Finite Group Databasebeta

← #4344520

Subgroups — #4344520

Order 512.

2402 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2563C211yesmaximal, characteristic
2C2×#2564C211yesmaximal, characteristic
3#16823C212yesmaximal, complemented
4#16507C212yesmaximal, complemented
5#16365C212yesmaximal, complemented
6#16303C212yesmaximal, complemented
7#15946C212yesmaximal, complemented
8#15944C212yesmaximal, complemented
9#15475C214yesmaximal, complemented
10#15101C212yesmaximal, complemented
11#15014C212yesmaximal, complemented
12#14840C212yesmaximal, complemented
13#14766C212yesmaximal, complemented
14#14407C214yesmaximal
15#13784C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19#2564—28no—
20#2563—28no—
21C2×#400C2211yescharacteristic
22C2×#399C2211yescharacteristic
23C2×#398C2212yes—
24C2×#398C2212yes—
25C2×#395C2211yescharacteristic
26C2×#395C2214yes—
27C2×#393C2211yescharacteristic
28C2×#393C2212yes—
29C2×#393C2212yes—
30C2×#390C2212yes—
31C2×#387C2212yes—
32C2×#386C2212yes—
33C2×#386C2212yes—
34C2×#385C2212yes—
35C2×#385C2214yes—
36C2×#382C2212yes—
37C2×#380C2212yes—
38C2×#380C2212yes—
39C2×#380C2212yes—
40C2×#380C2214yes—
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C4×C22⋊C4C2211yescharacteristic
43C22×#95C2211yescharacteristic
44C22×#95C2211yescharacteristic
45#1446C2214yes—
46#1446C2214yes—
47#1445C2212yes—
48#1445C2212yes—
49#1444C2214yes—
50#1444C2214yes—
51#1444C2212yes—
52#1444C2212yes—
53#1442C2214yes—
54#1442C2214yes—
55#1442C2214yes—
56#1442C2214yes—
57#1442C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2212yes—
61#1441C2212yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1437C2212yes—
67#1436C2214yes—
68#1436C2214yes—
69#1436C2212yes—
70#1436C2214yes—
71C4⋊#95C2214yes—
72#1434C2212yes—
73C22⋊#95C2214yes—
74C22⋊#95C2214yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C23×Q8C2312yes—
78C23×Q8C2311yescharacteristic
79C23×D4C2311yescharacteristic
80C24×C4C2312yes—
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2312yes—
84C2×C4⋊Q8—24no—
85C2×C4⋊Q8—42no—
86C2×C4⋊Q8—42no—
87C2×C22⋊Q8—24no—
88C2×C22⋊Q8—42no—
89C2×C22⋊Q8—42no—
90C2×C4×D4—42no—
91C2×C4×D4—42no—
92C2×C4×D4D418yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2311yescharacteristic
111C22×C22⋊C4C2311yescharacteristic
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C42C2311yescharacteristic
115C22×C42C2312yes—
116C22×C42C2312yes—
117#400—24no—
118#399—24no—
119#398—44no—
120#398—44no—
121#395—24no—
122#395—48no—
123#393—28no—
124#393—28no—
125#393—24no—
126#390—28no—
127#387—28no—
128#386—28no—
129#386Q8116yescomplemented
130#385—28no—
131#385—48no—
132#382—28no—
133#380—28no—
134#380—28no—
135#380—28no—
136#380—48no—
137C4×C4⋊C4—24no—
138C4×C22⋊C4—42no—
139C2×#95—24no—
140C2×#95—42no—
141C2×#95C2312yes—
142C2×#95C2311yescharacteristic
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2311yescharacteristic
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C25C2411yescharacteristic
170C25C2411yescharacteristic
171C22×Q8—22no—
172C22×Q8—24no—
173C22×Q8—41no—
174C22×Q8—41no—
175C22×Q8—42no—
176C22×Q8—42no—
177C22×Q8—44no—
178C22×Q8—44no—
179C22×Q8—44no—
180C22×D4—41no—
181C22×D4—41no—
182C22×D4—42no—
183C22×D4C2×D414yes—
184C22×D4C2×Q818yes—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—44no—
192C23×C4—44no—
193C23×C4—44no—
194C23×C4—42no—
195C23×C4—44no—
196C23×C4—42no—
197C23×C4—42no—
198C23×C4—44no—
199C23×C4—44no—
200C23×C4C2×D414yes—
201C23×C4C2×D414yes—
202C23×C4C2×Q818yes—
203C23×C4C2412yes—
204C23×C4C2414yes—
205C23×C4C2414yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2411yescharacteristic
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211C23×C4C2414yes—
212C23×C4C2414yes—
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C4⋊Q8—416no—
217C22⋊Q8—416no—
218C4×D4—416no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4C2×D414yes—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—41no—
263C2×C22⋊C4—41no—
264C2×C22⋊C4—41no—
265C2×C22⋊C4—41no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4C2×D414yes—
276C2×C22⋊C4C2×D414yes—
277C2×C22⋊C4C2×Q818yes—
278C2×C22⋊C4C2×Q8116yes—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—24no—
282C2×C42—44no—
283C2×C42—42no—
284C2×C42—44no—
285C2×C42C2×D414yes—
286#95—24no—
287#95—22no—
288#95—22no—
289#95—24no—
290#95—28no—
291#95—28no—
292#95—24no—
293#95—24no—
294#95—24no—
295#95—24no—
296#95—24no—
297#95—24no—
298#95—28no—
299#95—28no—
300#95—42no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95C2×Q8116yes—
314C24—22no—
315C24—22no—
316C24—21no—
317C24—21no—
318C24—21no—
319C24—21no—
320C24—22no—
321C24—22no—
322C24—24no—
323C24—24no—
324C24—22no—
325C24—22no—
326C24C22×D412yes—
327C24C22×D412yes—
328C24C22×Q814yes—
329C24C22×Q814yes—
330C24C2511yescharacteristic
331C2×Q8—44no—
332C2×Q8—48no—
333C2×Q8—84no—
334C2×Q8—88no—
335C2×Q8—84no—
336C2×Q8—84no—
337C2×Q8—88no—
338C2×Q8—84no—
339C2×Q8—84no—
340C2×D4—44no—
341C2×D4—44no—
342C2×D4—48no—
343C2×D4—48no—
344C2×D4C4⋊Q8116yes—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—28no—
376C22×C4—28no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
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—24no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—48no—
409C22×C4—48no—
410C22×C4—416no—
411C22×C4—48no—
412C22×C4—48no—
413C22×C4—48no—
414C22×C4—84no—
415C22×C4—84no—
416C22×C4—84no—
417C22×C4—84no—
418C22×C4—88no—
419C22×C4—84no—
420C22×C4—88no—
421C22×C4—84no—
422C22×C4—84no—
423C22×C4—88no—
424C22×C4C22×D412yes—
425C22×C4C22×D412yes—
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×Q818yes—
430C22×C4C22×Q818yes—
431C22×C4C22×Q814yes—
432C22×C4H(5,𝔽2)14yes—
433C22×C4H(5,𝔽2)12yes—
434C22×C4H(5,𝔽2)12yes—
435C4⋊C4—44no—
436C4⋊C4—48no—
437C4⋊C4—48no—
438C4⋊C4—44no—
439C4⋊C4—48no—
440C4⋊C4—48no—
441C4⋊C4—48no—
442C4⋊C4—84no—
443C4⋊C4—88no—
444C4⋊C4—88no—
445C4⋊C4—88no—
446C4⋊C4—88no—
447C22⋊C4—44no—
448C22⋊C4—44no—
449C22⋊C4—416no—
450C22⋊C4—48no—
451C22⋊C4—48no—
452C22⋊C4—48no—
453C22⋊C4—84no—
454C22⋊C4—88no—
455C22⋊C4—88no—
456C42—28no—
457C42—48no—
458C42—84no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—24no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—28no—
469C23—24no—
470C23—28no—
471C23—24no—
472C23—24no—
473C23—28no—
474C23—28no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—44no—
480C23—44no—
481C23—42no—
482C23—42no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—44no—
488C23—44no—
489C23C2×C4⋊Q814yes—
490C23C2×C4⋊Q814yes—
491C23C23×D411yescharacteristic
492C23C23×Q812yes—
493C23C2×H(5,𝔽2)11yescharacteristic
494C23C2×H(5,𝔽2)11yescharacteristic
495C23C2×H(5,𝔽2)12yes—
496C23#58512yes—
497C23#58514yes—
498C23#58512yes—
499Q8—816no—
500Q8—88no—
501D4—416no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—28no—
517C2×C4—28no—
518C2×C4—28no—
519C2×C4—28no—
520C2×C4—28no—
521C2×C4—28no—
522C2×C4—28no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—88no—
573C2×C4—88no—
574C2×C4—816no—
575C2×C4—816no—
576C2×C4C2×C4⋊Q814yes—
577C2×C4#53414yes—
578C22—28no—
579C22—28no—
580C22—24no—
581C22—24no—
582C22—44no—
583C22—44no—
584C22—42no—
585C22—42no—
586C22—48no—
587C22—48no—
588C22—44no—
589C22—44no—
590C22—44no—
591C22—44no—
592C22—42no—
593C22—42no—
594C22C22×C4⋊Q811yescharacteristic
595C22C2×#53411yescharacteristic
596C22#344911yescharacteristic
597C22#348512yes—
598C22#348812yes—
599C22#350112yes—
600C22#350114yes—
601C22#350112yes—
602C22#350414yes—
603C22#354512yes—
604C22#354512yes—
605C22#354514yes—
606C22#354612yes—
607C22#354911yescharacteristic
608C22#355111yescharacteristic
609C22#356614yes—
610C4—24no—
611C4—24no—
612C4—42no—
613C4—48no—
614C4—44no—
615C4—48no—
616C4—42no—
617C4—42no—
618C4—82no—
619C4—84no—
620C4—84no—
621C4—82no—
622C4—82no—
623C2—44no—
624C2—44no—
625C2#3501111yescharacteristic
626C2#3595012yes—
627C2#3711911yescharacteristic
628C2#3727311yescharacteristic
629C2#4142312yes—
630C2#4684212yes—
631C2#4684912yes—
632C2#4884414yes—
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
3. #16823C2semidirect
4. #16507C2semidirect
5. #16365C2semidirect
6. #16303C2semidirect
7. #15946C2semidirect
8. #15944C2semidirect
9. #15475C2semidirect
10. #15101C2semidirect
11. #15014C2semidirect
12. #14840C2semidirect
13. #14766C2semidirect
129. #386Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25632
C2×#25642
#168232
#165072
#163652
#163032
#159462
#159442
#154752
#151012
#150142
#148402
#147662
#144072
#137842