Finite Group Databasebeta

← #4388156

Subgroups — #4388156

Order 512.

2690 conjugacy classes of subgroups.

The subgroup lattice has 653 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2842C211yesmaximal, characteristic
2C2×#2312C211yesmaximal, characteristic
3#16302C214yesmaximal, complemented
4#15828C214yesmaximal, complemented
5#15587C214yesmaximal, complemented
6#15436C214yesmaximal, complemented
7#15294C214yesmaximal, complemented
8#14769C214yesmaximal, complemented
9#14338C211yesmaximal, characteristic
10#14338C211yesmaximal, characteristic
11#14127C211yesmaximal, characteristic
12#13869C211yesmaximal, characteristic
13#13869C211yesmaximal, characteristic
14C23×C22⋊C4C2211yescharacteristic
15C23×C22⋊C4C2211yescharacteristic
16C23×C22⋊C4C2211yescharacteristic
17#2842—28no—
18C2×#399C2214yes—
19C2×#398C2214yescomplemented
20C2×#396C2214yes—
21C2×#396C2214yes—
22#2312—28no—
23C2×#389C2214yescomplemented
24C2×#388C2214yes—
25C2×#388C2214yes—
26C2×#385C2212yes—
27C2×#385C2212yes—
28C2×#381C2212yes—
29C2×#381C2212yes—
30C2×#380C2212yes—
31C2×#380C2212yes—
32C2×#380C2212yes—
33C2×#379C2211yescharacteristic
34C2×#379C2211yescharacteristic
35C2×#379C2211yescharacteristic
36C2×#379C2211yescharacteristic
37C2×C4×C22⋊C4C2212yes—
38#1445C2214yescomplemented
39#1445C2214yescomplemented
40#1444C2214yescomplemented
41#1442C2214yescomplemented
42#1442C2214yescomplemented
43#1442C2214yescomplemented
44#1441C2214yescomplemented
45#1441C2214yescomplemented
46#1441C2214yescomplemented
47#1441C2214yescomplemented
48#1441C2214yescomplemented
49#1440C2214yes—
50#1440C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1436C2214yescomplemented
57#1436C2214yescomplemented
58#1434C2214yescomplemented
59C22⋊#95C2212yes—
60C22⋊#95C2212yes—
61C22⋊#95C2212yes—
62C22⋊#95C2212yes—
63C22⋊#95C2212yes—
64C22⋊#95C2212yes—
65C22⋊#95C2212yes—
66C22⋊#95C2212yes—
67C22⋊#95C2212yes—
68C22⋊#95C2212yes—
69C26C2311yescharacteristic
70C24×C4C2311yescharacteristic
71C24×C4C2311yescharacteristic
72C24×C4C2311yescharacteristic
73C24×C4C2311yescharacteristic
74C24×C4C2311yescharacteristic
75C24×C4C2311yescharacteristic
76C22×C4⋊C4C2314yes—
77C22×C4⋊C4C2314yes—
78C22×C4⋊C4C2314yes—
79C22×C4⋊C4C2314yes—
80C22×C4⋊C4C2314yes—
81C22×C4⋊C4C2314yes—
82C22×C4⋊C4C2314yes—
83C22×C22⋊C4—24no—
84C22×C22⋊C4—24no—
85C22×C22⋊C4—24no—
86C22×C22⋊C4—24no—
87C22×C22⋊C4—44no—
88C22×C22⋊C4—44no—
89C22×C22⋊C4Q818yes—
90C22×C22⋊C4C2312yes—
91C22×C22⋊C4C2312yes—
92C22×C22⋊C4C2312yes—
93C22×C22⋊C4C2312yes—
94C22×C22⋊C4C2312yes—
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C42C2314yes—
109#399—216no—
110#398—48no—
111#396—216no—
112#396—216no—
113#389—48no—
114#388—216no—
115#388—216no—
116#385—44no—
117#385—44no—
118#381—44no—
119#381—44no—
120#380—28no—
121#380—44no—
122#380—44no—
123#379—24no—
124#379—24no—
125#379—24no—
126#379D418yes—
127C4×C22⋊C4—28no—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
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—
148C25—42no—
149C25—42no—
150C25—42no—
151C25—42no—
152C25—42no—
153C25—42no—
154C25C2×D414yes—
155C25C2×Q814yes—
156C25#4114yes—
157C25C2411yescharacteristic
158C25C2411yescharacteristic
159C25C2411yescharacteristic
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—42no—
171C23×C4—42no—
172C23×C4—42no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4—42no—
177C23×C4—42no—
178C23×C4—42no—
179C23×C4—42no—
180C23×C4—42no—
181C23×C4—42no—
182C23×C4—42no—
183C23×C4—42no—
184C23×C4C2×Q814yes—
185C23×C4C2×Q814yes—
186C23×C4C2414yes—
187C23×C4C2414yes—
188C23×C4C2414yes—
189C23×C4C2412yes—
190C23×C4C2412yes—
191C23×C4C2414yes—
192C23×C4C2412yes—
193C23×C4C2412yes—
194C23×C4C2412yes—
195C23×C4C2412yes—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C22⋊C4—232no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—416no—
252C2×C22⋊C4—416no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4C2×D418yes—
270C2×C22⋊C4C2×D418yes—
271C2×C22⋊C4C2×D418yes—
272C2×C22⋊C4C2×Q818yes—
273C2×C22⋊C4C2×Q818yes—
274C2×C42—28no—
275C2×C42—44no—
276C2×C42—44no—
277#95—28no—
278#95—28no—
279#95—28no—
280#95—28no—
281#95—28no—
282#95—28no—
283#95—44no—
284#95—44no—
285#95—44no—
286#95—44no—
287#95—44no—
288#95—44no—
289#95—44no—
290#95—44no—
291#95—44no—
292#95—44no—
293#95—44no—
294#95—44no—
295#95—44no—
296#95—44no—
297C24—22no—
298C24—22no—
299C24—22no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24—22no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—44no—
316C24—44no—
317C24—48no—
318C24—48no—
319C24—48no—
320C24—48no—
321C24—48no—
322C24—48no—
323C24—44no—
324C24—44no—
325C24—44no—
326C24—44no—
327C24—88no—
328C24—88no—
329C24—88no—
330C24—88no—
331C24C22⋊Q8116yes—
332C24C22×D412yes—
333C24C22×D412yes—
334C24C22×D412yes—
335C24C22×Q812yes—
336C24C22×Q812yes—
337C24C22×Q812yes—
338C24C2×#4112yes—
339C24C2×#4112yes—
340C24C2×#4112yes—
341C24C2511yescharacteristic
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—28no—
365C22×C4—28no—
366C22×C4—28no—
367C22×C4—28no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
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—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—48no—
400C22×C4—48no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—88no—
404C22×C4—88no—
405C22×C4—82no—
406C22×C4—82no—
407C22×C4—82no—
408C22×C4—82no—
409C22×C4—82no—
410C22×C4—82no—
411C22×C4—82no—
412C22×C4—82no—
413C22×C4—84no—
414C22×C4—84no—
415C22×C4—82no—
416C22×C4—82no—
417C22×C4—82no—
418C22×C4—82no—
419C22×C4—88no—
420C22×C4—88no—
421C22×C4C22×D418yes—
422C22×C4C22×Q814yes—
423C22×C4C22×Q814yes—
424C22×C4H(5,𝔽2)14yes—
425C22×C4H(5,𝔽2)14yes—
426C4⋊C4—416no—
427C4⋊C4—88no—
428C4⋊C4—88no—
429C4⋊C4—88no—
430C4⋊C4—88no—
431C4⋊C4—88no—
432C4⋊C4—88no—
433C22⋊C4—216no—
434C22⋊C4—216no—
435C22⋊C4—48no—
436C22⋊C4—48no—
437C22⋊C4—48no—
438C22⋊C4—48no—
439C22⋊C4—48no—
440C22⋊C4—48no—
441C22⋊C4—48no—
442C22⋊C4—48no—
443C22⋊C4—84no—
444C22⋊C4—84no—
445C22⋊C4—84no—
446C22⋊C4—84no—
447C22⋊C4—84no—
448C22⋊C4—84no—
449C22⋊C4—84no—
450C22⋊C4—84no—
451C42—88no—
452C23—24no—
453C23—24no—
454C23—24no—
455C23—24no—
456C23—24no—
457C23—24no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—416no—
489C23—416no—
490C23—416no—
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—88no—
504C23—88no—
505C23—88no—
506C23—832no—
507C23—832no—
508C23—88no—
509C23C2×C22⋊Q814yes—
510C23C2×C22⋊Q814yes—
511C23C2×C22⋊Q814yes—
512C23C23×D411yescharacteristic
513C23C23×Q811yescharacteristic
514C23C22×#4111yescharacteristic
515C23C2×H(5,𝔽2)12yes—
516C23C2×H(5,𝔽2)12yes—
517C23#58512yes—
518C23#58512yes—
519C23#58512yes—
520C23#58512yes—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—28no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—28no—
527C2×C4—28no—
528C2×C4—28no—
529C2×C4—28no—
530C2×C4—28no—
531C2×C4—28no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
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—88no—
580C2×C4—88no—
581C2×C4—88no—
582C2×C4—88no—
583C2×C4—88no—
584C2×C4—88no—
585C22—24no—
586C22—24no—
587C22—24no—
588C22—24no—
589C22—24no—
590C22—24no—
591C22—24no—
592C22—24no—
593C22—24no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22—48no—
604C22—48no—
605C22—48no—
606C22—48no—
607C22—48no—
608C22—48no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—832no—
613C22C22×C22⋊Q811yescharacteristic
614C22#345812yes—
615C22#348512yes—
616C22#349011yescharacteristic
617C22#349011yescharacteristic
618C22#349111yescharacteristic
619C22#349111yescharacteristic
620C22#349111yescharacteristic
621C22#349111yescharacteristic
622C22#349612yes—
623C22#349612yes—
624C22#350112yes—
625C22#350112yes—
626C22#353414yes—
627C22#355114yes—
628C22#356214yes—
629C22#356614yes—
630C4—44no—
631C4—44no—
632C4—44no—
633C4—44no—
634C4—48no—
635C4—82no—
636C4—82no—
637C4—84no—
638C4—84no—
639C4—84no—
640C2—44no—
641C2—44no—
642C2—44no—
643C2#3509011yescharacteristic
644C2#3535911yescharacteristic
645C2#3535911yescharacteristic
646C2#4062614yes—
647C2#4638611yescharacteristic
648C2#4639011yescharacteristic
649C2#4683811yescharacteristic
650C2#4719511yescharacteristic
651C2#4785114yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #16302C2semidirect
4. #15828C2semidirect
5. #15587C2semidirect
6. #15436C2semidirect
7. #15294C2semidirect
8. #14769C2semidirect
19. C2×#398C22semidirect
23. C2×#389C22semidirect
38. #1445C22semidirect
39. #1445C22semidirect
40. #1444C22semidirect
41. #1442C22semidirect
42. #1442C22semidirect
43. #1442C22semidirect
44. #1441C22semidirect
45. #1441C22semidirect
46. #1441C22semidirect
47. #1441C22semidirect
48. #1441C22semidirect
56. #1436C22semidirect
57. #1436C22semidirect
58. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28422
C2×#23122
#163022
#158282
#155872
#154362
#152942
#147692
#143382
#141272
#138692