Finite Group Databasebeta

← #876511

Subgroups — #876511

Order 512.

3146 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2734C211yesmaximal, characteristic, complemented
2C2×#2733C211yesmaximal, characteristic
3C2×#2731C212yesmaximal, complemented
4C2×#2545C212yesmaximal, complemented
5#17071C212yesmaximal, complemented
6#17015C212yesmaximal, complemented
7#15825C214yesmaximal, complemented
8#15776C214yesmaximal, complemented
9#15163C212yesmaximal, complemented
10#14928C214yesmaximal, complemented
11#14700C212yesmaximal, complemented
12#14231C214yesmaximal, complemented
13C4×#391C211yesmaximal, characteristic, complemented
14C22×C4⋊Q8C2212yes—
15C22×#124C2212yes—
16C22×#124C2212yes—
17C22×C4×Q8C2211yescharacteristic
18#2734—28no—
19#2733—28no—
20#2731—216no—
21#2545—216no—
22C2×#399C2214yes—
23C2×#398C2212yes—
24C2×#396C2214yes—
25C2×#396C2214yes—
26C2×#396C2214yes—
27C2×#395C2214yes—
28C2×#394C2212yes—
29C2×#394C2214yes—
30C2×#393C2214yes—
31C2×#392C2214yes—
32C2×#391C2212yes—
33C2×#391C2211yescharacteristic
34C2×#391C2211yescharacteristic
35C2×#390C2212yes—
36C2×#390C2212yes—
37C2×#390C2212yes—
38C2×#390C2212yes—
39C2×#390C2212yes—
40C2×#389C2212yes—
41C2×#388C2214yes—
42C2×#388C2214yes—
43C2×#388C2214yes—
44C2×#388C2214yes—
45C2×#388C2214yes—
46C2×#387C2214yes—
47C2×#387C2214yes—
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C43C2211yescharacteristic
51#1445C2212yes—
52#1445C2214yes—
53#1444C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1438C2212yes—
59#1438C2214yes—
60#1438C2214yes—
61#1437C2214yes—
62#1437C2214yes—
63#1437C2214yes—
64#1436C2214yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2211yescharacteristic
69C4⋊#95C2211yescharacteristic
70C4×#95C2214yes—
71C23×Q8C2312yes—
72C23×Q8C2311yescharacteristic
73C23×D4C2312yes—
74C23×D4C2312yes—
75C23×D4C2312yes—
76C2×C4⋊Q8—28no—
77C2×C4⋊Q8—28no—
78C2×C4⋊Q8—28no—
79C2×#124—28no—
80C2×#124—28no—
81C2×#124—28no—
82C2×#124—28no—
83C2×#124—44no—
84C2×#124—44no—
85C2×C4×Q8—42no—
86C2×C4×Q8—42no—
87C2×C4×Q8D418yescomplemented
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C42C2312yes—
113C22×C42C2312yes—
114C22×C42C2312yes—
115C22×C42C2312yes—
116C22×C42C2311yescharacteristic
117C22×C42C2312yes—
118C22×C42C2312yes—
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121#399—216no—
122#398—28no—
123#396—216no—
124#396—216no—
125#396—216no—
126#395—216no—
127#394—28no—
128#394—216no—
129#393—216no—
130#392—216no—
131#391—28no—
132#391D418yes—
133#391D418yescomplemented
134#390—28no—
135#390—28no—
136#390—28no—
137#390—44no—
138#390—44no—
139#389—28no—
140#388—216no—
141#388—216no—
142#388—216no—
143#388—216no—
144#388—216no—
145#387—216no—
146#387—216no—
147C4×C4⋊C4—28no—
148C4×C4⋊C4—42no—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C43—42no—
165C25C2414yes—
166C22×Q8—24no—
167C22×Q8—24no—
168C22×Q8—24no—
169C22×Q8—41no—
170C22×Q8—41no—
171C22×Q8—41no—
172C22×Q8—41no—
173C22×Q8—42no—
174C22×Q8—42no—
175C22×Q8—42no—
176C22×Q8—42no—
177C22×Q8C2×D414yes—
178C22×Q8C2×D414yes—
179C22×Q8C2×D414yes—
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—24no—
183C22×D4—24no—
184C22×D4—24no—
185C22×D4—24no—
186C22×D4—24no—
187C22×D4—42no—
188C22×D4—42no—
189C22×D4—42no—
190C22×D4—42no—
191C22×D4—42no—
192C22×D4—42no—
193C22×D4—42no—
194C22×D4—44no—
195C22×D4—42no—
196C22×D4—42no—
197C22×D4—44no—
198C22×D4—42no—
199C23×C4C2414yes—
200C23×C4C2414yes—
201C23×C4C2412yes—
202C23×C4C2412yes—
203C23×C4C2411yescharacteristic
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2414yes—
207C23×C4C2411yescharacteristic
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2411yescharacteristic
211C4⋊Q8—432no—
212#124—432no—
213#124—432no—
214C4×Q8—416no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4C2×D418yes—
247C2×C4⋊C4C2×D414yes—
248C2×C4⋊C4C2×D414yes—
249C2×C4⋊C4C2×D414yes—
250C2×C4⋊C4#4118yes—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—41no—
302C2×C42—41no—
303C2×C42—42no—
304C2×C42—41no—
305C2×C42—41no—
306C2×C42—42no—
307C2×C42—42no—
308C2×C42—44no—
309C2×C42C2×D418yes—
310C2×C42C2×D414yes—
311C2×C42C2×D414yes—
312C2×C42C2×D414yes—
313#95—28no—
314#95—24no—
315#95—28no—
316#95—28no—
317#95—28no—
318#95—28no—
319#95—28no—
320#95—24no—
321#95—28no—
322#95—28no—
323#95—28no—
324#95—44no—
325#95—42no—
326#95C2×D4116yes—
327#95C2×D4116yes—
328C24—24no—
329C24—24no—
330C24—24no—
331C24—24no—
332C24—24no—
333C24—24no—
334C24—24no—
335C24—24no—
336C24—24no—
337C24—24no—
338C24—24no—
339C24—24no—
340C24—24no—
341C24—24no—
342C24—24no—
343C24C2511yescharacteristic
344C2×Q8—44no—
345C2×Q8—44no—
346C2×Q8—44no—
347C2×Q8—44no—
348C2×Q8—44no—
349C2×Q8—44no—
350C2×Q8—48no—
351C2×Q8—48no—
352C2×Q8—48no—
353C2×Q8—84no—
354C2×Q8—84no—
355C2×Q8—84no—
356C2×Q8—84no—
357C2×Q8#127116yes—
358C2×D4—48no—
359C2×D4—48no—
360C2×D4—48no—
361C2×D4—48no—
362C2×D4—48no—
363C2×D4—48no—
364C2×D4—48no—
365C2×D4—84no—
366C2×D4—84no—
367C2×D4—84no—
368C2×D4—84no—
369C2×D4—84no—
370C2×D4—84no—
371C2×D4—84no—
372C2×D4—84no—
373C2×D4—84no—
374C2×D4—88no—
375C2×D4—84no—
376C2×D4—88no—
377C22×C4—22no—
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—21no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4C22×D414yes—
434C22×C4C22×D414yes—
435C22×C4C22×D412yes—
436C22×C4C22×D414yes—
437C22×C4C22×D414yes—
438C22×C4C22×D414yes—
439C22×C4C22×D414yes—
440C22×C4C22×D414yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D414yes—
446C22×C4C2×#4118yes—
447C22×C4C2×#4114yes—
448C22×C4H(5,𝔽2)12yes—
449C22×C4H(5,𝔽2)12yes—
450C22×C42−1+414yes—
451C4⋊C4—216no—
452C4⋊C4—44no—
453C4⋊C4—44no—
454C4⋊C4—48no—
455C4⋊C4—44no—
456C4⋊C4—416no—
457C4⋊C4—416no—
458C4⋊C4—416no—
459C4⋊C4—416no—
460C4⋊C4—84no—
461C4⋊C4—84no—
462C4⋊C4—84no—
463C22⋊C4—416no—
464C22⋊C4—416no—
465C22⋊C4—416no—
466C22⋊C4—416no—
467C22⋊C4—416no—
468C22⋊C4—416no—
469C22⋊C4—416no—
470C22⋊C4—416no—
471C22⋊C4—416no—
472C22⋊C4—88no—
473C22⋊C4—88no—
474C22⋊C4—88no—
475C42—28no—
476C42—216no—
477C42—44no—
478C42—44no—
479C42—48no—
480C42—48no—
481C42—48no—
482C42—48no—
483C42—84no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23C23×D411yescharacteristic
520C23C23×D411yescharacteristic
521C23C23×D411yescharacteristic
522C23C22×#4112yes—
523C23C2×H(5,𝔽2)11yescharacteristic
524C23C2×H(5,𝔽2)11yescharacteristic
525C23C2×2−1+412yes—
526C23#58511yescharacteristic
527C23#58511yescharacteristic
528C23#58512yes—
529C23#58512yes—
530Q8—416no—
531Q8—816no—
532D4—816no—
533D4—816no—
534D4—816no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—28no—
560C2×C4—28no—
561C2×C4—28no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—28no—
565C2×C4—28no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—28no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4C2×#12718yes—
606C2×C4C2×#12714yes—
607C2×C4#53414yes—
608C2×C4#54818yes—
609C22—84no—
610C22—84no—
611C22—84no—
612C22—84no—
613C22—84no—
614C22—84no—
615C22—84no—
616C22—84no—
617C22—84no—
618C22—84no—
619C22—84no—
620C22—84no—
621C22—84no—
622C22—84no—
623C22—84no—
624C22C22×#12711yescharacteristic
625C22C2×#53411yescharacteristic
626C22#344911yescharacteristic
627C22#346012yes—
628C22C2×#54812yes—
629C22D4×#4112yes—
630C22D4×#4112yes—
631C22#348311yescharacteristic
632C22#349312yes—
633C22#349611yescharacteristic
634C22#349611yescharacteristic
635C22#349611yescharacteristic
636C22#349611yescharacteristic
637C22#349712yes—
638C22#349712yes—
639C22#350012yes—
640C22#351812yes—
641C22#352911yescharacteristic
642C22#353512yes—
643C22#354411yescharacteristic
644C22#354511yescharacteristic
645C22#356412yes—
646C22#356812yes—
647C4—24no—
648C4—28no—
649C4—24no—
650C4—28no—
651C4—44no—
652C4—48no—
653C4—44no—
654C4—44no—
655C4—44no—
656C4—42no—
657C4—84no—
658C4—84no—
659C2—164no—
660C2#3500111yescharacteristic
661C2#3646512yes—
662C2#3719011yescharacteristic
663C2#3843512yes—
664C2#3942312yes—
665C2#3945512yes—
666C2#4623711yescharacteristic
667C2#4631211yescharacteristic
668C2#5216711yescharacteristic
669C2#5965911yescharacteristic
670C2#5983411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2733C2
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×#2734C2semidirect
3. C2×#2731C2semidirect
4. C2×#2545C2semidirect
5. #17071C2semidirect
6. #17015C2semidirect
7. #15825C2semidirect
8. #15776C2semidirect
9. #15163C2semidirect
10. #14928C2semidirect
11. #14700C2semidirect
12. #14231C2semidirect
13. C4×#391C2semidirect
87. C2×C4×Q8D4semidirect
133. #391D4semidirect

Maximal subgroups

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

subgroupindex
C2×#27342
C2×#27332
C2×#27312
C2×#25452
#170712
#170152
#158252
#157762
#151632
#149282
#147002
#142312
C4×#3912