Finite Group Databasebeta

← #878104

Subgroups — #878104

Order 512.

2432 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16320C214yesmaximal
2#16248C212yesmaximal, complemented
3#16245C212yesmaximal, complemented
4#16242C212yesmaximal, complemented
5#15636C212yesmaximal, complemented
6#15232C212yesmaximal, complemented
7#15125C212yesmaximal, complemented
8#15125C212yesmaximal, complemented
9#14697C212yesmaximal, complemented
10#14553C214yesmaximal, complemented
11#14129C212yesmaximal, complemented
12#14053C212yesmaximal, complemented
13#13905C211yesmaximal, characteristic
14#13871C211yesmaximal, characteristic
15C4×#386C211yesmaximal, characteristic
16C22×C4⋊Q8C2212yes—
17C22×C4×D4C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C23×C42C2211yescharacteristic
20C2×#393C2214yes—
21C2×#393C2214yes—
22C2×#391C2214yes—
23C2×#389C2212yes—
24C2×#389C2212yes—
25C2×#386C2212yes—
26C2×#385C2214yes—
27C2×#382C2211yescharacteristic
28C2×#380C2212yes—
29C2×#380C2212yes—
30C2×#380C2211yescharacteristic
31C2×#380C2212yes—
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C22⋊C4C2211yescharacteristic
34C2×C4×C22⋊C4C2212yes—
35#1446C2214yes—
36#1446C2214yes—
37#1446C2214yes—
38#1445C2212yes—
39#1445C2214yes—
40#1443C2214yes—
41#1443C2214yes—
42#1442C2214yes—
43#1442C2214yes—
44#1442C2214yes—
45#1441C2214yes—
46#1440C2212yes—
47#1439C2214yes—
48#1439C2214yes—
49#1439C2214yes—
50#1439C2214yes—
51#1438C2212yes—
52#1438C2212yes—
53#1437C2211yescharacteristic
54#1437C2212yes—
55#1437C2212yes—
56#1437C2212yes—
57#1437C2211yescharacteristic
58#1436C2214yes—
59#1436C2212yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2211yescharacteristic
64#1436C2214yes—
65#1436C2211yescharacteristic
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C22⋊#95C2214yes—
73C22⋊#95C2212yes—
74C4×#95C2212yes—
75C23×Q8C2314yes—
76C23×D4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2312yes—
80C2×C4⋊Q8—28no—
81C2×C4⋊Q8—44no—
82C2×C4⋊Q8—44no—
83C2×C4×D4—42no—
84C2×C4×D4—42no—
85C2×C4×D4Q818yescomplemented
86C2×#117—44no—
87C2×#117—42no—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2311yescharacteristic
104C22×C22⋊C4C2311yescharacteristic
105C22×C22⋊C4C2312yes—
106C22×C42—44no—
107C22×C42—42no—
108C22×C42C2312yes—
109C22×C42C2312yes—
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112C22×C42C2311yescharacteristic
113C22×C42C2312yes—
114C22×C42C2312yes—
115#393—216no—
116#393—48no—
117#391—48no—
118#389—44no—
119#389—44no—
120#386—28no—
121#385—48no—
122#382—42no—
123#380—28no—
124#380—28no—
125#380—44no—
126#380—42no—
127C4×C4⋊C4—42no—
128C4×C22⋊C4—42no—
129C4×C22⋊C4—44no—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2312yes—
136C2×#95C2312yes—
137C2×#95C2312yes—
138C2×#95C2312yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2311yescharacteristic
154C2×#95C2312yes—
155C2×#95C2311yescharacteristic
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C25C2411yescharacteristic
160C25C2411yescharacteristic
161C22×Q8—28no—
162C22×Q8—28no—
163C22×Q8—44no—
164C22×Q8—44no—
165C22×Q8—44no—
166C22×Q8—44no—
167C22×Q8—44no—
168C22×D4—24no—
169C22×D4—41no—
170C22×D4—41no—
171C22×D4—42no—
172C22×D4C2×Q814yes—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—22no—
176C23×C4—24no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—42no—
185C23×C4—42no—
186C23×C4—42no—
187C23×C4—42no—
188C23×C4C2×Q814yes—
189C23×C4C2×Q814yes—
190C23×C4C2414yes—
191C23×C4C2414yes—
192C23×C4C2414yes—
193C23×C4C2411yescharacteristic
194C23×C4C2411yescharacteristic
195C23×C4C2411yescharacteristic
196C23×C4C2411yescharacteristic
197C23×C4C2414yes—
198C23×C4C2412yes—
199C23×C4C2412yes—
200C23×C4C2412yes—
201C23×C4C2412yes—
202C23×C4C2411yescharacteristic
203C4⋊Q8—432no—
204C4×D4—416no—
205#117—416no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4C2×Q814yes—
229C2×C4⋊C4#4118yes—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—41no—
242C2×C22⋊C4—41no—
243C2×C22⋊C4—41no—
244C2×C22⋊C4—41no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4C2×Q814yes—
256C2×C22⋊C4C2×Q814yes—
257C2×C22⋊C4#4118yes—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—24no—
261C2×C42—24no—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—44no—
267C2×C42—42no—
268C2×C42—41no—
269C2×C42—42no—
270C2×C42—42no—
271C2×C42—41no—
272C2×C42—416no—
273C2×C42—42no—
274C2×C42—44no—
275C2×C42C2×Q814yes—
276#95—28no—
277#95—28no—
278#95—24no—
279#95—24no—
280#95—28no—
281#95—28no—
282#95—28no—
283#95—28no—
284#95—24no—
285#95—28no—
286#95—24no—
287#95—44no—
288#95—44no—
289#95—44no—
290#95—42no—
291#95—42no—
292#95—41no—
293#95—41no—
294#95—42no—
295#95—44no—
296#95—42no—
297#95—42no—
298#95—42no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—44no—
305C24—22no—
306C24—22no—
307C24—21no—
308C24—22no—
309C24—21no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—21no—
317C24—21no—
318C24—22no—
319C24—22no—
320C24C22×Q812yes—
321C24C22×Q812yes—
322C24C2×#4114yes—
323C24C2511yescharacteristic
324C2×Q8—416no—
325C2×Q8—416no—
326C2×Q8—88no—
327C2×Q8—88no—
328C2×Q8—88no—
329C2×Q8—88no—
330C2×Q8—88no—
331C2×D4—28no—
332C2×D4—44no—
333C2×D4—44no—
334C2×D4—48no—
335C2×D4—48no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—28no—
383C22×C4—28no—
384C22×C4—216no—
385C22×C4—216no—
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—24no—
395C22×C4—24no—
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—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—48no—
408C22×C4—48no—
409C22×C4—48no—
410C22×C4—48no—
411C22×C4—48no—
412C22×C4—48no—
413C22×C4—48no—
414C22×C4—48no—
415C22×C4—48no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4C22×Q812yes—
421C22×C4C22×Q812yes—
422C22×C4C22×Q812yes—
423C22×C4C22×Q812yes—
424C22×C4C22×Q812yes—
425C22×C4C2×#4114yes—
426C22×C4C2×#4114yes—
427C22×C4C2×#4118yes—
428C22×C4C2×#4114yes—
429C22×C4H(5,𝔽2)14yes—
430C22×C4H(5,𝔽2)12yes—
431C22×C4H(5,𝔽2)14yes—
432C22×C4H(5,𝔽2)12yes—
433C22×C42−1+414yes—
434C4⋊C4—216no—
435C4⋊C4—44no—
436C4⋊C4—416no—
437C4⋊C4—48no—
438C4⋊C4—48no—
439C4⋊C4—48no—
440C4⋊C4—88no—
441C4⋊C4—88no—
442C22⋊C4—44no—
443C22⋊C4—44no—
444C22⋊C4—48no—
445C22⋊C4—48no—
446C22⋊C4—48no—
447C22⋊C4—416no—
448C22⋊C4—48no—
449C22⋊C4—88no—
450C22⋊C4—88no—
451C22⋊C4—84no—
452C42—28no—
453C42—44no—
454C42—44no—
455C42—48no—
456C42—48no—
457C42—48no—
458C42—84no—
459C23—22no—
460C23—22no—
461C23—24no—
462C23—24no—
463C23—24no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—24no—
473C23—24no—
474C23—42no—
475C23—41no—
476C23—42no—
477C23—42no—
478C23—41no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—41no—
483C23—41no—
484C23—42no—
485C23—41no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—41no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—41no—
499C23—41no—
500C23#53314yes—
501C23C23×Q811yescharacteristic
502C23C22×#4112yes—
503C23C22×#4112yes—
504C23C2×H(5,𝔽2)11yescharacteristic
505C23C2×H(5,𝔽2)12yes—
506C23C2×H(5,𝔽2)11yescharacteristic
507C23C2×H(5,𝔽2)12yes—
508C23C2×2−1+412yes—
509C23#58512yes—
510Q8—832no—
511D4—416no—
512C2×C4—22no—
513C2×C4—28no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—28no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—28no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—432no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—416no—
569C2×C4—416no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4#53818yes—
592C2×C4#54314yes—
593C2×C4#55418yes—
594C2×C4#55914yes—
595C2×C4H(3,𝔽4)14yes—
596C22—28no—
597C22—24no—
598C22—44no—
599C22—44no—
600C22—42no—
601C22—42no—
602C22—44no—
603C22—44no—
604C22—44no—
605C22—44no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—44no—
610C22—82no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22C2×#53311yescharacteristic
615C22C2×#53812yes—
616C22C2×#54311yescharacteristic
617C22#345811yescharacteristic
618C22C2×#55412yes—
619C22C2×#55911yescharacteristic
620C22#348112yes—
621C22#348212yes—
622C22#350312yes—
623C22#350512yes—
624C22#350612yes—
625C22#351412yes—
626C22#351812yes—
627C22#351912yes—
628C22#352012yes—
629C22C2×H(3,𝔽4)11yescharacteristic
630C22#353612yes—
631C22#354112yes—
632C22#354212yes—
633C22#354911yescharacteristic
634C22#355011yescharacteristic
635C4—44no—
636C4—44no—
637C4—44no—
638C4—42no—
639C4—42no—
640C4—42no—
641C4—42no—
642C4—48no—
643C4—44no—
644C4—84no—
645C4—84no—
646C4—84no—
647C4#280418yes—
648C2—44no—
649C2—82no—
650C2#3503311yescharacteristic
651C2#3636312yes—
652C2#3730611yescharacteristic
653C2#3810512yes—
654C2#4017312yes—
655C2#4019612yes—
656C2#4560112yes—
657C2C2×#280411yescharacteristic
658C2#5963012yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #16248C2semidirect
3. #16245C2semidirect
4. #16242C2semidirect
5. #15636C2semidirect
6. #15232C2semidirect
7. #15125C2semidirect
8. #15125C2semidirect
9. #14697C2semidirect
10. #14553C2semidirect
11. #14129C2semidirect
12. #14053C2semidirect
85. C2×C4×D4Q8semidirect

Maximal subgroups

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

subgroupindex
#163202
#162482
#162452
#162422
#156362
#152322
#151252
#146972
#145532
#141292
#140532
#139052
#138712
C4×#3862