Finite Group Databasebeta

← #4320007

Subgroups — #4320007

Order 512.

2950 conjugacy classes of subgroups.

The subgroup lattice has 665 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2610C212yesmaximal, complemented
2#16823C211yesmaximal, characteristic, complemented
3#16504C214yesmaximal, complemented
4#16357C212yesmaximal, complemented
5#16252C211yesmaximal, characteristic
6#15664C214yesmaximal, complemented
7#15655C214yesmaximal, complemented
8#15117C214yesmaximal, complemented
9#14886C212yesmaximal, complemented
10#14719C212yesmaximal, complemented
11#14521C212yesmaximal, complemented
12#14418C212yesmaximal, complemented
13#13795C211yesmaximal, characteristic, complemented
14C22×#124C2212yes—
15C22×#121C2212yes—
16C22×#117C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19#2610—216no—
20C2×#395C2212yes—
21C2×#394C2212yes—
22C2×#394C2212yes—
23C2×#394C2212yes—
24C2×#394C2212yes—
25C2×#393C2212yes—
26C2×#393C2214yes—
27C2×#392C2212yes—
28C2×#392C2212yes—
29C2×#392C2214yes—
30C2×#390C2214yes—
31C2×#390C2212yes—
32C2×#388C2214yes—
33C2×#388C2214yes—
34C2×#388C2214yes—
35C2×#387C2214yes—
36C2×#386C2214yes—
37C2×#386C2214yes—
38C2×#376C2212yes—
39#1446C2214yes—
40#1443C2214yes—
41#1442C2214yes—
42#1441C2212yes—
43#1441C2212yes—
44#1441C2212yes—
45#1440C2214yes—
46#1439C2214yes—
47#1439C2214yes—
48#1439C2214yes—
49#1439C2212yes—
50#1439C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1438C2214yes—
54#1438C2214yes—
55#1438C2214yes—
56#1438C2214yes—
57#1437C2214yes—
58#1437C2214yes—
59#1436C2214yes—
60C4⋊#95C2214yes—
61C4⋊#95C2214yes—
62#1434C2212yes—
63C22⋊#95C2211yescharacteristic
64C22⋊#95C2212yes—
65C22⋊#95C2211yescharacteristic
66C22⋊#95C2211yescharacteristic
67C22⋊#95C2211yescharacteristic
68C23×Q8C2312yes—
69C23×D4C2314yes—
70C23×D4C2312yes—
71C23×D4C2312yes—
72C23×D4C2312yes—
73C24×C4C2311yescharacteristic
74C24×C4C2311yescharacteristic
75C24×C4C2311yescharacteristic
76C2×#124—28no—
77C2×#124—44no—
78C2×#124—44no—
79C2×#121—28no—
80C2×#121—48no—
81C2×#117—24no—
82C2×#117—42no—
83C2×#117—42no—
84C2×#117—42no—
85C2×#117—42no—
86C2×#117D418yescomplemented
87C22×C4⋊C4—24no—
88C22×C4⋊C4—42no—
89C22×C4⋊C4—42no—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2311yescharacteristic
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2311yescharacteristic
111C22×C22⋊C4C2311yescharacteristic
112C22×C22⋊C4C2311yescharacteristic
113C22×C42C2312yes—
114C22×C42C2314yes—
115C22×C42C2312yes—
116#395—28no—
117#394—28no—
118#394—44no—
119#394—44no—
120#394—44no—
121#393—28no—
122#393—48no—
123#392—28no—
124#392—28no—
125#392—48no—
126#390—216no—
127#390—44no—
128#388—216no—
129#388—216no—
130#388—48no—
131#387—48no—
132#386—216no—
133#386—48no—
134#376—28no—
135C2×#95C2312yes—
136C2×#95C2312yes—
137C2×#95C2312yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C25C2414yes—
155C25C2411yescharacteristic
156C22×Q8—24no—
157C22×Q8—42no—
158C22×Q8—42no—
159C22×Q8—44no—
160C22×Q8—44no—
161C22×D4—28no—
162C22×D4—28no—
163C22×D4—28no—
164C22×D4—24no—
165C22×D4—24no—
166C22×D4—24no—
167C22×D4—28no—
168C22×D4—44no—
169C22×D4—44no—
170C22×D4—42no—
171C22×D4—42no—
172C22×D4—44no—
173C22×D4—44no—
174C22×D4—44no—
175C22×D4—42no—
176C22×D4—44no—
177C22×D4—44no—
178C22×D4—44no—
179C22×D4—44no—
180C22×D4—42no—
181C22×D4—42no—
182C22×D4—42no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—41no—
191C23×C4—41no—
192C23×C4—41no—
193C23×C4—41no—
194C23×C4—44no—
195C23×C4—44no—
196C23×C4C2×D414yes—
197C23×C4#4114yes—
198C23×C4C2414yes—
199C23×C4C2414yes—
200C23×C4C2412yes—
201C23×C4C2412yes—
202C23×C4C2412yes—
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2414yes—
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209#124—432no—
210#121—432no—
211#117—416no—
212#117—416no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—416no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4C2×D418yes—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—41no—
252C2×C22⋊C4—41no—
253C2×C22⋊C4—41no—
254C2×C22⋊C4—41no—
255C2×C22⋊C4—41no—
256C2×C22⋊C4—41no—
257C2×C22⋊C4—41no—
258C2×C22⋊C4—41no—
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⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
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⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4C2×D414yes—
279C2×C22⋊C4C2×D414yes—
280C2×C22⋊C4#4114yes—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—28no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—28no—
287C2×C42—42no—
288C2×C42—44no—
289C2×C42C2×D418yes—
290#95—28no—
291#95—24no—
292#95—24no—
293#95—28no—
294#95—28no—
295#95—24no—
296#95—28no—
297#95—28no—
298#95—28no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—42no—
307#95—42no—
308#95—42no—
309C24—21no—
310C24—21no—
311C24—21no—
312C24—24no—
313C24—21no—
314C24—24no—
315C24—24no—
316C24—24no—
317C24—24no—
318C24—24no—
319C24—24no—
320C24—24no—
321C24—24no—
322C24—24no—
323C24—24no—
324C24—24no—
325C24—24no—
326C24—24no—
327C24—24no—
328C24—24no—
329C24—24no—
330C24C22×D412yes—
331C24C2×#4112yes—
332C24C2×#4112yes—
333C24C2511yescharacteristic
334C2×Q8—48no—
335C2×Q8—88no—
336C2×Q8—88no—
337C2×Q8—84no—
338C2×Q8—84no—
339C2×D4—416no—
340C2×D4—416no—
341C2×D4—416no—
342C2×D4—48no—
343C2×D4—48no—
344C2×D4—48no—
345C2×D4—416no—
346C2×D4—88no—
347C2×D4—84no—
348C2×D4—84no—
349C2×D4—84no—
350C2×D4—88no—
351C2×D4—88no—
352C2×D4—88no—
353C2×D4—88no—
354C2×D4—88no—
355C2×D4—88no—
356C2×D4—84no—
357C2×D4—88no—
358C2×D4—88no—
359C2×D4—84no—
360C2×D4—84no—
361C22×C4—28no—
362C22×C4—28no—
363C22×C4—28no—
364C22×C4—28no—
365C22×C4—28no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—28no—
375C22×C4—28no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—44no—
423C22×C4—48no—
424C22×C4—44no—
425C22×C4—84no—
426C22×C4—84no—
427C22×C4—84no—
428C22×C4—84no—
429C22×C4—84no—
430C22×C4C22×D414yes—
431C22×C4C22×D414yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C2×#4114yes—
435C22×C4C2×#4114yes—
436C22×C4H(5,𝔽2)18yes—
437C22×C4H(5,𝔽2)12yes—
438C22×C4H(5,𝔽2)12yes—
439C22×C4H(5,𝔽2)12yes—
440C22×C42−1+412yes—
441C4⋊C4—216no—
442C4⋊C4—48no—
443C4⋊C4—48no—
444C4⋊C4—48no—
445C4⋊C4—48no—
446C4⋊C4—88no—
447C4⋊C4—84no—
448C22⋊C4—44no—
449C22⋊C4—44no—
450C22⋊C4—44no—
451C22⋊C4—416no—
452C22⋊C4—416no—
453C22⋊C4—416no—
454C22⋊C4—416no—
455C22⋊C4—44no—
456C22⋊C4—416no—
457C22⋊C4—416no—
458C22⋊C4—88no—
459C22⋊C4—88no—
460C22⋊C4—88no—
461C22⋊C4—88no—
462C22⋊C4—88no—
463C22⋊C4—88no—
464C42—216no—
465C42—416no—
466C42—48no—
467C23—22no—
468C23—22no—
469C23—28no—
470C23—22no—
471C23—22no—
472C23—24no—
473C23—22no—
474C23—22no—
475C23—24no—
476C23—28no—
477C23—42no—
478C23—44no—
479C23—44no—
480C23—42no—
481C23—42no—
482C23—42no—
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—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23#53214yes—
519C23C23×D411yescharacteristic
520C23C22×#4111yescharacteristic
521C23C22×#4111yescharacteristic
522C23C2×H(5,𝔽2)11yescharacteristic
523C23C2×H(5,𝔽2)11yescharacteristic
524C23C2×H(5,𝔽2)11yescharacteristic
525C23C2×H(5,𝔽2)14yes—
526C23C2×2−1+411yescharacteristic
527C23#58514yes—
528Q8—816no—
529D4—832no—
530D4—816no—
531D4—816no—
532D4—816no—
533C2×C4—232no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—28no—
549C2×C4—28no—
550C2×C4—28no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—28no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—28no—
557C2×C4—28no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—88no—
598C2×C4—88no—
599C2×C4#53414yes—
600C2×C4#53714yes—
601C22—28no—
602C22—24no—
603C22—44no—
604C22—44no—
605C22—42no—
606C22—42no—
607C22—44no—
608C22—44no—
609C22—44no—
610C22—84no—
611C22—84no—
612C22—84no—
613C22—84no—
614C22—84no—
615C22—84no—
616C22—84no—
617C22—84no—
618C22—84no—
619C22—84no—
620C22—84no—
621C22—84no—
622C22—84no—
623C22—84no—
624C22—84no—
625C22C2×#53211yescharacteristic
626C22C2×#53411yescharacteristic
627C22C2×#53711yescharacteristic
628C22#348814yes—
629C22#349012yes—
630C22#349011yescharacteristic
631C22#349011yescharacteristic
632C22#349211yescharacteristic
633C22#349211yescharacteristic
634C22#351012yes—
635C22#351314yes—
636C22#354012yes—
637C22#354614yes—
638C22#354714yes—
639C22#354912yes—
640C22#354912yes—
641C22#355112yes—
642C4—24no—
643C4—24no—
644C4—48no—
645C4—44no—
646C4—44no—
647C4—44no—
648C4—44no—
649C4—84no—
650C4—82no—
651C4—84no—
652C4—82no—
653C2—44no—
654C2—164no—
655C2#3505111yescharacteristic
656C2#3573212yes—
657C2#3727312yes—
658C2#4297112yes—
659C2#4703611yescharacteristic
660C2#4756311yescharacteristic
661C2#4756812yes—
662C2#4959212yes—
663C2#4963812yes—
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×#2610C2semidirect
2. #16823C2semidirect
3. #16504C2semidirect
4. #16357C2semidirect
6. #15664C2semidirect
7. #15655C2semidirect
8. #15117C2semidirect
9. #14886C2semidirect
10. #14719C2semidirect
11. #14521C2semidirect
12. #14418C2semidirect
13. #13795C2semidirect
86. C2×#117D4semidirect

Maximal subgroups

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

subgroupindex
C2×#26102
#168232
#165042
#163572
#162522
#156642
#156552
#151172
#148862
#147192
#145212
#144182
#137952