Finite Group Databasebeta

← #4598445

Subgroups — #4598445

Order 512.

2702 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2819C211yesmaximal, characteristic, complemented
2C2×#2503C211yesmaximal, characteristic, complemented
3#17152C214yesmaximal, complemented
4#16652C212yesmaximal, complemented
5#16559C212yesmaximal, complemented
6#16373C214yesmaximal, complemented
7#16258C214yesmaximal, complemented
8#15941C212yesmaximal, complemented
9#15468C214yesmaximal, complemented
10#15453C212yesmaximal, complemented
11#14915C212yesmaximal, complemented
12#14719C212yesmaximal, complemented
13#14476C211yesmaximal, characteristic, complemented
14C22×#124C2212yes—
15C22×C22⋊Q8C2211yescharacteristic
16C22×#121C2212yes—
17C23×C4⋊C4C2211yescharacteristic
18#2819—28no—
19#2503—28no—
20C2×#400C2214yes—
21C2×#394C2212yes—
22C2×#394C2212yes—
23C2×#394C2212yes—
24C2×#393C2214yes—
25C2×#393C2214yes—
26C2×#393C2212yes—
27C2×#392C2214yes—
28C2×#392C2212yes—
29C2×#392C2212yes—
30C2×#391C2211yescharacteristic
31C2×#391C2211yescharacteristic
32C2×#390C2212yes—
33C2×#386C2214yes—
34C2×#386C2214yes—
35C2×#382C2211yescharacteristic
36C2×#382C2211yescharacteristic
37C2×#381C2214yes—
38C2×C4×C22⋊C4C2211yescharacteristic
39#1446C2212yes—
40#1446C2214yes—
41#1445C2212yes—
42#1443C2214yes—
43#1443C2214yes—
44#1442C2214yes—
45#1441C2212yes—
46#1441C2212yes—
47#1440C2214yes—
48#1440C2214yes—
49#1440C2214yes—
50#1440C2214yes—
51#1440C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1438C2214yes—
57#1438C2214yes—
58#1438C2212yes—
59#1438C2214yes—
60#1438C2214yes—
61#1438C2214yes—
62#1437C2214yes—
63#1437C2214yes—
64#1437C2212yes—
65#1436C2212yes—
66#1434C2212yes—
67#1434C2214yes—
68C22⋊#95C2212yes—
69C23×Q8C2311yescharacteristic
70C23×Q8C2312yes—
71C23×D4C2314yes—
72C23×D4C2312yes—
73C23×D4C2312yes—
74C23×D4C2312yes—
75C24×C4C2311yescharacteristic
76C24×C4C2312yes—
77C2×#124—28no—
78C2×#124—48no—
79C2×C22⋊Q8—24no—
80C2×C22⋊Q8—42no—
81C2×C22⋊Q8—42no—
82C2×#121—44no—
83C2×#121—48no—
84C22×C4⋊C4—24no—
85C22×C4⋊C4—42no—
86C22×C4⋊C4—42no—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2312yes—
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C42C2311yescharacteristic
108C22×C42C2311yescharacteristic
109C22×C42C2312yes—
110#400—216no—
111#394—28no—
112#394—28no—
113#394—44no—
114#393—216no—
115#393—48no—
116#393—44no—
117#392—216no—
118#392—44no—
119#392—44no—
120#391—24no—
121#391—24no—
122#390—44no—
123#386—48no—
124#386—48no—
125#382—24no—
126#382D418yescomplemented
127#381—216no—
128C4×C22⋊C4—24no—
129C2×#95C2312yes—
130C2×#95C2312yes—
131C2×#95C2314yes—
132C2×#95C2312yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2312yes—
137C2×#95C2312yes—
138C2×#95C2312yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C25C2414yes—
152C25C2411yescharacteristic
153C22×Q8—22no—
154C22×Q8—24no—
155C22×Q8—24no—
156C22×Q8—41no—
157C22×Q8—41no—
158C22×Q8—44no—
159C22×Q8—42no—
160C22×Q8—44no—
161C22×Q8—42no—
162C22×Q8—42no—
163C22×D4—28no—
164C22×D4—28no—
165C22×D4—28no—
166C22×D4—24no—
167C22×D4—24no—
168C22×D4—24no—
169C22×D4—44no—
170C22×D4—42no—
171C22×D4—44no—
172C22×D4—44no—
173C22×D4—44no—
174C22×D4—44no—
175C22×D4—44no—
176C22×D4—44no—
177C22×D4—42no—
178C22×D4—44no—
179C22×D4—44no—
180C22×D4—44no—
181C22×D4—42no—
182C22×D4—42no—
183C22×D4—42no—
184C22×D4—42no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—24no—
189C23×C4—28no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—42no—
193C23×C4—42no—
194C23×C4—44no—
195C23×C4C2412yes—
196C23×C4C2414yes—
197C23×C4C2412yes—
198C23×C4C2414yes—
199C23×C4C2412yes—
200C23×C4C2414yes—
201C23×C4C2412yes—
202C23×C4C2412yes—
203C23×C4C2412yes—
204C23×C4C2411yescharacteristic
205C23×C4C2411yescharacteristic
206#124—432no—
207C22⋊Q8—416no—
208#121—432no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—416no—
225C2×C4⋊C4C2×D414yes—
226C2×C4⋊C4C2×D414yes—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
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—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⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C42—24no—
270C2×C42—22no—
271C2×C42—42no—
272C2×C42—42no—
273C2×C42—44no—
274C2×C42C2×D414yes—
275#95—24no—
276#95—28no—
277#95—28no—
278#95—28no—
279#95—24no—
280#95—24no—
281#95—28no—
282#95—44no—
283#95—44no—
284#95—44no—
285#95—44no—
286#95—44no—
287#95—44no—
288#95—44no—
289#95—42no—
290#95—42no—
291#95—42no—
292#95—44no—
293#95—44no—
294#95—44no—
295#95—42no—
296#95C2×D4116yes—
297C24—21no—
298C24—21no—
299C24—22no—
300C24—22no—
301C24—24no—
302C24—22no—
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—
316C24—24no—
317C24—24no—
318C24H(5,𝔽2)14yes—
319C24H(5,𝔽2)12yes—
320C24C2511yescharacteristic
321C2×Q8—44no—
322C2×Q8—48no—
323C2×Q8—44no—
324C2×Q8—44no—
325C2×Q8—84no—
326C2×Q8—84no—
327C2×Q8—84no—
328C2×Q8—84no—
329C2×Q8—88no—
330C2×Q8—88no—
331C2×D4—48no—
332C2×D4—416no—
333C2×D4—416no—
334C2×D4—48no—
335C2×D4—48no—
336C2×D4—416no—
337C2×D4—84no—
338C2×D4—84no—
339C2×D4—84no—
340C2×D4—88no—
341C2×D4—88no—
342C2×D4—84no—
343C2×D4—88no—
344C2×D4—88no—
345C2×D4—88no—
346C2×D4—88no—
347C2×D4—84no—
348C2×D4—88no—
349C2×D4—84no—
350C2×D4—88no—
351C2×D4—88no—
352C2×D4—88no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—28no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
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—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—48no—
410C22×C4—48no—
411C22×C4—44no—
412C22×C4—48no—
413C22×C4—416no—
414C22×C4—44no—
415C22×C4—48no—
416C22×C4—84no—
417C22×C4—88no—
418C22×C4C22×D414yes—
419C22×C4C22×D414yes—
420C22×C4C22×D414yes—
421C22×C4C22×D412yes—
422C22×C4H(5,𝔽2)12yes—
423C22×C4H(5,𝔽2)14yes—
424C4⋊C4—44no—
425C4⋊C4—44no—
426C4⋊C4—48no—
427C4⋊C4—48no—
428C4⋊C4—44no—
429C4⋊C4—48no—
430C22⋊C4—48no—
431C22⋊C4—48no—
432C22⋊C4—416no—
433C22⋊C4—416no—
434C22⋊C4—416no—
435C22⋊C4—88no—
436C22⋊C4—88no—
437C22⋊C4—88no—
438C22⋊C4—88no—
439C22⋊C4—88no—
440C22⋊C4—88no—
441C22⋊C4—88no—
442C22⋊C4—88no—
443C22⋊C4—88no—
444C42—28no—
445C42—44no—
446C42—84no—
447C23—22no—
448C23—24no—
449C23—28no—
450C23—24no—
451C23—28no—
452C23—22no—
453C23—24no—
454C23—24no—
455C23—41no—
456C23—41no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—44no—
461C23—44no—
462C23—44no—
463C23—44no—
464C23—44no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—44no—
473C23—44no—
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—
497C23H(3,𝔽4)14yes—
498C23C23×D411yescharacteristic
499C23C2×H(5,𝔽2)12yes—
500C23C2×H(5,𝔽2)11yescharacteristic
501C23C2×H(5,𝔽2)11yescharacteristic
502C23C2×H(5,𝔽2)12yes—
503C23#58512yes—
504C23#58512yes—
505C23#58514yes—
506Q8—88no—
507Q8—816no—
508D4—816no—
509D4—816no—
510D4—816no—
511D4—832no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—28no—
522C2×C4—28no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—416no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
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—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—816no—
574C2×C4#53514yes—
575C22—28no—
576C22—24no—
577C22—42no—
578C22—44no—
579C22—42no—
580C22—48no—
581C22—44no—
582C22—44no—
583C22—84no—
584C22—84no—
585C22—84no—
586C22—84no—
587C22—84no—
588C22—84no—
589C22—84no—
590C22—84no—
591C22—84no—
592C22—84no—
593C22—84no—
594C22—84no—
595C22—84no—
596C22—84no—
597C22—84no—
598C22C2×#53511yescharacteristic
599C22#348411yescharacteristic
600C22#348711yescharacteristic
601C22#348812yes—
602C22#350012yes—
603C22C2×H(3,𝔽4)11yescharacteristic
604C22#353314yes—
605C22#353412yes—
606C22#353412yes—
607C22#354012yes—
608C22#354214yes—
609C22#354512yes—
610C22#354511yescharacteristic
611C22#355114yes—
612C22#355812yes—
613C22#356614yes—
614C4—24no—
615C4—44no—
616C4—42no—
617C4—44no—
618C4—44no—
619C4—82no—
620C4—84no—
621C4—82no—
622C4—84no—
623C4—82no—
624C4—84no—
625C2—44no—
626C2—164no—
627C2#3671011yescharacteristic
628C2#4296212yes—
629C2#4597711yescharacteristic
630C2#5107012yes—
631C2#5174911yescharacteristic
632C2#5236712yes—
633C2#5542114yes—
634C2#5561212yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
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×#2819C2semidirect
2. C2×#2503C2semidirect
3. #17152C2semidirect
4. #16652C2semidirect
5. #16559C2semidirect
6. #16373C2semidirect
7. #16258C2semidirect
8. #15941C2semidirect
9. #15468C2semidirect
10. #15453C2semidirect
11. #14915C2semidirect
12. #14719C2semidirect
13. #14476C2semidirect
126. #382D4semidirect

Maximal subgroups

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

subgroupindex
C2×#28192
C2×#25032
#171522
#166522
#165592
#163732
#162582
#159412
#154682
#154532
#149152
#147192
#144762