Finite Group Databasebeta

← #394387

Subgroups — #394387

Order 512.

2528 conjugacy classes of subgroups.

The subgroup lattice has 651 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic
2C22×#391C211yesmaximal, characteristic
3C2×#2024C212yesmaximal, complemented
4C2×#1787C212yesmaximal
5#11067C214yesmaximal, complemented
6#11050C212yesmaximal, complemented
7#10730C212yesmaximal
8#10712C211yesmaximal, characteristic
9C24×Q8C2211yescharacteristic
10C22×C4⋊Q8C2211yescharacteristic
11C22×C22⋊Q8—21no—
12C23×C4⋊C4C2211yescharacteristic
13C23×C42C2211yescharacteristic
14C22×#103C2214yes—
15C22×#103C2214yes—
16C22×C22⋊C8C2212yes—
17#2857—28no—
18C2×#391—21no—
19C2×#391—21no—
20C2×#391—21no—
21C2×#391—28no—
22C2×#391C2212yes—
23C2×#391C2212yes—
24C2×#389—21no—
25C2×#389C2212yes—
26C2×#384C2211yescharacteristic
27C2×#380—21no—
28C2×#380C2212yes—
29C22×#95—21no—
30C22×#95—21no—
31#2024—28no—
32#1787—28no—
33C2×#340C2214yes—
34C2×#340C2214yes—
35C2×#336C2212yes—
36C2×#336C2212yes—
37C23×Q8—24no—
38C23×Q8—24no—
39C23×Q8—21no—
40C23×Q8—21no—
41C23×Q8—22no—
42C23×Q8—24no—
43C23×Q8—24no—
44C23×Q8—42no—
45C23×Q8—42no—
46C23×Q8Q814yescomplemented
47C23×Q8C2312yes—
48C24×C4—21no—
49C24×C4—21no—
50C24×C4D411yescharacteristic
51C24×C4D411yescharacteristic
52C24×C4C2311yescharacteristic
53C23×C8C2314yes—
54C2×C4⋊Q8—24no—
55C2×C4⋊Q8—24no—
56C2×C4⋊Q8—42no—
57C2×C22⋊Q8—28no—
58C2×C22⋊Q8—44no—
59C2×C22⋊Q8—44no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—24no—
66C22×C4⋊C4—24no—
67C22×C4⋊C4—42no—
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2312yes—
71C22×C22⋊C4—22no—
72C22×C42—24no—
73C22×C42—24no—
74C22×C42—42no—
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79C2×#103—28no—
80C2×#103—28no—
81C2×#103—28no—
82C2×#103—28no—
83C2×#103—44no—
84C2×#103—44no—
85C2×C22⋊C8—24no—
86C2×C22⋊C8—24no—
87C2×C22⋊C8—24no—
88#391—28no—
89#391—28no—
90#391—28no—
91#391—24no—
92#391—24no—
93#391—24no—
94#391D418yes—
95#389—28no—
96#389—24no—
97#389—24no—
98#384—24no—
99#380—28no—
100#380—24no—
101#380D418yes—
102C2×#95—28no—
103C2×#95—28no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110#340—28no—
111#340—44no—
112#336—24no—
113#336—42no—
114C25C2×D411yescharacteristic
115C22×Q8—24no—
116C22×Q8—216no—
117C22×Q8—24no—
118C22×Q8—216no—
119C22×Q8—24no—
120C22×Q8—24no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—216no—
125C22×Q8—24no—
126C22×Q8—24no—
127C22×Q8—24no—
128C22×Q8—24no—
129C22×Q8—48no—
130C22×Q8—416no—
131C22×Q8—416no—
132C22×Q8—416no—
133C22×Q8—416no—
134C22×Q8—42no—
135C22×Q8—42no—
136C22×Q8—48no—
137C22×Q8—42no—
138C22×Q8—42no—
139C22×Q8—42no—
140C22×Q8—42no—
141C22×Q8—42no—
142C22×Q8—42no—
143C22×Q8—48no—
144C22×Q8—42no—
145C22×Q8—42no—
146C22×Q8C2×D414yes—
147C22×Q8C2×Q814yes—
148C22×Q8#4114yes—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—24no—
154C23×C4—24no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—24no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—24no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—41no—
170C23×C4—41no—
171C23×C4—41no—
172C23×C4—41no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4—42no—
177C23×C4—42no—
178C23×C4—42no—
179C23×C4—42no—
180C23×C4—42no—
181C23×C4—42no—
182C23×C4—42no—
183C23×C4SD1614yes—
184C23×C4Q1614yes—
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D414yes—
191C23×C4C2×Q812yes—
192C23×C4#4114yes—
193C23×C4#4112yes—
194C23×C4C2411yescharacteristic
195C22×C8—24no—
196C22×C8—24no—
197C22×C8—24no—
198C22×C8—24no—
199C22×C8—24no—
200C22×C8—24no—
201C22×C8—24no—
202C4⋊Q8—416no—
203C22⋊Q8—432no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—416no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4C2×D414yes—
230C2×C4⋊C4C2×D414yes—
231C2×C4⋊C4C2×D414yes—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—416no—
245C2×C42—41no—
246C2×C42—41no—
247C2×C42C2×D414yes—
248#103—416no—
249#103—416no—
250C22⋊C8—48no—
251#95—28no—
252#95—28no—
253#95—28no—
254#95—28no—
255#95—28no—
256#95—28no—
257C24—21no—
258C24—21no—
259C24—21no—
260C24—21no—
261C24—21no—
262C24—21no—
263C24—21no—
264C24—21no—
265C24#12012yes—
266C24C22⋊Q812yes—
267C24#12412yes—
268C24C2×SD1612yes—
269C24C2×Q1612yes—
270C24AGL(1,ℤ/8)12yes—
271C24#13712yes—
272C24C22×D411yescharacteristic
273C2×Q8—216no—
274C2×Q8—24no—
275C2×Q8—24no—
276C2×Q8—24no—
277C2×Q8—28no—
278C2×Q8—28no—
279C2×Q8—28no—
280C2×Q8—28no—
281C2×Q8—28no—
282C2×Q8—28no—
283C2×Q8—464no—
284C2×Q8—48no—
285C2×Q8—48no—
286C2×Q8—48no—
287C2×Q8—48no—
288C2×Q8—48no—
289C2×Q8—464no—
290C2×Q8—48no—
291C2×Q8—48no—
292C2×Q8—48no—
293C2×Q8—44no—
294C2×Q8—44no—
295C2×Q8—44no—
296C2×Q8—44no—
297C2×Q8—44no—
298C2×Q8—44no—
299C2×Q8—48no—
300C2×Q8—84no—
301C2×Q8—84no—
302C2×Q8—84no—
303C2×Q8—84no—
304C2×Q8C22⋊Q818yescomplemented
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
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—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—28no—
322C22×C4—28no—
323C22×C4—28no—
324C22×C4—28no—
325C22×C4—216no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—24no—
330C22×C4—21no—
331C22×C4—24no—
332C22×C4—24no—
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—28no—
342C22×C4—28no—
343C22×C4—28no—
344C22×C4—28no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—44no—
349C22×C4—44no—
350C22×C4—44no—
351C22×C4—44no—
352C22×C4—44no—
353C22×C4—44no—
354C22×C4—44no—
355C22×C4—44no—
356C22×C4—44no—
357C22×C4—44no—
358C22×C4—48no—
359C22×C4—48no—
360C22×C4—48no—
361C22×C4—48no—
362C22×C4—48no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—48no—
368C22×C4—48no—
369C22×C4—48no—
370C22×C4—48no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—84no—
390C22×C4—84no—
391C22×C4—84no—
392C22×C4—84no—
393C22×C4#12012yes—
394C22×C4#12012yes—
395C22×C4#12012yes—
396C22×C4C22⋊Q818yes—
397C22×C4C22⋊Q812yes—
398C22×C4#12418yes—
399C22×C4#12412yes—
400C22×C4C2×SD1612yes—
401C22×C4C2×SD1612yes—
402C22×C4C2×Q1612yes—
403C22×C4C2×Q1612yes—
404C22×C4#13712yes—
405C22×C4#13712yes—
406C22×C4C22×D412yes—
407C22×C4C22×D412yes—
408C22×C4C22×Q811yescharacteristic
409C22×C4C2×#4112yes—
410C22×C4C2×#4112yes—
411C22×C4C2×#4111yescharacteristic
412C22×C4H(5,𝔽2)12yes—
413C22×C4H(5,𝔽2)12yes—
414C2×C8—44no—
415C2×C8—44no—
416C2×C8—44no—
417C2×C8—44no—
418C2×C8—44no—
419C2×C8—44no—
420C2×C8—44no—
421C4⋊C4—28no—
422C4⋊C4—28no—
423C4⋊C4—28no—
424C4⋊C4—48no—
425C4⋊C4—48no—
426C4⋊C4—48no—
427C4⋊C4—416no—
428C4⋊C4—48no—
429C4⋊C4—44no—
430C4⋊C4—44no—
431C4⋊C4—44no—
432C22⋊C4—416no—
433C42—28no—
434C42—44no—
435C42—44no—
436C42—44no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—22no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23#39314yes—
466C23#45014yes—
467C23#45114yes—
468C23#47514yes—
469C23#47714yes—
470C23#48714yes—
471C23#48914yes—
472C23C2×#12011yescharacteristic
473C23C2×#12011yescharacteristic
474C23C2×C22⋊Q811yescharacteristic
475C23#52511yescharacteristic
476C23C2×#12411yescharacteristic
477C23#53411yescharacteristic
478C23#53511yescharacteristic
479C23C22×SD1611yescharacteristic
480C23C22×Q1611yescharacteristic
481C23C2×AGL(1,ℤ/8)11yescharacteristic
482C23C2×#13711yescharacteristic
483C23C2×#13711yescharacteristic
484C23C2×#13711yescharacteristic
485C23#57511yescharacteristic
486C23#57511yescharacteristic
487Q8—216no—
488Q8—432no—
489Q8—48no—
490Q8—48no—
491Q8—48no—
492Q8—816no—
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—216no—
512C2×C4—216no—
513C2×C4—28no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—416no—
521C2×C4—416no—
522C2×C4—432no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—44no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—816no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4#39314yes—
560C2×C4#44814yes—
561C2×C4#45114yes—
562C2×C4C2×C22⋊Q812yes—
563C2×C4C2×C22⋊Q812yes—
564C2×C4#52512yes—
565C2×C4C2×#12412yes—
566C2×C4#53812yes—
567C2×C4#54312yes—
568C2×C4D4214yes—
569C2×C4#54614yes—
570C2×C4#54614yes—
571C2×C4#55114yes—
572C8—84no—
573C22—24no—
574C22—24no—
575C22—24no—
576C22—24no—
577C22—24no—
578C22—24no—
579C22—24no—
580C22—24no—
581C22—24no—
582C22—24no—
583C22—24no—
584C22—24no—
585C22—24no—
586C22—24no—
587C22#202418yescomplemented
588C22C2×#39311yescharacteristic
589C22#240311yescharacteristic
590C22#240411yescharacteristic
591C22#243511yescharacteristic
592C22#256111yescharacteristic
593C22#262611yescharacteristic
594C22#284811yescharacteristic
595C22C2×#45011yescharacteristic
596C22C2×#44811yescharacteristic
597C22C2×#45111yescharacteristic
598C22C2×#45111yescharacteristic
599C22#300611yescharacteristic
600C22#300711yescharacteristic
601C22#300711yescharacteristic
602C22#300711yescharacteristic
603C22C2×#47511yescharacteristic
604C22C2×#47711yescharacteristic
605C22#307811yescharacteristic
606C22#307811yescharacteristic
607C22#310311yescharacteristic
608C22#310411yescharacteristic
609C22#311211yescharacteristic
610C22#311311yescharacteristic
611C22C2×#48711yescharacteristic
612C22C2×#48911yescharacteristic
613C22#313911yescharacteristic
614C22#314011yescharacteristic
615C22#318711yescharacteristic
616C22#318911yescharacteristic
617C22#319311yescharacteristic
618C22#319511yescharacteristic
619C22#320611yescharacteristic
620C22#320911yescharacteristic
621C22#321311yescharacteristic
622C22#321511yescharacteristic
623C4—24no—
624C4—24no—
625C4—24no—
626C4—42no—
627C4—42no—
628C4—48no—
629C4—84no—
630C4—84no—
631C4—84no—
632C4#243714yes—
633C4#247214yes—
634C2—28no—
635C2#1471711yescharacteristic
636C2C2×#202411yescharacteristic
637C2#2345511yescharacteristic
638C2#2391911yescharacteristic
639C2#2392111yescharacteristic
640C2#2393111yescharacteristic
641C2#2393411yescharacteristic
642C2#2585611yescharacteristic
643C2#2586411yescharacteristic
644C2#2910511yescharacteristic
645C2#2911111yescharacteristic
646C2#2995111yescharacteristic
647C2#2995311yescharacteristic
648C2#3295411yescharacteristic
649C2#3295711yescharacteristic
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×#2024C2semidirect
5. #11067C2semidirect
6. #11050C2semidirect
46. C23×Q8Q8semidirect
304. C2×Q8C22⋊Q8semidirect
587. C22#2024semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
C22×#3912
C2×#20242
C2×#17872
#110672
#110502
#107302
#107122