Finite Group Databasebeta

← #877072

Subgroups — #877072

Order 512.

2328 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16398C214yesmaximal
2#16214C212yesmaximal, complemented
3#16171C212yesmaximal, complemented
4#16167C212yesmaximal, complemented
5#15649C212yesmaximal, complemented
6#15231C212yesmaximal, complemented
7#15220C212yesmaximal, complemented
8#15135C212yesmaximal, complemented
9#14686C212yesmaximal, complemented
10#14370C214yesmaximal, complemented
11#14029C212yesmaximal, complemented
12#13905C211yesmaximal, characteristic
13#13879C212yesmaximal, complemented
14#13871C211yesmaximal, characteristic
15C4×#386C211yesmaximal, characteristic
16C22×C4×D4C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C23×C42C2211yescharacteristic
19C2×#399C2214yes—
20C2×#389C2212yes—
21C2×#389C2212yes—
22C2×#387C2214yes—
23C2×#386C2212yes—
24C2×#384C2212yes—
25C2×#384C2212yes—
26C2×#382C2212yes—
27C2×#382C2211yescharacteristic
28C2×#380C2214yes—
29C2×#380C2214yes—
30C2×#380C2211yescharacteristic
31C2×#380C2212yes—
32C2×#380C2212yes—
33C2×#380C2212yes—
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C22⋊C4C2212yes—
36C2×C4×C22⋊C4C2211yescharacteristic
37#1445C2214yes—
38#1445C2214yes—
39#1445C2212yes—
40#1444C2214yes—
41#1443C2214yes—
42#1442C2214yes—
43#1442C2214yes—
44#1441C2214yes—
45#1441C2214yes—
46#1441C2214yes—
47#1441C2214yes—
48#1440C2212yes—
49#1439C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1438C2212yes—
55#1438C2212yes—
56#1437C2211yescharacteristic
57#1437C2212yes—
58#1437C2212yes—
59#1437C2212yes—
60#1437C2211yescharacteristic
61#1436C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2211yescharacteristic
65#1436C2212yes—
66#1436C2211yescharacteristic
67C4⋊#95C2212yes—
68C4⋊#95C2214yes—
69C4⋊#95C2214yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C22⋊#95C2212yes—
74C4×#95C2212yes—
75C23×D4C2311yescharacteristic
76C24×C4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2312yes—
79C2×C4×D4—42no—
80C2×C4×D4—42no—
81C2×C4×D4Q818yes—
82C2×#117—44no—
83C2×#117—42no—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2311yescharacteristic
104C22×C22⋊C4C2311yescharacteristic
105C22×C42—28no—
106C22×C42—42no—
107C22×C42C2312yes—
108C22×C42C2311yescharacteristic
109C22×C42C2311yescharacteristic
110C22×C42C2311yescharacteristic
111C22×C42C2312yes—
112C22×C42C2312yes—
113C22×C42C2312yes—
114#399—48no—
115#389—44no—
116#389—44no—
117#387—48no—
118#386—28no—
119#384—28no—
120#384—44no—
121#382—42no—
122#382—44no—
123#380—216no—
124#380—48no—
125#380—44no—
126#380—44no—
127#380—44no—
128#380—42no—
129C4×C4⋊C4—42no—
130C4×C22⋊C4—28no—
131C4×C22⋊C4—42no—
132C2×#95C2314yes—
133C2×#95C2312yes—
134C2×#95C2312yes—
135C2×#95C2312yes—
136C2×#95C2312yes—
137C2×#95C2312yes—
138C2×#95C2312yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2311yescharacteristic
150C2×#95C2311yescharacteristic
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C25C2411yescharacteristic
160C25C2411yescharacteristic
161C22×D4—24no—
162C22×D4—41no—
163C22×D4—41no—
164C22×D4—42no—
165C22×D4C2×Q814yes—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—24no—
169C23×C4—24no—
170C23×C4—22no—
171C23×C4—24no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—42no—
179C23×C4—42no—
180C23×C4C2×Q814yes—
181C23×C4C2×Q814yes—
182C23×C4#4118yes—
183C23×C4C2414yes—
184C23×C4C2414yes—
185C23×C4C2412yes—
186C23×C4C2414yes—
187C23×C4C2414yes—
188C23×C4C2412yes—
189C23×C4C2411yescharacteristic
190C23×C4C2411yescharacteristic
191C23×C4C2411yescharacteristic
192C23×C4C2411yescharacteristic
193C23×C4C2412yes—
194C23×C4C2412yes—
195C23×C4C2411yescharacteristic
196C4×D4—416no—
197#117—416no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
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—28no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
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⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4C2×Q814yes—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—41no—
244C2×C22⋊C4—41no—
245C2×C22⋊C4—41no—
246C2×C22⋊C4—41no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4C2×Q814yes—
258C2×C22⋊C4C2×Q814yes—
259C2×C42—24no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—416no—
269C2×C42—44no—
270C2×C42—42no—
271C2×C42—41no—
272C2×C42—41no—
273C2×C42—42no—
274C2×C42—44no—
275C2×C42C2×Q814yes—
276C2×C42#4118yes—
277#95—24no—
278#95—24no—
279#95—28no—
280#95—28no—
281#95—24no—
282#95—24no—
283#95—28no—
284#95—28no—
285#95—24no—
286#95—24no—
287#95—44no—
288#95—44no—
289#95—41no—
290#95—41no—
291#95—42no—
292#95—44no—
293#95—44no—
294#95—44no—
295#95—42no—
296#95—42no—
297#95—42no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304C24—22no—
305C24—22no—
306C24—22no—
307C24—21no—
308C24—21no—
309C24—22no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—21no—
316C24—21no—
317C24—22no—
318C24—22no—
319C24C22×Q812yes—
320C24C22×Q812yes—
321C24C2×#4114yes—
322C24C2511yescharacteristic
323C2×D4—28no—
324C2×D4—44no—
325C2×D4—44no—
326C2×D4—48no—
327C2×D4—48no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—28no—
353C22×C4—28no—
354C22×C4—216no—
355C22×C4—216no—
356C22×C4—216no—
357C22×C4—216no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—22no—
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—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—48no—
401C22×C4—48no—
402C22×C4—48no—
403C22×C4—48no—
404C22×C4—48no—
405C22×C4—48no—
406C22×C4—48no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4C22×Q812yes—
412C22×C4C22×Q812yes—
413C22×C4C22×Q812yes—
414C22×C4C22×Q812yes—
415C22×C4C22×Q812yes—
416C22×C4C2×#4114yes—
417C22×C4C2×#4114yes—
418C22×C4C2×#4114yes—
419C22×C4C2×#4114yes—
420C22×C4C2×#4114yes—
421C22×C4H(5,𝔽2)14yes—
422C22×C4H(5,𝔽2)12yes—
423C22×C4H(5,𝔽2)14yes—
424C22×C4H(5,𝔽2)12yes—
425C22×C42−1+414yes—
426C4⋊C4—44no—
427C4⋊C4—48no—
428C4⋊C4—48no—
429C4⋊C4—416no—
430C4⋊C4—48no—
431C4⋊C4—48no—
432C4⋊C4—88no—
433C4⋊C4—88no—
434C4⋊C4—88no—
435C4⋊C4—88no—
436C4⋊C4—88no—
437C22⋊C4—44no—
438C22⋊C4—44no—
439C22⋊C4—48no—
440C22⋊C4—48no—
441C22⋊C4—48no—
442C22⋊C4—48no—
443C22⋊C4—88no—
444C22⋊C4—88no—
445C22⋊C4—88no—
446C22⋊C4—84no—
447C42—28no—
448C42—216no—
449C42—44no—
450C42—44no—
451C42—48no—
452C42—48no—
453C42—84no—
454C23—22no—
455C23—22no—
456C23—24no—
457C23—24no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—24no—
462C23—24no—
463C23—24no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—24no—
468C23—24no—
469C23—42no—
470C23—41no—
471C23—42no—
472C23—42no—
473C23—41no—
474C23—42no—
475C23—42no—
476C23—41no—
477C23—41no—
478C23—42no—
479C23—42no—
480C23—41no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—41no—
490C23—42no—
491C23—42no—
492C23—41no—
493C23—41no—
494C23—42no—
495C23#53314yes—
496C23C23×Q811yescharacteristic
497C23C22×#4112yes—
498C23C22×#4112yes—
499C23C2×H(5,𝔽2)11yescharacteristic
500C23C2×H(5,𝔽2)12yes—
501C23C2×H(5,𝔽2)11yescharacteristic
502C23C2×H(5,𝔽2)12yes—
503C23C2×2−1+412yes—
504C23#58512yes—
505D4—416no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—432no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—416no—
540C2×C4—416no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4#54018yes—
584C2×C4#54314yes—
585C2×C4#55114yes—
586C2×C4#55218yes—
587C2×C4#55318yes—
588C2×C4#55418yes—
589C2×C4#56014yes—
590C22—28no—
591C22—24no—
592C22—44no—
593C22—44no—
594C22—44no—
595C22—42no—
596C22—44no—
597C22—42no—
598C22—44no—
599C22—44no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22—44no—
604C22—82no—
605C22—82no—
606C22—82no—
607C22—82no—
608C22C2×#53311yescharacteristic
609C22C2×#54012yes—
610C22C2×#54311yescharacteristic
611C22#345811yescharacteristic
612C22C2×#55111yescharacteristic
613C22C2×#55212yes—
614C22C2×#55312yes—
615C22C2×#55412yes—
616C22#349412yes—
617C22#349512yes—
618C22#350312yes—
619C22#350512yes—
620C22#350612yes—
621C22#350812yes—
622C22#351112yes—
623C22C2×#56011yescharacteristic
624C22#353612yes—
625C22#354612yes—
626C22#354911yescharacteristic
627C22#355011yescharacteristic
628C22#355412yes—
629C4—28no—
630C4—28no—
631C4—42no—
632C4—42no—
633C4—42no—
634C4—42no—
635C4—82no—
636C4—84no—
637C4—84no—
638C4—82no—
639C4—84no—
640C4—84no—
641C4#274818yes—
642C2—44no—
643C2—82no—
644C2#3503311yescharacteristic
645C2#3618912yes—
646C2#3730611yescharacteristic
647C2#3787812yes—
648C2#3948112yes—
649C2#3973512yes—
650C2#4552212yes—
651C2C2×#274811yescharacteristic
652C2#5919712yes—
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. #16214C2semidirect
3. #16171C2semidirect
4. #16167C2semidirect
5. #15649C2semidirect
6. #15231C2semidirect
7. #15220C2semidirect
8. #15135C2semidirect
9. #14686C2semidirect
10. #14370C2semidirect
11. #14029C2semidirect
13. #13879C2semidirect

Maximal subgroups

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

subgroupindex
#163982
#162142
#161712
#161672
#156492
#152312
#152202
#151352
#146862
#143702
#140292
#139052
#138792
#138712
C4×#3862