Finite Group Databasebeta

← #4933524

Subgroups — #4933524

Order 512.

3004 conjugacy classes of subgroups.

The subgroup lattice has 639 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2811C211yesmaximal, characteristic, complemented
2C2×#2436C211yesmaximal, characteristic
3#16559C212yesmaximal, complemented
4#16375C214yesmaximal, complemented
5#15854C212yesmaximal, complemented
6#15673C212yesmaximal
7#15636C214yesmaximal, complemented
8#15311C214yesmaximal, complemented
9#15050C212yesmaximal, complemented
10#14817C214yesmaximal, complemented
11#14492C212yesmaximal, complemented
12#13922C212yesmaximal, complemented
13#13834C211yesmaximal, characteristic, complemented
14C24×Q8C2211yescharacteristic
15C22×#124C2211yescharacteristic
16C22×#124C2211yescharacteristic
17C22×#124C2212yes—
18C22×#121C2211yescharacteristic
19#2811—28no—
20#2436—28no—
21C2×#397C2212yes—
22C2×#395C2212yes—
23C2×#395C2214yes—
24C2×#394C2214yes—
25C2×#393C2214yes—
26C2×#393C2214yes—
27C2×#393C2212yes—
28C2×#393C2212yes—
29C2×#391C2212yes—
30C2×#391C2214yes—
31C2×#391C2212yes—
32C2×#391C2212yes—
33C2×#391C2211yescharacteristic
34C2×#391C2212yes—
35C2×#391C2212yes—
36C2×#391C2211yescharacteristic
37C2×#386C2214yes—
38C2×#380C2212yes—
39C2×C4×C22⋊C4C2211yescharacteristic
40C2×C4×C22⋊C4C2212yes—
41#1446C2212yes—
42#1446C2214yes—
43#1446C2214yes—
44#1446C2214yes—
45#1445C2212yes—
46#1443C2214yes—
47#1443C2214yes—
48#1442C2214yes—
49#1440C2214yes—
50#1440C2214yes—
51#1440C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1438C2214yes—
57#1438C2212yes—
58#1437C2212yes—
59#1437C2214yes—
60#1437C2214yes—
61#1436C2214yes—
62#1436C2212yes—
63#1436C2214yes—
64#1434C2212yes—
65#1434C2214yes—
66#1434C2214yes—
67C22⋊#95C2214yes—
68C22⋊#95C2214yes—
69C23×Q8—24no—
70C23×Q8—28no—
71C23×Q8—42no—
72C23×Q8—42no—
73C23×Q8—44no—
74C23×Q8C2312yes—
75C23×Q8C2311yescharacteristic
76C23×Q8C2312yes—
77C23×Q8C2312yes—
78C23×Q8C2311yescharacteristic
79C23×D4C2312yes—
80C23×D4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2312yes—
83C2×#124—24no—
84C2×#124—28no—
85C2×#124—24no—
86C2×#124—44no—
87C2×#124—48no—
88C2×#124—44no—
89C2×#121—24no—
90C2×#121—42no—
91C2×#121—42no—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C42C2311yescharacteristic
108C22×C42C2312yes—
109C22×C42C2312yes—
110C22×C42C2312yes—
111C22×C42C2311yescharacteristic
112#397—44no—
113#395—28no—
114#395—216no—
115#394—48no—
116#393—216no—
117#393—48no—
118#393—44no—
119#393—44no—
120#391—28no—
121#391—28no—
122#391—24no—
123#391—28no—
124#391—216no—
125#391—44no—
126#391—44no—
127#391D418yes—
128#386—216no—
129#380—44no—
130C4×C22⋊C4—24no—
131C4×C22⋊C4—44no—
132C2×#95C2312yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
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—24no—
167C22×Q8—24no—
168C22×Q8—28no—
169C22×Q8—42no—
170C22×Q8—42no—
171C22×Q8—41no—
172C22×Q8—41no—
173C22×Q8—42no—
174C22×Q8—41no—
175C22×Q8—41no—
176C22×Q8—44no—
177C22×Q8—42no—
178C22×Q8—42no—
179C22×Q8—44no—
180C22×Q8—44no—
181C22×Q8—42no—
182C22×Q8—42no—
183C22×Q8—416no—
184C22×Q8—432no—
185C22×Q8—416no—
186C22×Q8—432no—
187C22×Q8—416no—
188C22×Q8C2×D414yes—
189C22×Q8#4118yes—
190C22×D4—24no—
191C22×D4—28no—
192C22×D4—22no—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—41no—
196C22×D4—41no—
197C22×D4—42no—
198C22×D4—42no—
199C22×D4—44no—
200C23×C4—22no—
201C23×C4—24no—
202C23×C4—22no—
203C23×C4—24no—
204C23×C4—22no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—42no—
210C23×C4—42no—
211C23×C4—42no—
212C23×C4C2414yes—
213C23×C4C2414yes—
214C23×C4C2412yes—
215C23×C4C2414yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2411yescharacteristic
220C23×C4C2414yes—
221C23×C4C2412yes—
222C23×C4C2411yescharacteristic
223#124—416no—
224#124—432no—
225#124—416no—
226#121—416no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4C2×D414yes—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—42no—
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×C42—22no—
273C2×C42—24no—
274C2×C42—24no—
275C2×C42—24no—
276C2×C42—42no—
277C2×C42—44no—
278C2×C42—42no—
279C2×C42—42no—
280C2×C42—42no—
281C2×C42—42no—
282C2×C42—42no—
283C2×C42C2×D414yes—
284#95—24no—
285#95—28no—
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—44no—
297#95—44no—
298#95—44no—
299#95—44no—
300#95—42no—
301#95—44no—
302#95—44no—
303#95—42no—
304#95—44no—
305#95—42no—
306#95—44no—
307#95C2×D4116yes—
308C24—21no—
309C24—22no—
310C24—22no—
311C24—21no—
312C24—24no—
313C24—22no—
314C24—22no—
315C24—24no—
316C24—24no—
317C24—22no—
318C24—24no—
319C24—22no—
320C24—24no—
321C24—24no—
322C24—24no—
323C24H(5,𝔽2)14yes—
324C24H(5,𝔽2)12yes—
325C24C2511yescharacteristic
326C2×Q8—28no—
327C2×Q8—216no—
328C2×Q8—216no—
329C2×Q8—44no—
330C2×Q8—44no—
331C2×Q8—48no—
332C2×Q8—48no—
333C2×Q8—48no—
334C2×Q8—48no—
335C2×Q8—48no—
336C2×Q8—48no—
337C2×Q8—48no—
338C2×Q8—48no—
339C2×Q8—44no—
340C2×Q8—48no—
341C2×Q8—4128no—
342C2×Q8—48no—
343C2×Q8—44no—
344C2×Q8—48no—
345C2×Q8—84no—
346C2×Q8—84no—
347C2×Q8—84no—
348C2×Q8—88no—
349C2×Q8—88no—
350C2×Q8—88no—
351C2×Q8—84no—
352C2×Q8—88no—
353C2×Q8#124116yescomplemented
354C2×D4—48no—
355C2×D4—416no—
356C2×D4—44no—
357C2×D4—44no—
358C2×D4—48no—
359C2×D4—44no—
360C2×D4—84no—
361C2×D4—84no—
362C2×D4—84no—
363C2×D4—88no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—28no—
372C22×C4—216no—
373C22×C4—24no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—48no—
429C22×C4—44no—
430C22×C4—48no—
431C22×C4—44no—
432C22×C4—48no—
433C22×C4C22×D414yes—
434C22×C4C22×D414yes—
435C22×C4C22×D412yes—
436C22×C4C22×D414yes—
437C22×C4C2×#4114yes—
438C22×C4C2×#4114yes—
439C22×C4C2×#4114yes—
440C22×C4H(5,𝔽2)12yes—
441C22×C4H(5,𝔽2)14yes—
442C22×C42−1+418yes—
443C4⋊C4—44no—
444C4⋊C4—84no—
445C4⋊C4—84no—
446C4⋊C4—84no—
447C4⋊C4—88no—
448C22⋊C4—48no—
449C22⋊C4—416no—
450C22⋊C4—48no—
451C22⋊C4—416no—
452C22⋊C4—416no—
453C22⋊C4—416no—
454C22⋊C4—416no—
455C22⋊C4—88no—
456C22⋊C4—88no—
457C22⋊C4—88no—
458C42—44no—
459C42—48no—
460C42—44no—
461C42—84no—
462C42—84no—
463C23—22no—
464C23—24no—
465C23—24no—
466C23—28no—
467C23—28no—
468C23—24no—
469C23—22no—
470C23—24no—
471C23—41no—
472C23—41no—
473C23—42no—
474C23—42no—
475C23—44no—
476C23—42no—
477C23—44no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—42no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23H(3,𝔽4)14yes—
500C23C23×D411yescharacteristic
501C23C22×#4112yes—
502C23C2×H(5,𝔽2)11yescharacteristic
503C23C2×H(5,𝔽2)12yes—
504C23C2×H(5,𝔽2)11yescharacteristic
505C23C2×H(5,𝔽2)12yes—
506C23C2×2−1+414yes—
507C23#58512yes—
508Q8—416no—
509Q8—416no—
510Q8—432no—
511Q8—816no—
512Q8—816no—
513D4—88no—
514D4—816no—
515C2×C4—22no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—28no—
531C2×C4—24no—
532C2×C4—28no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—28no—
537C2×C4—28no—
538C2×C4—416no—
539C2×C4—432no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—42no—
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—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4C2×#12414yes—
583C2×C4C2×#12418yes—
584C22—28no—
585C22—24no—
586C22—42no—
587C22—48no—
588C22—42no—
589C22—44no—
590C22—44no—
591C22—44no—
592C22—84no—
593C22—82no—
594C22—84no—
595C22—84no—
596C22—82no—
597C22—84no—
598C22—82no—
599C22—84no—
600C22—84no—
601C22C22×#12411yescharacteristic
602C22#348411yescharacteristic
603C22#348411yescharacteristic
604C22#348612yes—
605C22#348812yes—
606C22#349012yes—
607C22#349012yes—
608C22#351814yes—
609C22#352214yes—
610C22C2×H(3,𝔽4)11yescharacteristic
611C22#353614yes—
612C22#354412yes—
613C22#354511yescharacteristic
614C22#355012yes—
615C22#355812yes—
616C22#356114yes—
617C4—24no—
618C4—28no—
619C4—42no—
620C4—44no—
621C4—44no—
622C4—44no—
623C4—48no—
624C4—84no—
625C4—84no—
626C4—82no—
627C4—84no—
628C2—44no—
629C2—162no—
630C2#3536711yescharacteristic
631C2#3594812yes—
632C2#4597711yescharacteristic
633C2#4609912yes—
634C2#4783312yes—
635C2#5001414yes—
636C2#5168112yes—
637C2#5207211yescharacteristic
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
1. C2×#2811C2semidirect
3. #16559C2semidirect
4. #16375C2semidirect
5. #15854C2semidirect
7. #15636C2semidirect
8. #15311C2semidirect
9. #15050C2semidirect
10. #14817C2semidirect
11. #14492C2semidirect
12. #13922C2semidirect
13. #13834C2semidirect
353. C2×Q8#124semidirect

Maximal subgroups

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

subgroupindex
C2×#28112
C2×#24362
#165592
#163752
#158542
#156732
#156362
#153112
#150502
#148172
#144922
#139222
#138342