Finite Group Databasebeta

← #4941687

Subgroups — #4941687

Order 512.

3026 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×#2436C211yesmaximal, characteristic
2C2×#2379C211yesmaximal, characteristic, complemented
3#16694C212yesmaximal, complemented
4#16280C214yesmaximal, complemented
5#15784C212yesmaximal, complemented
6#15378C214yesmaximal, complemented
7#15269C214yesmaximal, complemented
8#15260C212yesmaximal
9#15016C212yesmaximal, complemented
10#14743C214yesmaximal, complemented
11#14492C211yesmaximal, characteristic, complemented
12#14467C212yesmaximal, complemented
13#13922C212yesmaximal, complemented
14C24×Q8C2211yescharacteristic
15C22×#124C2211yescharacteristic
16C22×#124C2211yescharacteristic
17C22×#123C2212yes—
18C22×#121C2211yescharacteristic
19#2436—28no—
20C2×#399C2212yes—
21C2×#397C2214yes—
22C2×#397C2212yes—
23C2×#394C2214yes—
24C2×#393C2214yes—
25#2379—28no—
26C2×#391C2212yes—
27C2×#391C2211yescharacteristic
28C2×#391C2212yes—
29C2×#391C2211yescharacteristic
30C2×#391C2212yes—
31C2×#391C2212yes—
32C2×#388C2214yes—
33C2×#386C2212yes—
34C2×#386C2214yes—
35C2×#385C2214yes—
36C2×#385C2212yes—
37C2×#382C2212yes—
38C2×#380C2212yes—
39C2×C4×C22⋊C4C2211yescharacteristic
40C2×C4×C22⋊C4C2212yes—
41#1446C2214yes—
42#1446C2214yes—
43#1445C2212yes—
44#1445C2212yes—
45#1444C2214yes—
46#1444C2214yes—
47#1444C2214yes—
48#1442C2214yes—
49#1442C2214yes—
50#1442C2214yes—
51#1442C2214yes—
52#1441C2214yes—
53#1440C2214yes—
54#1440C2212yes—
55#1440C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1438C2212yes—
60#1438C2212yes—
61#1437C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64C4⋊#95C2214yes—
65#1434C2212yes—
66C22⋊#95C2214yes—
67C22⋊#95C2214yes—
68C22⋊#95C2214yes—
69C23×Q8—24no—
70C23×Q8—28no—
71C23×Q8—42no—
72C23×Q8—42no—
73C23×Q8—44no—
74C23×Q8C2312yes—
75C23×Q8C2311yescharacteristic
76C23×Q8C2311yescharacteristic
77C23×D4C2312yes—
78C23×D4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2312yes—
81C24×C4C2312yes—
82C2×#124—28no—
83C2×#124—24no—
84C2×#124—44no—
85C2×#124D418yes—
86C2×#123—44no—
87C2×#123—48no—
88C2×#121—24no—
89C2×#121—42no—
90C2×#121—42no—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112C22×C42C2312yes—
113C22×C42C2312yes—
114#399—44no—
115#397—48no—
116#397—44no—
117#394—48no—
118#393—216no—
119#391—28no—
120#391—24no—
121#391—28no—
122#391—24no—
123#391—44no—
124#391—44no—
125#388—216no—
126#386—28no—
127#386—216no—
128#385—216no—
129#385—44no—
130#382—28no—
131#380—44no—
132C4×C22⋊C4—24no—
133C4×C22⋊C4—44no—
134C2×#95C2312yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C25C2412yes—
157C25C2411yescharacteristic
158C22×Q8—22no—
159C22×Q8—24no—
160C22×Q8—24no—
161C22×Q8—24no—
162C22×Q8—24no—
163C22×Q8—24no—
164C22×Q8—24no—
165C22×Q8—24no—
166C22×Q8—42no—
167C22×Q8—42no—
168C22×Q8—41no—
169C22×Q8—41no—
170C22×Q8—42no—
171C22×Q8—41no—
172C22×Q8—41no—
173C22×Q8—416no—
174C22×Q8—432no—
175C22×Q8—416no—
176C22×Q8—432no—
177C22×Q8—416no—
178C22×Q8C2×D414yes—
179C22×Q8#4118yes—
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—28no—
183C22×D4—24no—
184C22×D4—41no—
185C22×D4—41no—
186C22×D4—42no—
187C22×D4—42no—
188C22×D4—44no—
189C22×D4C2×D414yes—
190C23×C4—22no—
191C23×C4—24no—
192C23×C4—22no—
193C23×C4—24no—
194C23×C4—22no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—42no—
201C23×C4—42no—
202C23×C4—42no—
203C23×C4—42no—
204C23×C4—44no—
205C23×C4—44no—
206C23×C4—42no—
207C23×C4C2414yes—
208C23×C4C2412yes—
209C23×C4C2414yes—
210C23×C4C2414yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2414yes—
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218#124—232no—
219#124—416no—
220#123—432no—
221#121—416no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—44no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4C2×D4116yes—
276C2×C42—24no—
277C2×C42—22no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—42no—
281C2×C42—42no—
282C2×C42—42no—
283C2×C42—42no—
284C2×C42—42no—
285C2×C42C2×D414yes—
286#95—24no—
287#95—28no—
288#95—28no—
289#95—28no—
290#95—28no—
291#95—28no—
292#95—28no—
293#95—28no—
294#95—28no—
295#95—28no—
296#95—28no—
297#95—28no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—42no—
305#95—44no—
306#95—44no—
307#95—42no—
308C24—21no—
309C24—22no—
310C24—22no—
311C24—21no—
312C24—24no—
313C24—22no—
314C24—24no—
315C24—24no—
316C24—22no—
317C24—24no—
318C24—22no—
319C24—24no—
320C24—24no—
321C24—24no—
322C24C22×D414yes—
323C24H(5,𝔽2)14yes—
324C24H(5,𝔽2)12yes—
325C24C2511yescharacteristic
326C2×Q8—28no—
327C2×Q8—216no—
328C2×Q8—44no—
329C2×Q8—44no—
330C2×Q8—48no—
331C2×Q8—48no—
332C2×Q8—48no—
333C2×Q8—48no—
334C2×Q8—48no—
335C2×Q8—48no—
336C2×Q8—48no—
337C2×Q8—48no—
338C2×Q8—44no—
339C2×Q8—48no—
340C2×Q8—4128no—
341C2×Q8—44no—
342C2×Q8—48no—
343C2×Q8#123116yescomplemented
344C2×D4—28no—
345C2×D4—44no—
346C2×D4—44no—
347C2×D4—48no—
348C2×D4—416no—
349C2×D4—48no—
350C2×D4—84no—
351C2×D4—84no—
352C2×D4—84no—
353C2×D4—88no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—28no—
362C22×C4—216no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
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—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—48no—
415C22×C4—48no—
416C22×C4—48no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—44no—
421C22×C4—48no—
422C22×C4—44no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—84no—
426C22×C4—88no—
427C22×C4—84no—
428C22×C4—88no—
429C22×C4C22×D414yes—
430C22×C4C22×D412yes—
431C22×C4C22×D414yes—
432C22×C4C2×#4114yes—
433C22×C4C2×#4114yes—
434C22×C4C2×#4114yes—
435C22×C4H(5,𝔽2)12yes—
436C22×C4H(5,𝔽2)14yes—
437C22×C42−1+418yes—
438C4⋊C4—44no—
439C4⋊C4—416no—
440C4⋊C4—84no—
441C4⋊C4—84no—
442C4⋊C4—84no—
443C4⋊C4—84no—
444C4⋊C4—88no—
445C4⋊C4—88no—
446C22⋊C4—232no—
447C22⋊C4—48no—
448C22⋊C4—48no—
449C22⋊C4—416no—
450C22⋊C4—48no—
451C22⋊C4—416no—
452C22⋊C4—48no—
453C22⋊C4—416no—
454C22⋊C4—88no—
455C22⋊C4—84no—
456C22⋊C4—88no—
457C42—28no—
458C42—44no—
459C42—84no—
460C42—84no—
461C23—22no—
462C23—24no—
463C23—24no—
464C23—28no—
465C23—28no—
466C23—24no—
467C23—22no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—28no—
473C23—28no—
474C23—41no—
475C23—41no—
476C23—42no—
477C23—42no—
478C23—44no—
479C23—42no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—42no—
488C23—44no—
489C23—44no—
490C23—42no—
491C23—42no—
492C23—44no—
493C23—42no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23H(3,𝔽4)14yes—
498C23C23×D411yescharacteristic
499C23C22×#4112yes—
500C23C2×H(5,𝔽2)11yescharacteristic
501C23C2×H(5,𝔽2)12yes—
502C23C2×H(5,𝔽2)11yescharacteristic
503C23C2×H(5,𝔽2)12yes—
504C23C2×2−1+414yes—
505C23#58512yes—
506Q8—416no—
507Q8—416no—
508Q8—432no—
509D4—88no—
510D4—816no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—28no—
530C2×C4—28no—
531C2×C4—28no—
532C2×C4—416no—
533C2×C4—432no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—816no—
579C2×C4C2×#12314yes—
580C2×C4C2×#12318yes—
581C22—28no—
582C22—24no—
583C22—44no—
584C22—42no—
585C22—42no—
586C22—44no—
587C22—44no—
588C22—48no—
589C22—48no—
590C22—44no—
591C22—44no—
592C22—44no—
593C22—48no—
594C22—84no—
595C22—84no—
596C22—84no—
597C22—84no—
598C22C22×#12311yescharacteristic
599C22#348411yescharacteristic
600C22#348411yescharacteristic
601C22#348812yes—
602C22#349012yes—
603C22#349012yes—
604C22#349912yes—
605C22#351614yes—
606C22#352214yes—
607C22C2×H(3,𝔽4)11yescharacteristic
608C22#353614yes—
609C22#353812yes—
610C22#354412yes—
611C22#354511yescharacteristic
612C22#355414yes—
613C22#355812yes—
614C4—24no—
615C4—28no—
616C4—42no—
617C4—44no—
618C4—44no—
619C4—48no—
620C4—82no—
621C4—84no—
622C4—84no—
623C4—84no—
624C4—82no—
625C2—44no—
626C2—84no—
627C2#3535011yescharacteristic
628C2#3599612yes—
629C2#4597711yescharacteristic
630C2#4609912yes—
631C2#4855814yes—
632C2#5068912yes—
633C2#5154211yescharacteristic
634C2#5184012yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#2379C2semidirect
3. #16694C2semidirect
4. #16280C2semidirect
5. #15784C2semidirect
6. #15378C2semidirect
7. #15269C2semidirect
9. #15016C2semidirect
10. #14743C2semidirect
11. #14492C2semidirect
12. #14467C2semidirect
13. #13922C2semidirect
343. C2×Q8#123semidirect

Maximal subgroups

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

subgroupindex
C2×#24362
C2×#23792
#166942
#162802
#157842
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