Finite Group Databasebeta

← #360143

Subgroups — #360143

Order 512.

1794 conjugacy classes of subgroups.

The subgroup lattice has 713 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×Q16C211yesmaximal, characteristic
2C2×Q8×C4⋊C4C211yesmaximal, characteristic
3C2×#1878C212yesmaximal
4C2×#1803C211yesmaximal, characteristic
5C2×#1803C211yesmaximal, characteristic
6#13926C211yesmaximal, characteristic
7#11219C212yesmaximal
8#11067C212yesmaximal
9#10841C211yesmaximal, characteristic
10#10834C211yesmaximal, characteristic
11#10789C212yesmaximal
12C23×Q16C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4×Q8C2212yes—
15C22×C4×Q8C2211yescharacteristic
16C2×C4×Q16—24no—
17C2×C4×Q16—24no—
18C2×C4×Q16C418yescomplemented
19C22×#107C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C22×#103C2212yes—
22C22×#103C2212yes—
23C22×#103C2211yescharacteristic
24C22×#103C2211yescharacteristic
25C22×C4×C8C2211yescharacteristic
26Q8×C4⋊C4—24no—
27Q8×C4⋊C4—24no—
28C2×#391C2212yes—
29C2×#389—22no—
30C2×#389C2212yes—
31C2×#384—21no—
32C2×#384C2211yescharacteristic
33C2×#384C2211yescharacteristic
34C2×C4×C4⋊C4—21no—
35C2×C4×C4⋊C4C2211yescharacteristic
36#1878—28no—
37#1803—24no—
38#1803—24no—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#336C2212yes—
44C2×#336C2212yes—
45#1441—22no—
46C4⋊#95—21no—
47C4⋊#95—21no—
48C4⋊#95C2212yes—
49C23×Q8C2311yescharacteristic
50C23×Q8C2311yescharacteristic
51C22×Q16—22no—
52C22×Q16—22no—
53C22×Q16—22no—
54C22×Q16—22no—
55C22×Q16C2×C414yes—
56C22×Q16D414yes—
57C22×Q16Q814yes—
58C23×C8C2312yes—
59C23×C8C2311yescharacteristic
60C23×C8C2311yescharacteristic
61C2×C4×Q8—24no—
62C2×C4×Q8—24no—
63C2×C4×Q8—22no—
64C2×C4×Q8—22no—
65C2×C4×Q8—22no—
66C2×C4×Q8—22no—
67C2×C4×Q8—24no—
68C2×C4×Q8—24no—
69C2×C4×Q8—22no—
70C2×C4×Q8—22no—
71C2×C4×Q8—22no—
72C2×C4×Q8—22no—
73C2×C4×Q8—42no—
74C2×C4×Q8—42no—
75C2×C4×Q8C2×C414yes—
76C2×C4×Q8C2×C414yes—
77C2×C4×Q8D414yes—
78C2×C4×Q8D414yes—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—22no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4—22no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4—21no—
86C22×C4⋊C4—21no—
87C22×C4⋊C4D411yescharacteristic
88C22×C4⋊C4D411yescharacteristic
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C42—21no—
94C22×C42—21no—
95C22×C42—22no—
96C22×C42C2312yes—
97C22×C42C2311yescharacteristic
98C4×Q16—232no—
99C2×#107—22no—
100C2×#107—22no—
101C2×#107C2×C414yes—
102C2×C4⋊C8—22no—
103C2×C4⋊C8—22no—
104C2×C4⋊C8—22no—
105C2×#103—22no—
106C2×#103—22no—
107C2×#103—22no—
108C2×#103—22no—
109C2×#103—24no—
110C2×#103—24no—
111C2×#103—24no—
112C2×#103—24no—
113C2×#103—24no—
114C2×#103—42no—
115C2×#103C2×C414yes—
116C2×#103C2×C414yes—
117C2×C4×C8—22no—
118C2×C4×C8—22no—
119C2×C4×C8C2×C414yes—
120#391—24no—
121#391—42no—
122#389—216no—
123#389—24no—
124#389—24no—
125#384—28no—
126#384—22no—
127#384—22no—
128#384—22no—
129#384D414yes—
130C4×C4⋊C4—28no—
131C4×C4⋊C4—22no—
132C4×C4⋊C4—22no—
133C2×#95—22no—
134C2×#95—22no—
135C2×#95—22no—
136C2×#95—22no—
137C2×#95—22no—
138C2×#95D412yes—
139#340—24no—
140#340—24no—
141#340—24no—
142#340—42no—
143#336—24no—
144#336—42no—
145C22×Q8—21no—
146C22×Q8—21no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8—21no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×Q8—21no—
155C22×Q8—21no—
156C22×Q8—21no—
157C22×Q8—21no—
158C22×Q8—21no—
159C22×Q8—21no—
160C22×Q8—21no—
161C22×Q8—21no—
162C22×Q8—21no—
163C22×Q8C22×C412yes—
164C22×Q8C22×C412yes—
165C22×Q8C2×D412yes—
166C22×Q8C2×D412yes—
167C22×Q8C2×D412yes—
168C22×Q8C2×D412yes—
169C22×Q8C2×Q812yes—
170C22×Q8C2×Q812yes—
171C22×Q8#4112yes—
172C22×Q8#4112yes—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—22no—
177C23×C4—21no—
178C23×C4—22no—
179C23×C4C2×D412yes—
180C23×C4C2×D412yes—
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2411yescharacteristic
184C2×Q16—28no—
185C2×Q16—28no—
186C2×Q16—28no—
187C2×Q16—28no—
188C2×Q16—28no—
189C2×Q16—28no—
190C2×Q16C4⋊C4116yescomplemented
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8C22×C412yes—
209C22×C8C22×C412yes—
210C22×C8C2×D412yes—
211C22×C8C2×Q812yes—
212C4×Q8—28no—
213C4×Q8—28no—
214C4×Q8—28no—
215C4×Q8—28no—
216C4×Q8—28no—
217C4×Q8—48no—
218C4×Q8—48no—
219C4×Q8—48no—
220C4×Q8—48no—
221C4×Q8—44no—
222C4×Q8—44no—
223C4×Q8C22⋊C4116yes—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—81no—
267C2×C4⋊C4—81no—
268C2×C4⋊C4—81no—
269C2×C4⋊C4—81no—
270C2×C4⋊C4—82no—
271C2×C4⋊C4—82no—
272C2×C4⋊C4Q1614yescomplemented
273C2×C4⋊C4C22×C412yes—
274C2×C4⋊C4C22×C412yes—
275C2×C4⋊C4C2×D414yes—
276C2×C4⋊C4C2×D412yes—
277C2×C4⋊C4C2×D412yes—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—28no—
282C2×C42—24no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—44no—
290C2×C42—44no—
291C2×C42—42no—
292C2×C42—42no—
293C2×C42C22×C412yes—
294C2×C42C2×D412yes—
295C2×C42C2×D412yes—
296C2×C42C2×D412yes—
297C2×C42#4112yes—
298C2×C42#4112yes—
299#107—28no—
300C4⋊C8—44no—
301#103—28no—
302#103—28no—
303#103—48no—
304#103—48no—
305C4×C8—28no—
306#95—28no—
307#95—24no—
308#95—44no—
309#95—44no—
310#95—82no—
311#95—82no—
312C24C22×D411yescharacteristic
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—22no—
319C2×Q8—22no—
320C2×Q8—22no—
321C2×Q8—22no—
322C2×Q8—22no—
323C2×Q8—22no—
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—22no—
329C2×Q8—22no—
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8—22no—
336C2×Q8—22no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—22no—
343C2×Q8—22no—
344C2×Q8—22no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—22no—
349C2×Q8—41no—
350C2×Q8—41no—
351C2×Q8—41no—
352C2×Q8—41no—
353C2×Q8—41no—
354C2×Q8—41no—
355C2×Q8—41no—
356C2×Q8—41no—
357C2×Q8—41no—
358C2×Q8—41no—
359C2×Q8—41no—
360C2×Q8—41no—
361C2×Q8C2×C22⋊C414yes—
362C2×Q8C2×C4⋊C414yes—
363C2×Q8C2×C4⋊C414yes—
364C2×Q8#11714yes—
365C2×Q8#12114yes—
366C2×Q8#12114yes—
367C2×Q8C22⋊Q814yes—
368C2×Q8C22⋊Q814yes—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—41no—
416C22×C4—42no—
417C22×C4—41no—
418C22×C4—42no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4C4×D412yes—
429C22×C4C4×D412yes—
430C22×C4#12012yes—
431C22×C4#12012yes—
432C22×C4#12112yes—
433C22×C4#12112yes—
434C22×C4C2×Q1614yes—
435C22×C4C2×Q1612yes—
436C22×C4#13512yes—
437C22×C4#13712yes—
438C22×C4#13712yes—
439C22×C4C23×C411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×Q811yescharacteristic
445C22×C4C2×#4111yescharacteristic
446C22×C4C2×#4111yescharacteristic
447C22×C4C2×#4111yescharacteristic
448C22×C4C2×#4111yescharacteristic
449C22×C4H(5,𝔽2)11yescharacteristic
450C22×C4H(5,𝔽2)11yescharacteristic
451C22×C42−1+411yescharacteristic
452C22×C42−1+411yescharacteristic
453Q16—232no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—41no—
464C2×C8—41no—
465C2×C8—41no—
466C2×C8—41no—
467C2×C8—42no—
468C2×C8—42no—
469C2×C8—42no—
470C2×C8—42no—
471C2×C8—42no—
472C2×C8—42no—
473C2×C8—42no—
474C2×C8C2×C4⋊C414yes—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—216no—
479C4⋊C4—28no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—416no—
485C4⋊C4—42no—
486C4⋊C4—41no—
487C4⋊C4—41no—
488C4⋊C4—42no—
489C4⋊C4—48no—
490C4⋊C4—44no—
491C4⋊C4—48no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—84no—
497C4⋊C4—84no—
498C4⋊C4—88no—
499C4⋊C4—84no—
500C4⋊C4C2×C22⋊C414yes—
501C42—216no—
502C42—416no—
503C42—42no—
504C42—42no—
505C42—42no—
506C42—42no—
507C42—48no—
508C42C2×C22⋊C414yes—
509C42C4×D414yes—
510C42#12114yes—
511C42#12314yes—
512C23C2×C4×D411yescharacteristic
513C23C2×#12011yescharacteristic
514C23C2×#12111yescharacteristic
515C23D4211yescharacteristic
516C23#54611yescharacteristic
517C23#54811yescharacteristic
518C23D4×Q811yescharacteristic
519C23C22×Q1611yescharacteristic
520C23C2×#13511yescharacteristic
521C23C2×#13711yescharacteristic
522C23C2×#13711yescharacteristic
523C23#57611yescharacteristic
524C23#57711yescharacteristic
525C23#57711yescharacteristic
526C23#57811yescharacteristic
527Q8—24no—
528Q8—24no—
529Q8—24no—
530Q8—24no—
531Q8—24no—
532Q8—24no—
533Q8—24no—
534Q8—24no—
535Q8—24no—
536Q8—42no—
537Q8—42no—
538Q8—42no—
539Q8—42no—
540Q8—42no—
541Q8—42no—
542Q8#38018yes—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
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—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—82no—
591C2×C4—81no—
592C2×C4—81no—
593C2×C4—82no—
594C2×C4—82no—
595C2×C4—81no—
596C2×C4—81no—
597C2×C4—82no—
598C2×C4—82no—
599C2×C4—82no—
600C2×C4—82no—
601C2×C4—82no—
602C2×C4—82no—
603C2×C4—82no—
604C2×C4—82no—
605C2×C4#38614yes—
606C2×C4#38614yes—
607C2×C4C4×Q1614yes—
608C2×C4#44114yes—
609C2×C4#45114yes—
610C2×C4#45214yes—
611C2×C4#46214yes—
612C2×C4#46414yes—
613C2×C4C22×C22⋊C411yescharacteristic
614C2×C4C22×C4⋊C411yescharacteristic
615C2×C4C2×#11711yescharacteristic
616C2×C4C2×C4×D411yescharacteristic
617C2×C4#52011yescharacteristic
618C2×C4#52011yescharacteristic
619C2×C4C2×#12111yescharacteristic
620C2×C4C2×#12111yescharacteristic
621C2×C4C2×#12111yescharacteristic
622C2×C4C2×C22⋊Q811yescharacteristic
623C2×C4C2×C22⋊Q811yescharacteristic
624C2×C4C2×#12311yescharacteristic
625C2×C4#53511yescharacteristic
626C2×C4#53611yescharacteristic
627C2×C4#53711yescharacteristic
628C2×C4#53711yescharacteristic
629C2×C4D4211yescharacteristic
630C2×C4#54611yescharacteristic
631C2×C4#54711yescharacteristic
632C2×C4#54711yescharacteristic
633C2×C4#54811yescharacteristic
634C2×C4D4×Q811yescharacteristic
635C2×C4#55211yescharacteristic
636C2×C4#55211yescharacteristic
637C2×C4#55311yescharacteristic
638C2×C4#55311yescharacteristic
639C2×C4#55411yescharacteristic
640C2×C4#55411yescharacteristic
641C8—24no—
642C8—42no—
643C8—82no—
644C22C2×#38611yescharacteristic
645C22D4×C4⋊C411yescharacteristic
646C22#235211yescharacteristic
647C22#240811yescharacteristic
648C22#240911yescharacteristic
649C22#241611yescharacteristic
650C22#241711yescharacteristic
651C22C2×C4×Q1611yescharacteristic
652C22C2×#44111yescharacteristic
653C22#295111yescharacteristic
654C22#295211yescharacteristic
655C22C2×#45111yescharacteristic
656C22C2×#45211yescharacteristic
657C22#301211yescharacteristic
658C22#301311yescharacteristic
659C22C2×#46211yescharacteristic
660C22C2×#46411yescharacteristic
661C22#304511yescharacteristic
662C22#304611yescharacteristic
663C22#327811yescharacteristic
664C22#327811yescharacteristic
665C22D4×Q1611yescharacteristic
666C22#328711yescharacteristic
667C22#329911yescharacteristic
668C22#330411yescharacteristic
669C22#332811yescharacteristic
670C22#333311yescharacteristic
671C22#336311yescharacteristic
672C22#336511yescharacteristic
673C22#337411yescharacteristic
674C22#337511yescharacteristic
675C22Q8×Q1611yescharacteristic
676C22#338311yescharacteristic
677C22#338711yescharacteristic
678C22#338711yescharacteristic
679C4—28no—
680C4—24no—
681C4—44no—
682C4—44no—
683C4—44no—
684C4—82no—
685C4—84no—
686C4—82no—
687C4—162no—
688C4#187818yescomplemented
689C4C2×#38011yescharacteristic
690C4#231111yescharacteristic
691C4D4×C4⋊C411yescharacteristic
692C4#235211yescharacteristic
693C4#241611yescharacteristic
694C4#241711yescharacteristic
695C4#248611yescharacteristic
696C4#248711yescharacteristic
697C2#1387211yescharacteristic
698C2C2×#187811yescharacteristic
699C2#2093711yescharacteristic
700C2#2267411yescharacteristic
701C2C4⋊C4×Q1611yescharacteristic
702C2#2276311yescharacteristic
703C2#2277111yescharacteristic
704C2#2424211yescharacteristic
705C2#2424311yescharacteristic
706C2#2433211yescharacteristic
707C2#2433311yescharacteristic
708C2#2473211yescharacteristic
709C2#2473311yescharacteristic
710C2#2481711yescharacteristic
711C2#2481811yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
18. C2×C4×Q16C4semidirect
190. C2×Q16C4⋊C4semidirect
272. C2×C4⋊C4Q16semidirect
688. C4#1878semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Q162
C2×Q8×C4⋊C42
C2×#18782
C2×#18032
#139262
#112192
#110672
#108412
#108342
#107892