Finite Group Databasebeta

← #4320390

Subgroups — #4320390

Order 512.

2998 conjugacy classes of subgroups.

The subgroup lattice has 647 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2760C212yesmaximal, complemented
2#16823C211yesmaximal, characteristic, complemented
3#16741C214yesmaximal, complemented
4#16660C212yesmaximal, complemented
5#16252C211yesmaximal, characteristic
6#15484C214yesmaximal, complemented
7#15314C214yesmaximal, complemented
8#14915C212yesmaximal, complemented
9#14730C214yesmaximal, complemented
10#14719C212yesmaximal, complemented
11#14417C212yesmaximal, complemented
12#14175C212yesmaximal, complemented
13#13776C211yesmaximal, characteristic, complemented
14C22×#124C2212yes—
15C22×#121C2212yes—
16C22×C4×D4C2212yes—
17C23×C4⋊C4C2211yescharacteristic
18#2760—216no—
19C2×#401C2214yes—
20C2×#394C2212yes—
21C2×#394C2212yes—
22C2×#393C2212yes—
23C2×#393C2214yes—
24C2×#392C2212yes—
25C2×#392C2212yes—
26C2×#392C2214yes—
27C2×#386C2214yes—
28C2×#386C2214yes—
29C2×#386C2214yes—
30C2×#386C2214yes—
31C2×#386C2214yes—
32C2×#381C2211yescharacteristic
33C2×#381C2214yes—
34C2×#381C2211yescharacteristic
35C2×C4×C22⋊C4C2211yescharacteristic
36C2×C4×C22⋊C4C2211yescharacteristic
37#1446C2212yes—
38#1446C2214yes—
39#1445C2214yes—
40#1443C2214yes—
41#1442C2212yes—
42#1442C2214yes—
43#1441C2212yes—
44#1441C2212yes—
45#1440C2214yes—
46#1440C2214yes—
47#1440C2214yes—
48#1439C2212yes—
49#1439C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1438C2214yes—
54#1438C2214yes—
55#1438C2212yes—
56#1438C2212yes—
57#1437C2214yes—
58#1437C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61#1434C2212yes—
62#1434C2212yes—
63C22⋊#95C2211yescharacteristic
64C22⋊#95C2214yes—
65C22⋊#95C2212yes—
66C22⋊#95C2214yes—
67C22⋊#95C2211yescharacteristic
68C23×Q8C2312yes—
69C23×D4C2314yes—
70C23×D4C2312yes—
71C23×D4C2312yes—
72C23×D4C2312yes—
73C24×C4C2311yescharacteristic
74C24×C4C2314yes—
75C24×C4C2311yescharacteristic
76C24×C4C2311yescharacteristic
77C2×#124—44no—
78C2×#124—48no—
79C2×#121—28no—
80C2×#121—48no—
81C2×C4×D4—44no—
82C2×C4×D4—48no—
83C22×C4⋊C4—24no—
84C22×C4⋊C4—42no—
85C22×C4⋊C4—42no—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C22⋊C4C2314yes—
91C22×C22⋊C4C2314yes—
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2311yescharacteristic
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2311yescharacteristic
103C22×C22⋊C4C2311yescharacteristic
104C22×C22⋊C4C2311yescharacteristic
105C22×C42C2312yes—
106C22×C42C2312yes—
107#401—216no—
108#394—44no—
109#394—44no—
110#393—28no—
111#393—216no—
112#392—28no—
113#392—28no—
114#392—48no—
115#386—216no—
116#386—216no—
117#386—216no—
118#386—48no—
119#386—48no—
120#381—24no—
121#381—216no—
122#381D418yescomplemented
123C4×C22⋊C4—42no—
124C4×C22⋊C4—42no—
125C2×#95C2312yes—
126C2×#95C2312yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2312yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2312yes—
136C2×#95C2312yes—
137C2×#95C2312yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C25C2414yes—
149C25C2411yescharacteristic
150C22×Q8—24no—
151C22×Q8—42no—
152C22×Q8—44no—
153C22×Q8—44no—
154C22×Q8—42no—
155C22×D4—28no—
156C22×D4—28no—
157C22×D4—28no—
158C22×D4—28no—
159C22×D4—28no—
160C22×D4—28no—
161C22×D4—28no—
162C22×D4—24no—
163C22×D4—24no—
164C22×D4—24no—
165C22×D4—42no—
166C22×D4—44no—
167C22×D4—42no—
168C22×D4—44no—
169C22×D4—42no—
170C22×D4—44no—
171C22×D4—44no—
172C22×D4—44no—
173C22×D4—42no—
174C22×D4—44no—
175C22×D4—42no—
176C22×D4—42no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—28no—
183C23×C4—28no—
184C23×C4—41no—
185C23×C4—41no—
186C23×C4—41no—
187C23×C4—41no—
188C23×C4—41no—
189C23×C4—41no—
190C23×C4—44no—
191C23×C4—44no—
192C23×C4—44no—
193C23×C4—44no—
194C23×C4—44no—
195C23×C4—44no—
196C23×C4—44no—
197C23×C4C2×D414yes—
198C23×C4#4114yes—
199C23×C4C2414yes—
200C23×C4C2412yes—
201C23×C4C2414yes—
202C23×C4C2414yes—
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2411yescharacteristic
209C23×C4C2411yescharacteristic
210#124—432no—
211#121—432no—
212C4×D4—432no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—416no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—28no—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—41no—
239C2×C22⋊C4—41no—
240C2×C22⋊C4—41no—
241C2×C22⋊C4—41no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—41no—
248C2×C22⋊C4—41no—
249C2×C22⋊C4—41no—
250C2×C22⋊C4—41no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4C2×D414yes—
267C2×C22⋊C4C2×D414yes—
268C2×C22⋊C4#4114yes—
269C2×C42—24no—
270C2×C42—24no—
271C2×C42—44no—
272C2×C42—44no—
273#95—24no—
274#95—28no—
275#95—28no—
276#95—28no—
277#95—28no—
278#95—28no—
279#95—24no—
280#95—28no—
281#95—24no—
282#95—44no—
283#95—42no—
284#95—44no—
285#95—44no—
286#95—42no—
287#95—42no—
288#95—44no—
289#95—44no—
290#95—44no—
291#95—44no—
292#95—44no—
293#95—42no—
294#95—42no—
295#95C2×D4116yes—
296C24—21no—
297C24—21no—
298C24—24no—
299C24—21no—
300C24—21no—
301C24—24no—
302C24—24no—
303C24—24no—
304C24—24no—
305C24—24no—
306C24—24no—
307C24—24no—
308C24—24no—
309C24—24no—
310C24—24no—
311C24—24no—
312C24—24no—
313C24—24no—
314C24—24no—
315C24—24no—
316C24C22×D412yes—
317C24C2×#4112yes—
318C24C2×#4112yes—
319C24H(5,𝔽2)18yes—
320C24C2511yescharacteristic
321C2×Q8—48no—
322C2×Q8—88no—
323C2×Q8—84no—
324C2×Q8—88no—
325C2×Q8—84no—
326C2×D4—216no—
327C2×D4—416no—
328C2×D4—48no—
329C2×D4—416no—
330C2×D4—48no—
331C2×D4—416no—
332C2×D4—48no—
333C2×D4—416no—
334C2×D4—48no—
335C2×D4—416no—
336C2×D4—416no—
337C2×D4—416no—
338C2×D4—84no—
339C2×D4—84no—
340C2×D4—84no—
341C2×D4—88no—
342C2×D4—88no—
343C2×D4—88no—
344C2×D4—88no—
345C2×D4—88no—
346C2×D4—88no—
347C2×D4—84no—
348C22×C4—28no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—28no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
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—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—44no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4—48no—
408C22×C4—416no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—416no—
414C22×C4—88no—
415C22×C4—88no—
416C22×C4—84no—
417C22×C4—84no—
418C22×C4—84no—
419C22×C4—88no—
420C22×C4—84no—
421C22×C4—84no—
422C22×C4—88no—
423C22×C4—88no—
424C22×C4C22×D414yes—
425C22×C4C22×D414yes—
426C22×C4C22×D414yes—
427C22×C4C2×#4114yes—
428C22×C4C2×#4114yes—
429C22×C4H(5,𝔽2)12yes—
430C22×C4H(5,𝔽2)12yes—
431C4⋊C4—48no—
432C4⋊C4—48no—
433C4⋊C4—48no—
434C4⋊C4—48no—
435C22⋊C4—44no—
436C22⋊C4—44no—
437C22⋊C4—44no—
438C22⋊C4—416no—
439C22⋊C4—416no—
440C22⋊C4—416no—
441C22⋊C4—44no—
442C22⋊C4—416no—
443C22⋊C4—88no—
444C22⋊C4—88no—
445C22⋊C4—88no—
446C22⋊C4—88no—
447C22⋊C4—88no—
448C22⋊C4—88no—
449C22⋊C4—88no—
450C42—48no—
451C42—48no—
452C23—22no—
453C23—22no—
454C23—28no—
455C23—22no—
456C23—24no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—24no—
461C23—28no—
462C23—28no—
463C23—28no—
464C23—28no—
465C23—28no—
466C23—28no—
467C23—28no—
468C23—28no—
469C23—42no—
470C23—42no—
471C23—44no—
472C23—42no—
473C23—42no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
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#53214yes—
504C23C23×D411yescharacteristic
505C23C22×#4111yescharacteristic
506C23C22×#4111yescharacteristic
507C23C2×H(5,𝔽2)11yescharacteristic
508C23C2×H(5,𝔽2)11yescharacteristic
509C23C2×H(5,𝔽2)14yes—
510C23#58511yescharacteristic
511C23#58511yescharacteristic
512C23#58514yes—
513Q8—816no—
514D4—432no—
515D4—816no—
516D4—816no—
517D4—832no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—28no—
521C2×C4—28no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—28no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—416no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—88no—
581C2×C4—88no—
582C2×C4—832no—
583C22—28no—
584C22—24no—
585C22—44no—
586C22—44no—
587C22—42no—
588C22—44no—
589C22—42no—
590C22—44no—
591C22—44no—
592C22—48no—
593C22—48no—
594C22—48no—
595C22—48no—
596C22—48no—
597C22—48no—
598C22—48no—
599C22—84no—
600C22—84no—
601C22—84no—
602C22—84no—
603C22—84no—
604C22—84no—
605C22—84no—
606C22—84no—
607C22C2×#53211yescharacteristic
608C22#345911yescharacteristic
609C22#346011yescharacteristic
610C22#348814yes—
611C22#349012yes—
612C22#349011yescharacteristic
613C22#349311yescharacteristic
614C22#349611yescharacteristic
615C22#349611yescharacteristic
616C22#351012yes—
617C22#351314yes—
618C22#353212yes—
619C22#353212yes—
620C22#353314yes—
621C22#354012yes—
622C22#355112yes—
623C22#356014yes—
624C4—44no—
625C4—42no—
626C4—42no—
627C4—44no—
628C4—44no—
629C4—44no—
630C4—44no—
631C4—84no—
632C4—84no—
633C4—82no—
634C4—84no—
635C2—44no—
636C2—88no—
637C2#3500311yescharacteristic
638C2#3573212yes—
639C2#3684012yes—
640C2#4114212yes—
641C2#4683612yes—
642C2#4699611yescharacteristic
643C2#4703611yescharacteristic
644C2#4788712yes—
645C2#5106012yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radical#16252C2
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×#2760C2semidirect
2. #16823C2semidirect
3. #16741C2semidirect
4. #16660C2semidirect
6. #15484C2semidirect
7. #15314C2semidirect
8. #14915C2semidirect
9. #14730C2semidirect
10. #14719C2semidirect
11. #14417C2semidirect
12. #14175C2semidirect
13. #13776C2semidirect
122. #381D4semidirect

Maximal subgroups

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

subgroupindex
C2×#27602
#168232
#167412
#166602
#162522
#154842
#153142
#149152
#147302
#147192
#144172
#141752
#137762