Finite Group Databasebeta

← #2037176

Subgroups — #2037176

Order 512.

2734 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2804C211yesmaximal, characteristic, complemented
2C2×#2850C211yesmaximal, characteristic
3#17080C212yesmaximal, complemented
4#16280C214yesmaximal, complemented
5#15946C212yesmaximal, complemented
6#15784C212yesmaximal, complemented
7#15670C212yesmaximal, complemented
8#15613C214yesmaximal, complemented
9#15269C214yesmaximal, complemented
10#15260C212yesmaximal
11#14702C212yesmaximal, complemented
12#14445C211yesmaximal, characteristic, complemented
13#14403C214yesmaximal, complemented
14C22×#124C2211yescharacteristic
15C22×#123C2212yes—
16C22×C22⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C22×#121C2211yescharacteristic
19C23×C42C2211yescharacteristic
20#2850—28no—
21#2804—28no—
22C2×#400C2212yes—
23C2×#397C2212yes—
24C2×#397C2214yes—
25C2×#394C2214yes—
26C2×#393C2214yes—
27C2×#391C2211yescharacteristic
28C2×#391C2212yes—
29C2×#391C2212yes—
30C2×#390C2211yescharacteristic
31C2×#388C2214yes—
32C2×#388C2214yes—
33C2×#386C2214yes—
34C2×#386C2212yes—
35C2×#385C2214yes—
36C2×#385C2212yes—
37C2×#382C2211yescharacteristic
38C2×#382C2212yes—
39C2×#380C2212yes—
40#1446C2214yes—
41#1446C2214yes—
42#1445C2212yes—
43#1444C2214yes—
44#1444C2214yes—
45#1444C2214yes—
46#1442C2214yes—
47#1442C2214yes—
48#1442C2214yes—
49#1441C2214yes—
50#1441C2214yes—
51#1440C2214yes—
52#1440C2212yes—
53#1440C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1438C2212yes—
58#1438C2212yes—
59#1437C2212yes—
60#1436C2212yes—
61#1436C2214yes—
62#1436C2214yes—
63C4⋊#95C2212yes—
64C4⋊#95C2214yes—
65C4⋊#95C2212yes—
66C22⋊#95C2214yes—
67C22⋊#95C2214yes—
68C22⋊#95C2214yes—
69C23×Q8C2312yes—
70C23×Q8C2311yescharacteristic
71C23×D4C2312yes—
72C23×D4C2311yescharacteristic
73C24×C4C2311yescharacteristic
74C24×C4C2312yes—
75C24×C4C2312yes—
76C2×#124—24no—
77C2×#124—44no—
78C2×#123—48no—
79C2×#123—44no—
80C2×C22⋊Q8—24no—
81C2×C22⋊Q8—28no—
82C2×C22⋊Q8—44no—
83C2×C22⋊Q8—42no—
84C2×C22⋊Q8—42no—
85C2×C22⋊Q8—44no—
86C2×#121—24no—
87C2×#121—42no—
88C2×#121—42no—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2312yes—
110C22×C42—24no—
111C22×C42—44no—
112C22×C42C2311yescharacteristic
113C22×C42C2312yes—
114C22×C42C2312yes—
115C22×C42C2311yescharacteristic
116#400—28no—
117#397—48no—
118#397—44no—
119#394—48no—
120#393—216no—
121#391—24no—
122#391—28no—
123#391—44no—
124#390D418yes—
125#388—216no—
126#388—216no—
127#386—28no—
128#386—216no—
129#385—216no—
130#385—44no—
131#382—24no—
132#382—44no—
133#380—44no—
134C2×#95C2312yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C25C2412yes—
156C25C2411yescharacteristic
157C22×Q8—22no—
158C22×Q8—24no—
159C22×Q8—24no—
160C22×Q8—24no—
161C22×Q8—24no—
162C22×Q8—42no—
163C22×Q8—42no—
164C22×Q8—41no—
165C22×Q8—41no—
166C22×Q8—42no—
167C22×Q8—42no—
168C22×Q8—42no—
169C22×D4—24no—
170C22×D4—24no—
171C22×D4—28no—
172C22×D4—24no—
173C22×D4—41no—
174C22×D4—41no—
175C22×D4—42no—
176C22×D4—42no—
177C22×D4—44no—
178C22×D4C2×D414yes—
179C23×C4—22no—
180C23×C4—24no—
181C23×C4—22no—
182C23×C4—24no—
183C23×C4—22no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—42no—
190C23×C4—42no—
191C23×C4—42no—
192C23×C4—42no—
193C23×C4—44no—
194C23×C4—44no—
195C23×C4—42no—
196C23×C4C2414yes—
197C23×C4C2414yes—
198C23×C4C2412yes—
199C23×C4C2414yes—
200C23×C4C2412yes—
201C23×C4C2412yes—
202C23×C4C2414yes—
203C23×C4C2412yes—
204C23×C4C2411yescharacteristic
205C23×C4C2412yes—
206C23×C4C2411yescharacteristic
207#124—416no—
208#123—432no—
209C22⋊Q8—416no—
210C22⋊Q8—432no—
211#121—416no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4C2×D414yes—
241C2×C4⋊C4#4118yes—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4C2×D4116yes—
272C2×C42—22no—
273C2×C42—24no—
274C2×C42—24no—
275C2×C42—28no—
276C2×C42—24no—
277C2×C42—416no—
278C2×C42—42no—
279C2×C42—42no—
280C2×C42—42no—
281C2×C42C2×D414yes—
282#95—28no—
283#95—28no—
284#95—28no—
285#95—28no—
286#95—28no—
287#95—24no—
288#95—28no—
289#95—28no—
290#95—28no—
291#95—28no—
292#95—28no—
293#95—44no—
294#95—44no—
295#95—44no—
296#95—44no—
297#95—44no—
298#95—44no—
299#95—42no—
300#95—44no—
301#95—44no—
302#95—44no—
303C24—21no—
304C24—22no—
305C24—22no—
306C24—21no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—24no—
311C24—24no—
312C24—22no—
313C24—24no—
314C24—22no—
315C24—24no—
316C24—24no—
317C24—24no—
318C24C22×D414yes—
319C24H(5,𝔽2)12yes—
320C24H(5,𝔽2)14yes—
321C24C2511yescharacteristic
322C2×Q8—44no—
323C2×Q8—48no—
324C2×Q8—44no—
325C2×Q8—48no—
326C2×Q8—44no—
327C2×Q8—48no—
328C2×Q8—84no—
329C2×Q8—84no—
330C2×Q8—84no—
331C2×Q8—84no—
332C2×Q8—84no—
333C2×Q8—84no—
334C2×D4—28no—
335C2×D4—44no—
336C2×D4—44no—
337C2×D4—48no—
338C2×D4—416no—
339C2×D4—48no—
340C2×D4—84no—
341C2×D4—84no—
342C2×D4—84no—
343C2×D4—88no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—28no—
352C22×C4—216no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—21no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—48no—
405C22×C4—48no—
406C22×C4—48no—
407C22×C4—48no—
408C22×C4—48no—
409C22×C4—48no—
410C22×C4—44no—
411C22×C4—48no—
412C22×C4—44no—
413C22×C4—48no—
414C22×C4—48no—
415C22×C4—84no—
416C22×C4—88no—
417C22×C4—84no—
418C22×C4—88no—
419C22×C4C22×D414yes—
420C22×C4C22×D412yes—
421C22×C4C22×D414yes—
422C22×C4C2×#4118yes—
423C22×C4C2×#4114yes—
424C22×C4H(5,𝔽2)12yes—
425C22×C4H(5,𝔽2)14yes—
426C22×C42−1+414yes—
427C22×C42−1+414yes—
428C4⋊C4—44no—
429C4⋊C4—48no—
430C4⋊C4—48no—
431C4⋊C4—416no—
432C4⋊C4—44no—
433C4⋊C4—416no—
434C4⋊C4—84no—
435C4⋊C4—84no—
436C4⋊C4—84no—
437C4⋊C4—88no—
438C22⋊C4—232no—
439C22⋊C4—48no—
440C22⋊C4—48no—
441C22⋊C4—48no—
442C22⋊C4—416no—
443C22⋊C4—416no—
444C22⋊C4—48no—
445C22⋊C4—416no—
446C22⋊C4—88no—
447C22⋊C4—84no—
448C22⋊C4—88no—
449C42—28no—
450C42—44no—
451C42—48no—
452C42—48no—
453C23—22no—
454C23—24no—
455C23—24no—
456C23—24no—
457C23—24no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—22no—
462C23—24no—
463C23—24no—
464C23—28no—
465C23—24no—
466C23—24no—
467C23—28no—
468C23—41no—
469C23—42no—
470C23—42no—
471C23—41no—
472C23—41no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—41no—
477C23—42no—
478C23—44no—
479C23—42no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—42no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—42no—
491C23—42no—
492C23—44no—
493C23—44no—
494C23H(3,𝔽4)14yes—
495C23C23×D411yescharacteristic
496C23C22×#4112yes—
497C23C2×H(5,𝔽2)12yes—
498C23C2×H(5,𝔽2)11yescharacteristic
499C23C2×H(5,𝔽2)11yescharacteristic
500C23C2×H(5,𝔽2)12yes—
501C23C2×2−1+412yes—
502C23C2×2−1+412yes—
503C23#58512yes—
504Q8—88no—
505Q8—816no—
506D4—88no—
507D4—816no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—28no—
519C2×C4—28no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—28no—
524C2×C4—24no—
525C2×C4—28no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—416no—
529C2×C4—432no—
530C2×C4—41no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—816no—
576C2×C4#53614yes—
577C2×C4#54718yes—
578C22—28no—
579C22—24no—
580C22—44no—
581C22—44no—
582C22—44no—
583C22—42no—
584C22—44no—
585C22—42no—
586C22—44no—
587C22—48no—
588C22—44no—
589C22—48no—
590C22—44no—
591C22—44no—
592C22—84no—
593C22—84no—
594C22—84no—
595C22—84no—
596C22C2×#53611yescharacteristic
597C22#346012yes—
598C22C2×#54712yes—
599C22#348111yescharacteristic
600C22#348411yescharacteristic
601C22#348411yescharacteristic
602C22#348812yes—
603C22#349012yes—
604C22#351512yes—
605C22#352212yes—
606C22#352212yes—
607C22C2×H(3,𝔽4)11yescharacteristic
608C22#353512yes—
609C22#353612yes—
610C22#353612yes—
611C22#353612yes—
612C22#354412yes—
613C22#354511yescharacteristic
614C22#355011yescharacteristic
615C22#355512yes—
616C22#355812yes—
617C4—24no—
618C4—28no—
619C4—48no—
620C4—42no—
621C4—44no—
622C4—44no—
623C4—84no—
624C4—82no—
625C4—84no—
626C4—84no—
627C4—82no—
628C2—44no—
629C2—84no—
630C2#3671811yescharacteristic
631C2#3716712yes—
632C2#3776112yes—
633C2#3876212yes—
634C2#3892112yes—
635C2#4597711yescharacteristic
636C2#4609912yes—
637C2#5319712yes—
638C2#5907011yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2804C2semidirect
3. #17080C2semidirect
4. #16280C2semidirect
5. #15946C2semidirect
6. #15784C2semidirect
7. #15670C2semidirect
8. #15613C2semidirect
9. #15269C2semidirect
11. #14702C2semidirect
12. #14445C2semidirect
13. #14403C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28042
C2×#28502
#170802
#162802
#159462
#157842
#156702
#156132
#152692
#152602
#147022
#144452
#144032