Finite Group Databasebeta

← #380340

Subgroups — #380340

Order 512.

1380 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2295C211yesmaximal, characteristic
2C2×#1948C211yesmaximal, characteristic
3C2×#1803C211yesmaximal, characteristic
4C2×#1803C211yesmaximal, characteristic
5#14712C211yesmaximal, characteristic
6#11327C212yesmaximal
7#11292C212yesmaximal
8#11122C212yesmaximal
9#11067C212yesmaximal
10#10849C211yesmaximal, characteristic
11#10832C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C22×#106C2212yes—
14C22×C4⋊C8C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×#103C2212yes—
17C22×#103C2212yes—
18#2295—24no—
19#2295—24no—
20C2×#391C2211yescharacteristic
21C2×#391C2211yescharacteristic
22C2×#389—21no—
23C2×#389—21no—
24C2×#389C2211yescharacteristic
25C2×#389C2211yescharacteristic
26C2×#389C2212yes—
27C2×#387—21no—
28C2×#387—21no—
29C2×#384C2211yescharacteristic
30C2×#384C2211yescharacteristic
31C2×C4×C4⋊C4—21no—
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×#376—21no—
34C2×C43C2211yescharacteristic
35#1948—24no—
36#1803—24no—
37#1803C418yescomplemented
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#336C2212yes—
44C2×#336C2212yes—
45C4⋊#95—21no—
46C4⋊#95—21no—
47C4⋊#95C2212yes—
48C23×Q8C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×C4×Q8—22no—
52C2×C4×Q8—22no—
53C2×C4×Q8—22no—
54C2×C4×Q8C2×C414yes—
55C2×C4×Q8D414yes—
56C2×C4×Q8Q814yes—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C42—21no—
71C22×C42—21no—
72C22×C42D411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79C2×#106—24no—
80C2×#106—42no—
81C2×#106—42no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8C2×C414yes—
88C2×#103—24no—
89C2×#103—42no—
90C2×#103—42no—
91C2×#103—42no—
92C2×#103—42no—
93C2×#103C2×C418yes—
94#391—22no—
95#391—22no—
96#391—22no—
97#391—22no—
98#389—22no—
99#389—22no—
100#389—22no—
101#389—22no—
102#389—42no—
103#389—42no—
104#389—44no—
105#389—44no—
106#387—28no—
107#387—28no—
108#384—22no—
109#384—22no—
110#384—22no—
111#384C2×C414yes—
112C4×C4⋊C4—28no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4D414yes—
115#376—28no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95—22no—
126C43—22no—
127C43—22no—
128#340—24no—
129#340—24no—
130#340—42no—
131#340—42no—
132#340C2×C418yes—
133#336—42no—
134#336—42no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8—21no—
144C22×Q8—21no—
145C22×Q8—21no—
146C22×Q8C22×C412yes—
147C22×Q8C2×D412yes—
148C22×Q8C2×Q812yes—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—22no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—22no—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2411yescharacteristic
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×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8C22×C414yes—
176C4×Q8—28no—
177C4×Q8—28no—
178C4×Q8—28no—
179C4×Q8C4⋊C4116yescomplemented
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
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⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4C22×C414yes—
229C2×C4⋊C4C22×C412yes—
230C2×C4⋊C4C2×D412yes—
231C2×C4⋊C4C2×D412yes—
232C2×C4⋊C4C2×D412yes—
233C2×C4⋊C4C2×Q812yes—
234C2×C42—24no—
235C2×C42—24no—
236C2×C42—22no—
237C2×C42—24no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—24no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—41no—
260C2×C42—41no—
261C2×C42—42no—
262C2×C42—42no—
263C2×C42SD1614yes—
264C2×C42C22×C412yes—
265C2×C42C2×D412yes—
266C2×C42C2×D412yes—
267C2×C42C2×Q812yes—
268C2×C42#4112yes—
269C2×C42#4112yes—
270#106—48no—
271C4⋊C8—28no—
272C4⋊C8—44no—
273#103—48no—
274#103—48no—
275#95—24no—
276#95—24no—
277#95—24no—
278#95—24no—
279#95—24no—
280#95—24no—
281#95—24no—
282#95—44no—
283#95—44no—
284#95—44no—
285C24C22×D411yescharacteristic
286C2×Q8—22no—
287C2×Q8—22no—
288C2×Q8—22no—
289C2×Q8—22no—
290C2×Q8—22no—
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8—22no—
294C2×Q8—22no—
295C2×Q8—22no—
296C2×Q8—22no—
297C2×Q8—22no—
298C2×Q8—22no—
299C2×Q8—22no—
300C2×Q8—22no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8—41no—
308C2×Q8—41no—
309C2×Q8—41no—
310C2×Q8—41no—
311C2×Q8—41no—
312C2×Q8—41no—
313C2×Q8C2×C4⋊C414yes—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—42no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—42no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—42no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—42no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4C4×D412yes—
385C22×C4C4×D412yes—
386C22×C4#12012yes—
387C22×C4#12012yes—
388C22×C4C22⋊Q812yes—
389C22×C4C22⋊Q812yes—
390C22×C4C2×SD1612yes—
391C22×C4C2×SD1612yes—
392C22×C4C2×SD1612yes—
393C22×C4#13512yes—
394C22×C4AGL(1,ℤ/8)12yes—
395C22×C4#13712yes—
396C22×C4#13712yes—
397C22×C4#13712yes—
398C22×C4C23×C411yescharacteristic
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C22×Q811yescharacteristic
402C22×C4C2×#4111yescharacteristic
403C22×C4C2×#4111yescharacteristic
404C22×C4H(5,𝔽2)11yescharacteristic
405C22×C4H(5,𝔽2)11yescharacteristic
406C22×C4H(5,𝔽2)11yescharacteristic
407C22×C4H(5,𝔽2)11yescharacteristic
408C22×C42−1+411yescharacteristic
409C22×C42−1+411yescharacteristic
410C22×C42−1+411yescharacteristic
411C22×C42−1+411yescharacteristic
412C2×C8—24no—
413C2×C8—24no—
414C2×C8—24no—
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—42no—
425C2×C8—42no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—216no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—48no—
437C4⋊C4—48no—
438C4⋊C4—48no—
439C4⋊C4—41no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4—41no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—48no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—84no—
451C4⋊C4—88no—
452C4⋊C4—84no—
453C4⋊C4C2×C4⋊C414yes—
454C42—216no—
455C42—28no—
456C42—44no—
457C42—44no—
458C42—41no—
459C42—41no—
460C42—41no—
461C42—41no—
462C42—41no—
463C42—41no—
464C42—41no—
465C42—41no—
466C42—44no—
467C42—48no—
468C42C2×C4⋊C414yes—
469C42C4×D414yes—
470C42C22⋊Q814yes—
471C42#12314yes—
472C23C2×C4×D411yescharacteristic
473C23C2×#12011yescharacteristic
474C23C2×C22⋊Q811yescharacteristic
475C23#53511yescharacteristic
476C23#53611yescharacteristic
477C23#53711yescharacteristic
478C23#53711yescharacteristic
479C23C22×SD1611yescharacteristic
480C23C2×#13511yescharacteristic
481C23C2×AGL(1,ℤ/8)11yescharacteristic
482C23C2×#13711yescharacteristic
483C23C2×#13711yescharacteristic
484C23C2×#13711yescharacteristic
485C23#57511yescharacteristic
486C23#57511yescharacteristic
487Q8—24no—
488Q8—24no—
489Q8—42no—
490Q8—42no—
491Q8—42no—
492Q8—42no—
493Q8—42no—
494Q8—42no—
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—22no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—81no—
555C2×C4—81no—
556C2×C4—81no—
557C2×C4—81no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—81no—
561C2×C4—81no—
562C2×C4—81no—
563C2×C4—81no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4#38514yes—
572C2×C4#38514yes—
573C2×C4C4×SD1614yes—
574C2×C4#44114yes—
575C2×C4#44814yes—
576C2×C4#45214yes—
577C2×C4#47514yes—
578C2×C4#47914yes—
579C2×C4C22×C4⋊C411yescharacteristic
580C2×C4C2×C4×D411yescharacteristic
581C2×C4#51811yescharacteristic
582C2×C4#51911yescharacteristic
583C2×C4#52011yescharacteristic
584C2×C4#52011yescharacteristic
585C2×C4C2×C22⋊Q811yescharacteristic
586C2×C4C2×#12311yescharacteristic
587C2×C4#53511yescharacteristic
588C2×C4#53511yescharacteristic
589C2×C4#53611yescharacteristic
590C2×C4#53611yescharacteristic
591C2×C4#53711yescharacteristic
592C2×C4#53711yescharacteristic
593C2×C4#53711yescharacteristic
594C2×C4#53711yescharacteristic
595C2×C4#54011yescharacteristic
596C2×C4#54011yescharacteristic
597C2×C4#54111yescharacteristic
598C2×C4#54111yescharacteristic
599C2×C4#54211yescharacteristic
600C2×C4#54211yescharacteristic
601C2×C4#54211yescharacteristic
602C2×C4#54211yescharacteristic
603C2×C4#54411yescharacteristic
604C2×C4#54411yescharacteristic
605C2×C4#54411yescharacteristic
606C2×C4#54411yescharacteristic
607C8—44no—
608C8—82no—
609C22C2×#38511yescharacteristic
610C22#232311yescharacteristic
611C22#232411yescharacteristic
612C22#261511yescharacteristic
613C22#261611yescharacteristic
614C22#262811yescharacteristic
615C22#262911yescharacteristic
616C22C2×C4×SD1611yescharacteristic
617C22C2×#44111yescharacteristic
618C22C4×AGL(1,ℤ/8)11yescharacteristic
619C22C4×#13711yescharacteristic
620C22C2×#44811yescharacteristic
621C22C2×#45211yescharacteristic
622C22#300611yescharacteristic
623C22#300711yescharacteristic
624C22C2×#47511yescharacteristic
625C22C2×#47911yescharacteristic
626C22#307911yescharacteristic
627C22#308011yescharacteristic
628C22#320811yescharacteristic
629C22#321211yescharacteristic
630C22#321611yescharacteristic
631C22#321811yescharacteristic
632C22#322911yescharacteristic
633C22#323311yescharacteristic
634C22#323711yescharacteristic
635C22#323811yescharacteristic
636C22#324611yescharacteristic
637C22#324611yescharacteristic
638C22#324711yescharacteristic
639C22#324811yescharacteristic
640C22#325211yescharacteristic
641C22#325211yescharacteristic
642C22#325411yescharacteristic
643C22#325511yescharacteristic
644C4—24no—
645C4—44no—
646C4—44no—
647C4—42no—
648C4—42no—
649C4—42no—
650C4—42no—
651C4—84no—
652C4—82no—
653C4—82no—
654C4—162no—
655C4#186518yes—
656C4#231611yescharacteristic
657C4#231711yescharacteristic
658C4#232311yescharacteristic
659C4#232411yescharacteristic
660C4#262411yescharacteristic
661C4#262511yescharacteristic
662C4#262811yescharacteristic
663C4#262911yescharacteristic
664C2#1426411yescharacteristic
665C2C2×#186511yescharacteristic
666C2#2087511yescharacteristic
667C2#2211211yescharacteristic
668C2#2211911yescharacteristic
669C2#2214711yescharacteristic
670C2#2215411yescharacteristic
671C2#2966111yescharacteristic
672C2#2966511yescharacteristic
673C2#2968811yescharacteristic
674C2#2969211yescharacteristic
675C2#2998311yescharacteristic
676C2#2998811yescharacteristic
677C2#2999611yescharacteristic
678C2#2999711yescharacteristic
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
37. #1803C4semidirect
179. C4×Q8C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22952
C2×#19482
C2×#18032
#147122
#113272
#112922
#111222
#110672
#108492
#108322