharacteristic
649C4#235711yescharacteristic
650C4#236611yescharacteristic
651C4#236711yescharacteristic
652C4#244511yescharacteristic
653C4#244811yescharacteristic
654C4#251211yescharacteristic
655C4#251311yescharacteristic
656C2#1402211yescharacteristic
657C2C2×#193311yescharacteristic
658C2#2109411yescharacteristic
659C2C4×#46111yescharacteristic
660C2C4×#46211yescharacteristic
661C2#2214611yescharacteristic
662C2#2214811yescharacteristic
663C2#2465111yescharacteristic
664C2#2465211yescharacteristic
665C2#2475111yescharacteristic
666C2#2475211yescharacteristic
667C2#2874811yescharacteristic
668C2#2875211yescharacteristic
669C2#2934711yescharacteristic
670C2#2934911yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
30. #1981C4semidirect
647. C4#1933semidirect

Maximal subgroups

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

subgroupindex
C2×#19812
C2×#19332
C2×C4×C4⋊C82
#147142
C4×#3892
#111902
#110672
#109262
#108492