Finite Group Databasebeta

← #4388023

Subgroups — #4388023

Order 512.

2530 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2842C211yesmaximal, characteristic
2C2×#2483C211yesmaximal, characteristic
3#16832C214yesmaximal, complemented
4#15922C214yesmaximal, complemented
5#15916C214yesmaximal, complemented
6#15470C214yesmaximal, complemented
7#15469C214yesmaximal, complemented
8#14833C214yesmaximal, complemented
9#14341C211yesmaximal, characteristic
10#14341C211yesmaximal, characteristic
11#13869C211yesmaximal, characteristic
12#13869C211yesmaximal, characteristic
13#13775C211yesmaximal, characteristic
14C23×C22⋊C4C2211yescharacteristic
15C23×C22⋊C4C2211yescharacteristic
16C23×C22⋊C4C2211yescharacteristic
17#2842—28no—
18#2483—28no—
19C2×#400C2214yescomplemented
20C2×#399C2214yes—
21C2×#397C2214yes—
22C2×#396C2214yes—
23C2×#396C2214yes—
24C2×#385C2212yes—
25C2×#385C2212yes—
26C2×#381C2212yes—
27C2×#381C2212yes—
28C2×#381C2212yes—
29C2×#381C2212yes—
30C2×#379C2211yescharacteristic
31C2×#379C2211yescharacteristic
32C2×#379C2211yescharacteristic
33C2×#379C2211yescharacteristic
34C2×C4×C22⋊C4C2212yes—
35#1445C2214yescomplemented
36#1443C2214yescomplemented
37#1443C2214yescomplemented
38#1442C2214yescomplemented
39#1442C2214yescomplemented
40#1442C2214yescomplemented
41#1442C2214yescomplemented
42#1442C2214yescomplemented
43#1442C2214yescomplemented
44#1442C2214yescomplemented
45#1441C2214yescomplemented
46#1440C2214yes—
47#1440C2214yes—
48#1440C2214yes—
49#1440C2214yes—
50#1440C2214yes—
51#1440C2214yes—
52#1437C2214yes—
53#1437C2214yes—
54#1436C2214yescomplemented
55#1436C2214yescomplemented
56C4⋊#95C2214yescomplemented
57#1434C2214yescomplemented
58C22⋊#95C2212yes—
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×C22⋊C4—24no—
81C22×C22⋊C4—24no—
82C22×C22⋊C4—24no—
83C22×C22⋊C4—24no—
84C22×C22⋊C4—24no—
85C22×C22⋊C4—24no—
86C22×C22⋊C4—24no—
87C22×C22⋊C4—44no—
88C22×C22⋊C4C2312yes—
89C22×C22⋊C4C2312yes—
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×C42C2314yes—
107#400—48no—
108#399—216no—
109#397—216no—
110#396—216no—
111#396—216no—
112#385—28no—
113#385—28no—
114#381—28no—
115#381—28no—
116#381—44no—
117#381—44no—
118#379—24no—
119#379—24no—
120#379—24no—
121#379D418yes—
122C4×C22⋊C4—44no—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
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—
146C25—42no—
147C25—42no—
148C25—42no—
149C25—42no—
150C25—42no—
151C25—42no—
152C25C2×D414yes—
153C25#4114yes—
154C25#4114yes—
155C25C2411yescharacteristic
156C25C2411yescharacteristic
157C25C2411yescharacteristic
158C23×C4—22no—
159C23×C4—24no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—22no—
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×C4—42no—
185C23×C4—42no—
186C23×C4C2414yes—
187C23×C4C2414yes—
188C23×C4C2414yes—
189C23×C4C2412yes—
190C23×C4C2412yes—
191C23×C4C2412yes—
192C23×C4C2412yes—
193C23×C4C2412yes—
194C23×C4C2414yes—
195C23×C4C2412yes—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
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—42no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—416no—
244C2×C22⋊C4—416no—
245C2×C22⋊C4—416no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—42no—
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⋊C4C2×D418yes—
263C2×C22⋊C4C2×D418yes—
264C2×C22⋊C4C2×D418yes—
265C2×C42—44no—
266C2×C42—44no—
267C2×C42—44no—
268#95—28no—
269#95—28no—
270#95—28no—
271#95—28no—
272#95—28no—
273#95—28no—
274#95—28no—
275#95—44no—
276#95—44no—
277#95—44no—
278#95—44no—
279#95—44no—
280#95—44no—
281#95—44no—
282#95—44no—
283#95—44no—
284#95—44no—
285#95—44no—
286#95—44no—
287#95—44no—
288#95—44no—
289#95—44no—
290#95—44no—
291C24—22no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—22no—
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—44no—
310C24—48no—
311C24—48no—
312C24—48no—
313C24—48no—
314C24—48no—
315C24—48no—
316C24—44no—
317C24—44no—
318C24—44no—
319C24—44no—
320C24—44no—
321C24—88no—
322C24—88no—
323C24—88no—
324C24—88no—
325C24#124116yes—
326C24C22×D412yes—
327C24C22×D412yes—
328C24C22×D412yes—
329C24C2×#4112yes—
330C24C2×#4112yes—
331C24C2×#4112yes—
332C24C2×#4112yes—
333C24C2×#4112yes—
334C24C2×#4112yes—
335C24C2511yescharacteristic
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—28no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
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—44no—
394C22×C4—48no—
395C22×C4—48no—
396C22×C4—48no—
397C22×C4—48no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—88no—
403C22×C4—82no—
404C22×C4—82no—
405C22×C4—82no—
406C22×C4—82no—
407C22×C4—82no—
408C22×C4—82no—
409C22×C4—82no—
410C22×C4—82no—
411C22×C4—84no—
412C22×C4—84no—
413C22×C4—82no—
414C22×C4—82no—
415C22×C4—82no—
416C22×C4—82no—
417C22×C4—84no—
418C22×C4—84no—
419C22×C4—88no—
420C22×C4C22×D418yes—
421C22×C4H(5,𝔽2)14yes—
422C4⋊C4—88no—
423C4⋊C4—88no—
424C4⋊C4—88no—
425C4⋊C4—88no—
426C22⋊C4—48no—
427C22⋊C4—48no—
428C22⋊C4—48no—
429C22⋊C4—48no—
430C22⋊C4—48no—
431C22⋊C4—48no—
432C22⋊C4—48no—
433C22⋊C4—48no—
434C22⋊C4—48no—
435C22⋊C4—48no—
436C22⋊C4—84no—
437C22⋊C4—84no—
438C22⋊C4—84no—
439C22⋊C4—84no—
440C22⋊C4—84no—
441C22⋊C4—84no—
442C22⋊C4—84no—
443C22⋊C4—84no—
444C42—88no—
445C23—24no—
446C23—24no—
447C23—24no—
448C23—24no—
449C23—24no—
450C23—24no—
451C23—24no—
452C23—24no—
453C23—24no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—24no—
458C23—24no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—24no—
464C23—24no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—24no—
476C23—24no—
477C23—22no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—416no—
482C23—416no—
483C23—416no—
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—88no—
497C23—88no—
498C23—832no—
499C23—832no—
500C23—88no—
501C23—88no—
502C23C2×#12414yes—
503C23C2×#12414yes—
504C23C2×#12414yes—
505C23C23×D411yescharacteristic
506C23C22×#4111yescharacteristic
507C23C22×#4111yescharacteristic
508C23C2×H(5,𝔽2)12yes—
509C23#58512yes—
510C23#58512yes—
511C23#58512yes—
512C23#58512yes—
513C23#58512yes—
514C2×C4—28no—
515C2×C4—28no—
516C2×C4—24no—
517C2×C4—28no—
518C2×C4—28no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—42no—
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—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—88no—
573C2×C4—88no—
574C2×C4—88no—
575C2×C4—88no—
576C2×C4—88no—
577C2×C4—88no—
578C22—24no—
579C22—24no—
580C22—24no—
581C22—24no—
582C22—24no—
583C22—24no—
584C22—24no—
585C22—24no—
586C22—24no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22—48no—
594C22—48no—
595C22—48no—
596C22—48no—
597C22—48no—
598C22—48no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22—42no—
604C22—42no—
605C22—832no—
606C22C22×#12411yescharacteristic
607C22#345912yes—
608C22#348512yes—
609C22#348512yes—
610C22#349011yescharacteristic
611C22#349111yescharacteristic
612C22#349111yescharacteristic
613C22#349111yescharacteristic
614C22#349111yescharacteristic
615C22#349112yes—
616C22#349112yes—
617C22#349111yescharacteristic
618C22#349612yes—
619C22#356014yes—
620C22#356214yes—
621C22#356614yes—
622C22#356614yes—
623C4—44no—
624C4—48no—
625C4—82no—
626C4—82no—
627C4—82no—
628C4—84no—
629C4—84no—
630C4—84no—
631C4—82no—
632C4—82no—
633C2—44no—
634C2—44no—
635C2—44no—
636C2#3499311yescharacteristic
637C2#3545511yescharacteristic
638C2#3545511yescharacteristic
639C2#4160614yes—
640C2#4638811yescharacteristic
641C2#4639011yescharacteristic
642C2#4683811yescharacteristic
643C2#4683811yescharacteristic
644C2#4881014yes—
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. #16832C2semidirect
4. #15922C2semidirect
5. #15916C2semidirect
6. #15470C2semidirect
7. #15469C2semidirect
8. #14833C2semidirect
19. C2×#400C22semidirect
35. #1445C22semidirect
36. #1443C22semidirect
37. #1443C22semidirect
38. #1442C22semidirect
39. #1442C22semidirect
40. #1442C22semidirect
41. #1442C22semidirect
42. #1442C22semidirect
43. #1442C22semidirect
44. #1442C22semidirect
45. #1441C22semidirect
54. #1436C22semidirect
55. #1436C22semidirect
56. C4⋊#95C22semidirect
57. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28422
C2×#24832
#168322
#159222
#159162
#154702
#154692
#148332
#143412
#138692
#137752