Finite Group Databasebeta

← #397887

Subgroups — #397887

Order 512.

2252 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#467C212yesmaximal
2C2×#2857C211yesmaximal, characteristic
3C2×#2016C211yesmaximal, characteristic, complemented
4C2×#1892C212yesmaximal
5C22×#336C211yesmaximal, characteristic
6#14702C211yesmaximal, characteristic
7#11324C211yesmaximal, characteristic, complemented
8#11124C212yesmaximal, complemented
9#11050C211yesmaximal, characteristic, complemented
10#11024C212yesmaximal, complemented
11#11010C211yesmaximal, characteristic, complemented
12C23×Q16C2212yes—
13C24×C8C2212yes—
14C22×C4⋊Q8C2211yescharacteristic
15C22×C22⋊Q8C2212yes—
16C22×C22⋊Q8C2211yescharacteristic
17C23×C42C2211yescharacteristic
18C2×#467—28no—
19C2×#467—28no—
20C2×#467—28no—
21C22×#107C2212yes—
22C22×#103C2212yes—
23C22×#103C2212yes—
24C22×#103C2212yes—
25C22×#103C2212yes—
26#2857—28no—
27C2×#391C2212yes—
28C2×#391C2211yescharacteristic
29C2×#391C2211yescharacteristic
30C2×#389—22no—
31C2×#384C2211yescharacteristic
32C2×#380—22no—
33C2×#380C2211yescharacteristic
34#2016—24no—
35#1892—28no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336—24no—
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41C2×#336C2211yescharacteristic
42#1436—22no—
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2211yescharacteristic
45C22⋊#95—22no—
46C23×Q8C2312yes—
47C23×Q8C2311yescharacteristic
48C24×C4D412yes—
49C24×C4C2311yescharacteristic
50C22×Q16—24no—
51C22×Q16—24no—
52C22×Q16—24no—
53C22×Q16—24no—
54C22×Q16—24no—
55C22×Q16—42no—
56C22×Q16—42no—
57C23×C8—24no—
58C23×C8—24no—
59C23×C8—24no—
60C23×C8—24no—
61C23×C8—42no—
62C23×C8—42no—
63C23×C8D418yes—
64C23×C8C2312yes—
65C23×C8C2312yes—
66C2×C4⋊Q8—24no—
67C2×C4⋊Q8—22no—
68C2×C4⋊Q8—22no—
69C2×C4⋊Q8—42no—
70C2×C22⋊Q8—24no—
71C2×C22⋊Q8—22no—
72C2×C22⋊Q8—24no—
73C2×C22⋊Q8—24no—
74C2×C22⋊Q8—24no—
75C2×C22⋊Q8—24no—
76C2×C22⋊Q8—24no—
77C2×C22⋊Q8—24no—
78C2×C22⋊Q8D418yes—
79C2×C22⋊Q8D414yes—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—22no—
82C22×C4⋊C4—22no—
83C22×C4⋊C4—22no—
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2312yes—
88C22×C22⋊C4—21no—
89C22×C22⋊C4—22no—
90C22×C22⋊C4—21no—
91C22×C42—24no—
92C22×C42—22no—
93C22×C42—22no—
94C22×C42—42no—
95C22×C42D412yes—
96C22×C42C2311yescharacteristic
97C22×C42C2311yescharacteristic
98#467—264no—
99C2×#107—24no—
100C2×#107—24no—
101C2×#107—24no—
102C2×#103—24no—
103C2×#103—24no—
104C2×#103—24no—
105C2×#103—24no—
106C2×#103—24no—
107C2×#103—24no—
108C2×#103—24no—
109C2×#103—24no—
110C2×#103—24no—
111C2×#103—24no—
112C2×#103—42no—
113C2×#103—42no—
114#391—24no—
115#391—24no—
116#391—22no—
117#391—22no—
118#391—42no—
119#389—216no—
120#384—24no—
121#380—216no—
122#380—24no—
123C2×#95—24no—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95—22no—
127C2×#95—22no—
128C2×#95—21no—
129#340—24no—
130#340—24no—
131#336—24no—
132#336—22no—
133#336D414yes—
134C25C2×D411yescharacteristic
135C22×Q8—22no—
136C22×Q8—21no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—21no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—21no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8—21no—
153C22×Q8—22no—
154C22×Q8—22no—
155C22×Q8—22no—
156C22×Q8—21no—
157C22×Q8C2×D414yes—
158C22×Q8C2×D412yes—
159C23×C4—21no—
160C23×C4—22no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—22no—
164C23×C4—21no—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—24no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—42no—
176C23×C4—42no—
177C23×C4Q1618yescomplemented
178C23×C4C2×D412yes—
179C23×C4C2×D412yes—
180C23×C4C2×D414yes—
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D414yes—
183C23×C4C2×D412yes—
184C23×C4#4112yes—
185C23×C4C2411yescharacteristic
186C2×Q16—216no—
187C2×Q16—48no—
188C2×Q16—48no—
189C2×Q16—48no—
190C2×Q16—48no—
191C2×Q16—48no—
192C2×Q16—48no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—216no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—216no—
205C22×C8—48no—
206C22×C8—48no—
207C22×C8—48no—
208C22×C8—48no—
209C22×C8—48no—
210C22×C8C2×D414yes—
211C22×C8C2×D414yes—
212C22×C8C2×D414yes—
213C22×C8#4114yes—
214C4⋊Q8—48no—
215C4⋊Q8—48no—
216C22⋊Q8—216no—
217C22⋊Q8—216no—
218C22⋊Q8—216no—
219C22⋊Q8—48no—
220C22⋊Q8—48no—
221C22⋊Q8—48no—
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—28no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4C2×D414yes—
251C2×C4⋊C4C2×D412yes—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—44no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—21no—
265C2×C42—22no—
266C2×C42—21no—
267C2×C42—22no—
268C2×C42—48no—
269C2×C42—48no—
270C2×C42—42no—
271C2×C42C2×D412yes—
272C2×C42#4112yes—
273#107—216no—
274#103—216no—
275#103—216no—
276#103—48no—
277#103—48no—
278#95—216no—
279#95—44no—
280#95—44no—
281#95—44no—
282#95—42no—
283#95—44no—
284C24—21no—
285C24—21no—
286C24—21no—
287C24—21no—
288C24—22no—
289C24—22no—
290C24#12012yes—
291C24#12112yes—
292C24#12712yes—
293C24C2×Q1614yes—
294C24#13714yes—
295C24C22×D411yescharacteristic
296C2×Q8—24no—
297C2×Q8—24no—
298C2×Q8—24no—
299C2×Q8—24no—
300C2×Q8—22no—
301C2×Q8—24no—
302C2×Q8—24no—
303C2×Q8—24no—
304C2×Q8—22no—
305C2×Q8—24no—
306C2×Q8—42no—
307C2×Q8—42no—
308C2×Q8—42no—
309C2×Q8—42no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—41no—
314C2×Q8—41no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8—42no—
326C2×Q8—42no—
327C2×Q8—42no—
328C2×Q8—42no—
329C2×Q8—42no—
330C2×Q8—42no—
331C2×Q8—42no—
332C2×Q8—42no—
333C2×Q8—42no—
334C2×Q8—42no—
335C2×Q8—42no—
336C2×Q8—42no—
337C2×Q8—42no—
338C2×Q8—42no—
339C2×Q8—42no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—216no—
369C22×C4—216no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—48no—
375C22×C4—48no—
376C22×C4—48no—
377C22×C4—48no—
378C22×C4—48no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—41no—
387C22×C4—42no—
388C22×C4—41no—
389C22×C4—42no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—42no—
393C22×C4—41no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—44no—
397C22×C4—42no—
398C22×C4—44no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4#12018yes—
410C22×C4#12012yes—
411C22×C4#12118yes—
412C22×C4#12112yes—
413C22×C4#12114yes—
414C22×C4#12712yes—
415C22×C4C2×Q1614yes—
416C22×C4C2×Q1614yes—
417C22×C4#13514yes—
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D412yes—
420C22×C4C22×D412yes—
421C22×C4C22×D411yescharacteristic
422C22×C4C2×#4111yescharacteristic
423C22×C4C2×#4112yes—
424C22×C4C2×#4111yescharacteristic
425C22×C4H(5,𝔽2)11yescharacteristic
426C22×C4H(5,𝔽2)12yes—
427C22×C42−1+411yescharacteristic
428Q16—432no—
429C2×C8—24no—
430C2×C8—24no—
431C2×C8—24no—
432C2×C8—24no—
433C2×C8—24no—
434C2×C8—24no—
435C2×C8—24no—
436C2×C8—24no—
437C2×C8—24no—
438C2×C8—432no—
439C2×C8—42no—
440C2×C8—42no—
441C2×C8—42no—
442C2×C8—42no—
443C2×C8#12118yes—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—24no—
447C4⋊C4—416no—
448C4⋊C4—416no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—416no—
453C4⋊C4—48no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C22⋊C4—416no—
459C22⋊C4—48no—
460C22⋊C4—84no—
461C42—24no—
462C42—24no—
463C42—48no—
464C42—42no—
465C42—42no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—24no—
471C23—22no—
472C23—24no—
473C23—22no—
474C23—24no—
475C23—24no—
476C23—22no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—22no—
483C23—24no—
484C23#39214yes—
485C23#45118yes—
486C23#46218yes—
487C23#49414yes—
488C23#49714yes—
489C23C2×#12011yescharacteristic
490C23C2×#12111yescharacteristic
491C23C2×#12111yescharacteristic
492C23C2×#12711yescharacteristic
493C23#53211yescharacteristic
494C23#53411yescharacteristic
495C23#53711yescharacteristic
496C23C22×Q1612yes—
497C23C2×#13512yes—
498C23C2×#13712yes—
499C23#57512yes—
500Q8—44no—
501Q8—44no—
502Q8—44no—
503Q8—44no—
504Q8—44no—
505Q8—44no—
506Q8—82no—
507Q8—82no—
508Q8—82no—
509Q8—82no—
510Q8—82no—
511Q8—82no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—28no—
525C2×C4—28no—
526C2×C4—28no—
527C2×C4—28no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—432no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—48no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—81no—
547C2×C4—82no—
548C2×C4—81no—
549C2×C4—82no—
550C2×C4—81no—
551C2×C4—82no—
552C2×C4—81no—
553C2×C4—82no—
554C2×C4—81no—
555C2×C4—82no—
556C2×C4—81no—
557C2×C4—82no—
558C2×C4—84no—
559C2×C4—82no—
560C2×C4—81no—
561C2×C4—81no—
562C2×C4—81no—
563C2×C4—81no—
564C2×C4—81no—
565C2×C4—81no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4#39214yes—
571C2×C4#46718yescomplemented
572C2×C4C2×#12012yes—
573C2×C4C2×#12112yes—
574C2×C4C2×#12112yes—
575C2×C4#53212yes—
576C2×C4#53412yes—
577C2×C4#53712yes—
578C2×C4D4211yescharacteristic
579C2×C4D4212yes—
580C2×C4#54612yes—
581C2×C4#54612yes—
582C2×C4#54612yes—
583C2×C4#54712yes—
584C2×C4#54711yescharacteristic
585C2×C4#54811yescharacteristic
586C2×C4#55312yes—
587C2×C4#55311yescharacteristic
588C8—28no—
589C8—44no—
590C22—24no—
591C22—24no—
592C22—28no—
593C22—24no—
594C22—24no—
595C22—28no—
596C22—28no—
597C22—24no—
598C22—24no—
599C22—28no—
600C22#201618yescomplemented
601C22C2×#39211yescharacteristic
602C22#241011yescharacteristic
603C22#241611yescharacteristic
604C22#243011yescharacteristic
605C22#254311yescharacteristic
606C22#265611yescharacteristic
607C22#285611yescharacteristic
608C22C2×#45112yes—
609C22#300212yes—
610C22C2×#46212yes—
611C22#303512yes—
612C22C2×#46712yes—
613C22#305612yes—
614C22C2×#49411yescharacteristic
615C22#314711yescharacteristic
616C22C2×#49711yescharacteristic
617C22#315011yescharacteristic
618C22#317012yes—
619C22#318112yes—
620C22#318912yes—
621C22#319012yes—
622C22#324812yes—
623C22#324912yes—
624C4—28no—
625C4—24no—
626C4—44no—
627C4—84no—
628C4—82no—
629C4—84no—
630C4—82no—
631C4—161no—
632C4—161no—
633C4#241012yes—
634C4#241612yes—
635C4#243012yes—
636C4#254312yes—
637C2—28no—
638C2#1475711yescharacteristic
639C2C2×#201611yescharacteristic
640C2#2342111yescharacteristic
641C2#2438012yes—
642C2#2475212yes—
643C2#2549812yes—
644C2#2857512yes—
645C2#3024211yescharacteristic
646C2#3024411yescharacteristic
647C2#3317211yescharacteristic
648C2#3318411yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#2016C2semidirect
7. #11324C2semidirect
8. #11124C2semidirect
9. #11050C2semidirect
10. #11024C2semidirect
11. #11010C2semidirect
177. C23×C4Q16semidirect
571. C2×C4#467semidirect
600. C22#2016semidirect

Maximal subgroups

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

subgroupindex
C22×#4672
C2×#28572
C2×#20162
C2×#18922
C22×#3362
#147022
#113242
#111242
#110502
#110242
#110102