Finite Group Databasebeta

← #878100

Subgroups — #878100

Order 512.

2256 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16893C214yesmaximal
2#16246C212yesmaximal, complemented
3#16244C212yesmaximal, complemented
4#16242C212yesmaximal, complemented
5#16214C212yesmaximal, complemented
6#15232C212yesmaximal, complemented
7#15216C212yesmaximal, complemented
8#15216C212yesmaximal, complemented
9#14697C212yesmaximal, complemented
10#14457C214yesmaximal, complemented
11#14129C212yesmaximal, complemented
12#14054C212yesmaximal, complemented
13#13905C211yesmaximal, characteristic
14#13871C211yesmaximal, characteristic
15C4×#386C211yesmaximal, characteristic
16C22×#125C2212yes—
17C22×C4×D4C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C23×C42C2211yescharacteristic
20C2×#389C2212yes—
21C2×#389C2212yes—
22C2×#386C2212yes—
23C2×#385C2214yes—
24C2×#382C2211yescharacteristic
25C2×#380C2212yes—
26C2×#380C2212yes—
27C2×#380C2212yes—
28C2×#380C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C22⋊C4C2212yes—
31C2×C4×C22⋊C4C2211yescharacteristic
32#1445C2212yes—
33#1445C2214yes—
34#1445C2214yes—
35#1443C2214yes—
36#1443C2214yes—
37#1442C2214yes—
38#1442C2214yes—
39#1442C2214yes—
40#1442C2214yes—
41#1442C2214yes—
42#1441C2214yes—
43#1440C2212yes—
44#1439C2214yes—
45#1439C2214yes—
46#1439C2214yes—
47#1439C2214yes—
48#1439C2214yes—
49#1439C2214yes—
50#1438C2212yes—
51#1438C2212yes—
52#1437C2212yes—
53#1437C2212yes—
54#1437C2212yes—
55#1437C2211yescharacteristic
56#1437C2211yescharacteristic
57#1436C2212yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2211yescharacteristic
63#1436C2211yescharacteristic
64#1436C2214yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2214yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C22⋊#95C2214yes—
73C22⋊#95C2212yes—
74C4×#95C2212yes—
75C23×D4C2311yescharacteristic
76C24×C4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2312yes—
79C2×#125—28no—
80C2×#125—44no—
81C2×#125—44no—
82C2×C4×D4—42no—
83C2×C4×D4—42no—
84C2×C4×D4Q818yes—
85C2×#117—44no—
86C2×#117—42no—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2311yescharacteristic
105C22×C22⋊C4C2311yescharacteristic
106C22×C42—44no—
107C22×C42—42no—
108C22×C42C2312yes—
109C22×C42C2312yes—
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112C22×C42C2311yescharacteristic
113C22×C42C2312yes—
114C22×C42C2312yes—
115#389—44no—
116#389—44no—
117#386—28no—
118#385—216no—
119#382—42no—
120#380—28no—
121#380—28no—
122#380—44no—
123#380—42no—
124C4×C4⋊C4—42no—
125C4×C22⋊C4—42no—
126C4×C22⋊C4—44no—
127C2×#95C2312yes—
128C2×#95C2312yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2312yes—
134C2×#95C2312yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2312yes—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2311yescharacteristic
151C2×#95C2311yescharacteristic
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C25C2411yescharacteristic
157C25C2411yescharacteristic
158C22×D4—24no—
159C22×D4—41no—
160C22×D4—41no—
161C22×D4—42no—
162C22×D4C2×Q814yes—
163C23×C4—24no—
164C23×C4—24no—
165C23×C4—22no—
166C23×C4—24no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—24no—
171C23×C4—42no—
172C23×C4—42no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4—42no—
177C23×C4C2×Q814yes—
178C23×C4C2×Q814yes—
179C23×C4#4118yes—
180C23×C4C2414yes—
181C23×C4C2414yes—
182C23×C4C2414yes—
183C23×C4C2412yes—
184C23×C4C2412yes—
185C23×C4C2411yescharacteristic
186C23×C4C2411yescharacteristic
187C23×C4C2411yescharacteristic
188C23×C4C2411yescharacteristic
189C23×C4C2414yes—
190C23×C4C2412yes—
191C23×C4C2412yes—
192C23×C4C2411yescharacteristic
193#125—432no—
194C4×D4—416no—
195#117—416no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
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—28no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4C2×Q814yes—
222C2×C4⋊C4#4118yes—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—28no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—41no—
237C2×C22⋊C4—41no—
238C2×C22⋊C4—41no—
239C2×C22⋊C4—41no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4C2×Q814yes—
250C2×C22⋊C4C2×Q814yes—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—24no—
254C2×C42—24no—
255C2×C42—24no—
256C2×C42—24no—
257C2×C42—24no—
258C2×C42—24no—
259C2×C42—44no—
260C2×C42—42no—
261C2×C42—41no—
262C2×C42—42no—
263C2×C42—42no—
264C2×C42—41no—
265C2×C42—416no—
266C2×C42—42no—
267C2×C42—44no—
268C2×C42C2×Q814yes—
269#95—28no—
270#95—28no—
271#95—28no—
272#95—28no—
273#95—24no—
274#95—24no—
275#95—42no—
276#95—44no—
277#95—44no—
278#95—44no—
279#95—44no—
280#95—44no—
281#95—42no—
282#95—44no—
283#95—44no—
284#95—44no—
285#95—44no—
286#95—44no—
287#95—42no—
288#95—42no—
289#95—41no—
290#95—41no—
291#95—42no—
292#95—44no—
293#95—42no—
294#95—42no—
295#95—42no—
296#95—44no—
297#95—44no—
298C24—22no—
299C24—21no—
300C24—22no—
301C24—21no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24—22no—
306C24—22no—
307C24—22no—
308C24—21no—
309C24—21no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24C22×Q812yes—
314C24C22×Q812yes—
315C24C2×#4114yes—
316C24C2511yescharacteristic
317C2×D4—28no—
318C2×D4—44no—
319C2×D4—44no—
320C2×D4—48no—
321C2×D4—48no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—28no—
357C22×C4—28no—
358C22×C4—216no—
359C22×C4—216no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—48no—
396C22×C4—48no—
397C22×C4—48no—
398C22×C4—48no—
399C22×C4—48no—
400C22×C4—48no—
401C22×C4—48no—
402C22×C4—48no—
403C22×C4—48no—
404C22×C4—44no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4C22×Q812yes—
409C22×C4C22×Q812yes—
410C22×C4C22×Q812yes—
411C22×C4C22×Q812yes—
412C22×C4C22×Q812yes—
413C22×C4C2×#4114yes—
414C22×C4C2×#4114yes—
415C22×C4C2×#4118yes—
416C22×C4C2×#4114yes—
417C22×C4H(5,𝔽2)12yes—
418C22×C4H(5,𝔽2)14yes—
419C22×C4H(5,𝔽2)12yes—
420C4⋊C4—216no—
421C4⋊C4—44no—
422C4⋊C4—416no—
423C4⋊C4—416no—
424C4⋊C4—48no—
425C4⋊C4—48no—
426C4⋊C4—48no—
427C4⋊C4—88no—
428C4⋊C4—88no—
429C22⋊C4—44no—
430C22⋊C4—44no—
431C22⋊C4—416no—
432C22⋊C4—48no—
433C22⋊C4—48no—
434C22⋊C4—48no—
435C22⋊C4—48no—
436C22⋊C4—88no—
437C22⋊C4—88no—
438C22⋊C4—84no—
439C42—28no—
440C42—44no—
441C42—44no—
442C42—48no—
443C42—48no—
444C42—48no—
445C42—84no—
446C23—22no—
447C23—22no—
448C23—24no—
449C23—24no—
450C23—24no—
451C23—24no—
452C23—24no—
453C23—24no—
454C23—24no—
455C23—24no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—24no—
460C23—24no—
461C23—42no—
462C23—41no—
463C23—42no—
464C23—42no—
465C23—41no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—41no—
470C23—41no—
471C23—42no—
472C23—42no—
473C23—41no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—41no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—41no—
486C23—41no—
487C23#53314yes—
488C23C23×Q811yescharacteristic
489C23C22×#4112yes—
490C23C22×#4112yes—
491C23C2×H(5,𝔽2)11yescharacteristic
492C23C2×H(5,𝔽2)12yes—
493C23C2×H(5,𝔽2)11yescharacteristic
494C23#58512yes—
495C23#58512yes—
496C23#58512yes—
497D4—416no—
498C2×C4—22no—
499C2×C4—28no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—28no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—28no—
516C2×C4—432no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—41no—
538C2×C4—44no—
539C2×C4—41no—
540C2×C4—416no—
541C2×C4—416no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4#53818yes—
578C2×C4#54314yes—
579C2×C4#55418yes—
580C2×C4#55914yes—
581C2×C4H(3,𝔽4)14yes—
582C22—28no—
583C22—24no—
584C22—42no—
585C22—42no—
586C22—44no—
587C22—44no—
588C22—44no—
589C22—44no—
590C22—44no—
591C22—44no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—44no—
596C22—82no—
597C22—82no—
598C22—82no—
599C22—82no—
600C22C2×#53311yescharacteristic
601C22C2×#53812yes—
602C22C2×#54311yescharacteristic
603C22#345811yescharacteristic
604C22C2×#55412yes—
605C22#347512yes—
606C22C2×#55911yescharacteristic
607C22#348012yes—
608C22#350412yes—
609C22#350512yes—
610C22#350512yes—
611C22#350512yes—
612C22#350612yes—
613C22#352112yes—
614C22#352412yes—
615C22C2×H(3,𝔽4)11yescharacteristic
616C22#353912yes—
617C22#353912yes—
618C22#353912yes—
619C22#355111yescharacteristic
620C22#355111yescharacteristic
621C4—42no—
622C4—42no—
623C4—42no—
624C4—48no—
625C4—44no—
626C4—44no—
627C4—42no—
628C4—82no—
629C4—82no—
630C4—84no—
631C4—84no—
632C4—84no—
633C4#280418yes—
634C2—44no—
635C2—82no—
636C2#3503311yescharacteristic
637C2#3638712yes—
638C2#3742511yescharacteristic
639C2#3810712yes—
640C2#3953712yes—
641C2#3979512yes—
642C2#4550912yes—
643C2C2×#280411yescharacteristic
644C2#5962712yes—
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. #16246C2semidirect
3. #16244C2semidirect
4. #16242C2semidirect
5. #16214C2semidirect
6. #15232C2semidirect
7. #15216C2semidirect
8. #15216C2semidirect
9. #14697C2semidirect
10. #14457C2semidirect
11. #14129C2semidirect
12. #14054C2semidirect

Maximal subgroups

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

subgroupindex
#168932
#162462
#162442
#162422
#162142
#152322
#152162
#146972
#144572
#141292
#140542
#139052
#138712
C4×#3862