Finite Group Databasebeta

← #412404

Subgroups — #412404

Order 512.

1664 conjugacy classes of subgroups.

The subgroup lattice has 641 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#467C211yesmaximal, characteristic
2C2×#2036C212yesmaximal
3C2×#1969C211yesmaximal, characteristic, complemented
4C2×#1969C211yesmaximal, characteristic, complemented
5C2×#1845C211yesmaximal, characteristic, complemented
6C2×#1808C211yesmaximal, characteristic
7C2×C8×C22⋊C4C211yesmaximal, characteristic
8#15182C212yesmaximal
9#11124C212yesmaximal, complemented
10#11050C212yesmaximal, complemented
11#10862C211yesmaximal, characteristic, complemented
12C23×Q16C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×C22⋊Q8C2212yes—
15C2×#467—24no—
16C2×#467—24no—
17C2×#467—24no—
18C22×#107C2211yescharacteristic
19C22×#107C2211yescharacteristic
20C22×#107C2212yes—
21C22×#107C2211yescharacteristic
22C22×#103C2212yes—
23C22×#103C2212yes—
24C22×#103C2212yes—
25C22×C22⋊C8C2211yescharacteristic
26C22×C4×C8C2211yescharacteristic
27C22×C4×C8C2211yescharacteristic
28C2×#391C2212yes—
29C2×#391C2212yes—
30C2×#384C2212yes—
31C2×#380C2212yes—
32C2×C4×C22⋊C4C2211yescharacteristic
33#2036—28no—
34#1969—24no—
35#1969—24no—
36#1845—24no—
37#1808—24no—
38C8×C22⋊C4—24no—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2211yescharacteristic
42C2×#336C2211yescharacteristic
43#1441—24no—
44#1439—24no—
45C4⋊#95C2212yes—
46C23×Q8C2312yes—
47C24×C4C2311yescharacteristic
48C22×Q16—22no—
49C22×Q16—22no—
50C22×Q16—22no—
51C22×Q16—22no—
52C22×Q16—22no—
53C22×Q16—22no—
54C22×Q16D414yes—
55C23×C8—22no—
56C23×C8—22no—
57C23×C8—22no—
58C23×C8—22no—
59C23×C8—22no—
60C23×C8—22no—
61C23×C8D414yes—
62C23×C8C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C23×C8C2311yescharacteristic
66C2×C22⋊Q8—24no—
67C2×C22⋊Q8—24no—
68C2×C22⋊Q8—24no—
69C2×C22⋊Q8—24no—
70C2×C22⋊Q8—42no—
71C2×C22⋊Q8—42no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—24no—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C22⋊C4—22no—
78C22×C22⋊C4—22no—
79C22×C22⋊C4D412yes—
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82#467—232no—
83C2×#107—22no—
84C2×#107—22no—
85C2×#107—22no—
86C2×#107—22no—
87C2×#107—22no—
88C2×#107—22no—
89C2×#107—22no—
90C2×#107—24no—
91C2×#107—22no—
92C2×#107—24no—
93C2×#107—42no—
94C2×#107D414yes—
95C2×#103—24no—
96C2×#103—24no—
97C2×#103—24no—
98C2×#103—24no—
99C2×#103—24no—
100C2×#103—24no—
101C2×#103—24no—
102C2×#103—24no—
103C2×#103—42no—
104C2×C22⋊C8—22no—
105C2×C22⋊C8—22no—
106C2×C22⋊C8D414yes—
107C2×C4×C8—22no—
108C2×C4×C8—22no—
109C2×C4×C8—22no—
110C2×C4×C8—22no—
111C2×C4×C8—22no—
112C2×C4×C8D414yes—
113#391—24no—
114#391—24no—
115#391—24no—
116#391—42no—
117#384—24no—
118#384—42no—
119#380—24no—
120#380—24no—
121C4×C22⋊C4—22no—
122C4×C22⋊C4—22no—
123C2×#95—24no—
124C2×#95—24no—
125C2×#95—22no—
126C2×#95—24no—
127C2×#95D412yes—
128#340—24no—
129#340—24no—
130#336—22no—
131#336—22no—
132C25C2×D411yescharacteristic
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×Q8—22no—
145C22×Q8—22no—
146C22×Q8C2×D414yes—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D412yes—
165C23×C4C2×D412yes—
166C23×C4C2×D412yes—
167C23×C4C2×D412yes—
168C23×C4C2411yescharacteristic
169C2×Q16—28no—
170C2×Q16—28no—
171C2×Q16—28no—
172C2×Q16—44no—
173C2×Q16—44no—
174C2×Q16—44no—
175C2×Q16—44no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—28no—
193C22×C8—28no—
194C22×C8—28no—
195C22×C8—44no—
196C22×C8—44no—
197C22×C8—44no—
198C22×C8—44no—
199C22×C8C2×D412yes—
200C22×C8C2×D412yes—
201C22×C8C2×D412yes—
202C22×C8C2×D412yes—
203C22×C8C2×D412yes—
204C22×C8C2×D412yes—
205C22×C8C2×D412yes—
206C22×C8C2×D412yes—
207C22×C8#4112yes—
208C22×C8#4112yes—
209C22×C8#4112yes—
210C22×C8#4112yes—
211C22⋊Q8—216no—
212C22⋊Q8—48no—
213C22⋊Q8—48no—
214C22⋊Q8—48no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—48no—
236C2×C4⋊C4—48no—
237C2×C4⋊C4—84no—
238C2×C4⋊C4C2×D414yes—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—84no—
246C2×C22⋊C4Q1618yescomplemented
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×C42C2×D412yes—
258C2×C42#4112yes—
259#107—28no—
260#107—48no—
261#107—44no—
262#107—44no—
263#103—216no—
264#103—48no—
265#103—48no—
266C22⋊C8—28no—
267C4×C8—28no—
268C4×C8—44no—
269#95—24no—
270#95—48no—
271#95—48no—
272#95—44no—
273#95—84no—
274C24—21no—
275C24—21no—
276C24—21no—
277C24—21no—
278C24—21no—
279C24—21no—
280C24—22no—
281C24—22no—
282C24—22no—
283C24#12712yes—
284C24C2×Q1614yes—
285C24C22×D411yescharacteristic
286C2×Q8—24no—
287C2×Q8—24no—
288C2×Q8—24no—
289C2×Q8—24no—
290C2×Q8—24no—
291C2×Q8—24no—
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—
308C2×Q8—42no—
309C2×Q8—42no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—42no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
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—41no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4#12114yes—
381C22×C4#12712yes—
382C22×C4C2×Q1614yes—
383C22×C4C2×Q1614yes—
384C22×C4#13514yes—
385C22×C4C22×D411yescharacteristic
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4C22×D411yescharacteristic
389C22×C4C22×D411yescharacteristic
390C22×C4C2×#4111yescharacteristic
391C22×C4C2×#4111yescharacteristic
392C22×C4C2×#4111yescharacteristic
393C22×C4C2×#4111yescharacteristic
394C22×C4C2×#4111yescharacteristic
395C22×C4H(5,𝔽2)11yescharacteristic
396C22×C4H(5,𝔽2)11yescharacteristic
397C22×C42−1+411yescharacteristic
398C22×C42−1+411yescharacteristic
399Q16—416no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—416no—
421C2×C8#12014yes—
422C2×C8#12114yes—
423C2×C8#12114yes—
424C2×C8#12114yes—
425C2×C8#12114yes—
426C2×C8#12314yes—
427C2×C8#12314yes—
428C2×C8#12414yes—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—416no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—816no—
443C22⋊C4—216no—
444C22⋊C4—416no—
445C22⋊C4—88no—
446C42—22no—
447C42—22no—
448C42—22no—
449C42—22no—
450C42—41no—
451C42—41no—
452C42—41no—
453C42—41no—
454C23—24no—
455C23—24no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—24no—
461C23—22no—
462C23—22no—
463C23—24no—
464C23—24no—
465C23—24no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—42no—
473C23—42no—
474C23—41no—
475C23—42no—
476C23#49414yes—
477C23C2×#12111yescharacteristic
478C23C2×#12711yescharacteristic
479C23#53211yescharacteristic
480C23D4211yescharacteristic
481C23D4211yescharacteristic
482C23#54711yescharacteristic
483C23#54711yescharacteristic
484C23C22×Q1612yes—
485C23C2×#13512yes—
486C23#57612yes—
487C23#57812yes—
488Q8—44no—
489Q8—44no—
490Q8—44no—
491Q8—44no—
492Q8—82no—
493Q8—82no—
494Q8—82no—
495Q8—82no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—48no—
528C2×C4—44no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—84no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—84no—
542C2×C4—82no—
543C2×C4—84no—
544C2×C4—84no—
545C2×C4—84no—
546C2×C4#46718yescomplemented
547C2×C4C2×#12011yescharacteristic
548C2×C4C2×#12111yescharacteristic
549C2×C4C2×#12111yescharacteristic
550C2×C4C2×#12111yescharacteristic
551C2×C4C2×#12111yescharacteristic
552C2×C4C2×#12311yescharacteristic
553C2×C4C2×#12311yescharacteristic
554C2×C4C2×#12411yescharacteristic
555C2×C4#53211yescharacteristic
556C2×C4#53211yescharacteristic
557C2×C4#53511yescharacteristic
558C2×C4#53611yescharacteristic
559C2×C4#53711yescharacteristic
560C2×C4#53711yescharacteristic
561C2×C4#54211yescharacteristic
562C2×C4#54211yescharacteristic
563C2×C4D4211yescharacteristic
564C2×C4D4211yescharacteristic
565C2×C4#54711yescharacteristic
566C2×C4#54711yescharacteristic
567C2×C4#54711yescharacteristic
568C2×C4#54711yescharacteristic
569C2×C4#54811yescharacteristic
570C2×C4#54811yescharacteristic
571C2×C4#55311yescharacteristic
572C2×C4#55311yescharacteristic
573C2×C4#55511yescharacteristic
574C2×C4#55511yescharacteristic
575C8—24no—
576C8—42no—
577C8—42no—
578C8#39418yes—
579C22—24no—
580C22—24no—
581C22—24no—
582C22—42no—
583C22—42no—
584C22—44no—
585C22—44no—
586C22—42no—
587C22—42no—
588C22—44no—
589C22—44no—
590C22#241611yescharacteristic
591C22#242211yescharacteristic
592C22#243311yescharacteristic
593C22#249111yescharacteristic
594C22#250111yescharacteristic
595C22#253711yescharacteristic
596C22#285611yescharacteristic
597C22C2×#46712yes—
598C22#306112yes—
599C22C2×#49411yescharacteristic
600C22#314711yescharacteristic
601C22#315511yescharacteristic
602C22#315611yescharacteristic
603C22#317012yes—
604C22#317512yes—
605C22#328012yes—
606C22D4×Q1612yes—
607C22D4×Q1612yes—
608C22#328712yes—
609C22#333612yes—
610C22#333812yes—
611C22#333812yes—
612C22#334212yes—
613C4—28no—
614C4—42no—
615C4—44no—
616C4—84no—
617C4—84no—
618C4—162no—
619C4—162no—
620C4C2×#39411yescharacteristic
621C4#241611yescharacteristic
622C4#242211yescharacteristic
623C4#243211yescharacteristic
624C4#249111yescharacteristic
625C4#250011yescharacteristic
626C4#253711yescharacteristic
627C4#262011yescharacteristic
628C2—44no—
629C2#1513011yescharacteristic
630C2#2478112yes—
631C2#2503512yes—
632C2#2577711yescharacteristic
633C2#2578011yescharacteristic
634C2#2779212yes—
635C2#2792711yescharacteristic
636C2#2793011yescharacteristic
637C2#2822812yes—
638C2#3317211yescharacteristic
639C2#3318211yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1969C2semidirect
4. C2×#1969C2semidirect
5. C2×#1845C2semidirect
9. #11124C2semidirect
10. #11050C2semidirect
11. #10862C2semidirect
246. C2×C22⋊C4Q16semidirect
546. C2×C4#467semidirect

Maximal subgroups

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

subgroupindex
C22×#4672
C2×#20362
C2×#19692
C2×#18452
C2×#18082
C2×C8×C22⋊C42
#151822
#111242
#110502
#108622