Finite Group Databasebeta

← #4607234

Subgroups — #4607234

Order 512.

2558 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2683C211yesmaximal, characteristic, complemented
2C2×#2504C211yesmaximal, characteristic
3#17170C214yesmaximal, complemented
4#16960C212yesmaximal, complemented
5#16686C212yesmaximal, complemented
6#16348C214yesmaximal, complemented
7#16262C214yesmaximal, complemented
8#15984C212yesmaximal
9#15951C214yesmaximal, complemented
10#15342C212yesmaximal, complemented
11#15045C212yesmaximal, complemented
12#14720C212yesmaximal, complemented
13#14667C211yesmaximal, characteristic, complemented
14C22×C22⋊Q8C2214yes—
15C22×C22⋊Q8C2212yes—
16C22×#121C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18#2683—28no—
19#2504—28no—
20C2×#397C2214yes—
21C2×#397C2214yes—
22C2×#396C2214yes—
23C2×#395C2212yes—
24C2×#395C2212yes—
25C2×#390C2211yescharacteristic
26C2×#390C2211yescharacteristic
27C2×#389C2212yes—
28C2×#386C2214yes—
29C2×#386C2212yes—
30C2×#386C2212yes—
31C2×#385C2214yes—
32C2×#385C2214yes—
33C2×#382C2211yescharacteristic
34C2×#382C2211yescharacteristic
35C2×#381C2212yes—
36C2×C4×C22⋊C4C2211yescharacteristic
37#1446C2214yes—
38#1445C2214yes—
39#1444C2214yes—
40#1444C2214yes—
41#1444C2214yes—
42#1443C2214yes—
43#1442C2212yes—
44#1442C2214yes—
45#1442C2214yes—
46#1442C2214yes—
47#1442C2212yes—
48#1441C2212yes—
49#1441C2214yes—
50#1441C2214yes—
51#1441C2212yes—
52#1441C2214yes—
53#1440C2212yes—
54#1440C2214yes—
55#1440C2214yes—
56#1440C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2212yes—
60#1439C2214yes—
61#1439C2212yes—
62#1438C2212yes—
63#1438C2214yes—
64#1438C2212yes—
65#1434C2214yes—
66#1434C2212yes—
67C22⋊#95C2214yes—
68C22⋊#95C2212yes—
69C23×Q8C2312yes—
70C23×D4C2312yes—
71C23×D4C2311yescharacteristic
72C24×C4C2311yescharacteristic
73C24×C4C2312yes—
74C24×C4C2312yes—
75C2×C22⋊Q8—216no—
76C2×C22⋊Q8—44no—
77C2×C22⋊Q8—48no—
78C2×C22⋊Q8—48no—
79C2×C22⋊Q8—48no—
80C2×#121—24no—
81C2×#121—42no—
82C2×#121—42no—
83C22×C4⋊C4—24no—
84C22×C4⋊C4—42no—
85C22×C4⋊C4—42no—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2314yes—
108C22×C42C2311yescharacteristic
109C22×C42C2311yescharacteristic
110#397—216no—
111#397—48no—
112#396—216no—
113#395—28no—
114#395—44no—
115#390—24no—
116#390D418yes—
117#389—28no—
118#386—28no—
119#386—216no—
120#386—44no—
121#385—48no—
122#385—48no—
123#382—24no—
124#382—24no—
125#381—44no—
126C4×C22⋊C4—24no—
127C2×#95C2312yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2312yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2312yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C25C2412yes—
151C25C2411yescharacteristic
152C22×Q8—24no—
153C22×Q8—28no—
154C22×Q8—42no—
155C22×Q8—44no—
156C22×Q8—42no—
157C22×D4—24no—
158C22×D4—24no—
159C22×D4—28no—
160C22×D4—24no—
161C22×D4—41no—
162C22×D4—41no—
163C22×D4—42no—
164C22×D4—42no—
165C22×D4—44no—
166C22×D4C2×D414yes—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—28no—
174C23×C4—24no—
175C23×C4—42no—
176C23×C4—42no—
177C23×C4—44no—
178C23×C4—44no—
179C23×C4—42no—
180C23×C4—44no—
181C23×C4—42no—
182C23×C4C2414yes—
183C23×C4C2414yes—
184C23×C4C2414yes—
185C23×C4C2412yes—
186C23×C4C2412yes—
187C23×C4C2412yes—
188C23×C4C2412yes—
189C23×C4C2412yes—
190C23×C4C2414yes—
191C23×C4C2411yescharacteristic
192C23×C4C2411yescharacteristic
193C22⋊Q8—464no—
194C22⋊Q8—432no—
195#121—416no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—416no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4C2×D414yes—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—44no—
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—42no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4C2×D4116yes—
257C2×C42—22no—
258C2×C42—24no—
259C2×C42—42no—
260C2×C42C2×D414yes—
261#95—24no—
262#95—28no—
263#95—28no—
264#95—28no—
265#95—24no—
266#95—28no—
267#95—28no—
268#95—28no—
269#95—42no—
270#95—42no—
271#95—44no—
272#95—44no—
273#95—44no—
274#95—44no—
275#95—44no—
276#95—44no—
277#95—44no—
278#95—44no—
279#95—44no—
280#95—44no—
281#95—42no—
282#95—44no—
283#95—44no—
284C24—21no—
285C24—24no—
286C24—21no—
287C24—22no—
288C24—22no—
289C24—22no—
290C24—24no—
291C24—22no—
292C24—24no—
293C24—22no—
294C24—24no—
295C24—24no—
296C24—24no—
297C24—24no—
298C24C22×D414yes—
299C24H(5,𝔽2)12yes—
300C242−1+414yes—
301C24C2511yescharacteristic
302C2×Q8—416no—
303C2×Q8—48no—
304C2×Q8—84no—
305C2×Q8—88no—
306C2×Q8—84no—
307C2×D4—28no—
308C2×D4—44no—
309C2×D4—44no—
310C2×D4—48no—
311C2×D4—416no—
312C2×D4—48no—
313C2×D4—84no—
314C2×D4—84no—
315C2×D4—84no—
316C2×D4—88no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—28no—
321C22×C4—21no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—21no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—48no—
376C22×C4—48no—
377C22×C4—48no—
378C22×C4—44no—
379C22×C4—48no—
380C22×C4—416no—
381C22×C4—44no—
382C22×C4—48no—
383C22×C4—88no—
384C22×C4—88no—
385C22×C4—84no—
386C22×C4—84no—
387C22×C4—84no—
388C22×C4—88no—
389C22×C4C22×D414yes—
390C22×C4C22×D414yes—
391C22×C4C22×D412yes—
392C22×C4H(5,𝔽2)12yes—
393C22×C4H(5,𝔽2)18yes—
394C22×C42−1+414yes—
395C4⋊C4—216no—
396C4⋊C4—44no—
397C4⋊C4—48no—
398C4⋊C4—416no—
399C4⋊C4—44no—
400C4⋊C4—44no—
401C4⋊C4—416no—
402C4⋊C4—416no—
403C4⋊C4—48no—
404C4⋊C4—84no—
405C4⋊C4—88no—
406C22⋊C4—416no—
407C22⋊C4—48no—
408C22⋊C4—48no—
409C22⋊C4—416no—
410C22⋊C4—416no—
411C22⋊C4—416no—
412C22⋊C4—416no—
413C22⋊C4—88no—
414C22⋊C4—84no—
415C22⋊C4—84no—
416C22⋊C4—84no—
417C42—44no—
418C42—44no—
419C23—22no—
420C23—24no—
421C23—24no—
422C23—24no—
423C23—28no—
424C23—22no—
425C23—28no—
426C23—24no—
427C23—24no—
428C23—28no—
429C23—24no—
430C23—24no—
431C23—28no—
432C23—41no—
433C23—42no—
434C23—41no—
435C23—42no—
436C23—42no—
437C23—44no—
438C23—44no—
439C23—42no—
440C23—44no—
441C23—44no—
442C23—44no—
443C23—42no—
444C23—42no—
445C23—44no—
446C23—44no—
447C23—44no—
448C23—44no—
449C23—42no—
450C23—44no—
451C23—44no—
452C23—44no—
453C23—44no—
454C23—44no—
455C23#56314yes—
456C23C23×D411yescharacteristic
457C23C2×H(5,𝔽2)11yescharacteristic
458C23C2×H(5,𝔽2)14yes—
459C23C2×H(5,𝔽2)11yescharacteristic
460C23C2×2−1+412yes—
461C23C2×2−1+412yes—
462C23#58512yes—
463C23#58512yes—
464Q8—816no—
465D4—88no—
466D4—816no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—22no—
473C2×C4—28no—
474C2×C4—28no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—28no—
479C2×C4—24no—
480C2×C4—28no—
481C2×C4—416no—
482C2×C4—41no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—816no—
531C2×C4—816no—
532C2×C4#53614yes—
533C22—24no—
534C22—28no—
535C22—48no—
536C22—42no—
537C22—42no—
538C22—44no—
539C22—44no—
540C22—44no—
541C22—44no—
542C22—44no—
543C22—44no—
544C22—48no—
545C22—48no—
546C22—84no—
547C22—84no—
548C22—84no—
549C22—84no—
550C22C2×#53611yescharacteristic
551C22#348411yescharacteristic
552C22#348412yes—
553C22#348511yescharacteristic
554C22#348912yes—
555C22C2×#56311yescharacteristic
556C22#353212yes—
557C22#353512yes—
558C22#353512yes—
559C22#353712yes—
560C22#353914yes—
561C22#354114yes—
562C22#354511yescharacteristic
563C22#354714yes—
564C22#355214yes—
565C22#356112yes—
566C4—24no—
567C4—48no—
568C4—44no—
569C4—42no—
570C4—44no—
571C4—84no—
572C4—82no—
573C4—82no—
574C4—84no—
575C4—82no—
576C4—84no—
577C2—44no—
578C2—84no—
579C2#3672011yescharacteristic
580C2#4248512yes—
581C2#4613611yescharacteristic
582C2#5016612yes—
583C2#5200411yescharacteristic
584C2#5241412yes—
585C2#5813114yes—
586C2#5880212yes—
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×#2683C2semidirect
3. #17170C2semidirect
4. #16960C2semidirect
5. #16686C2semidirect
6. #16348C2semidirect
7. #16262C2semidirect
9. #15951C2semidirect
10. #15342C2semidirect
11. #15045C2semidirect
12. #14720C2semidirect
13. #14667C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26832
C2×#25042
#171702
#169602
#166862
#163482
#162622
#159842
#159512
#153422
#150452
#147202
#146672