Finite Group Databasebeta

← #4414135

Subgroups — #4414135

Order 512.

3234 conjugacy classes of subgroups.

The subgroup lattice has 675 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2842C211yesmaximal, characteristic
2C2×#2280C211yesmaximal, characteristic
3#16667C214yesmaximal, complemented
4#16302C214yesmaximal, complemented
5#16300C214yesmaximal, complemented
6#14891C214yesmaximal, complemented
7#14773C214yesmaximal, complemented
8#14772C214yesmaximal, complemented
9#14681C211yesmaximal, characteristic
10#14681C211yesmaximal, characteristic
11#14673C211yesmaximal, characteristic
12#14673C211yesmaximal, characteristic
13#14673C211yesmaximal, characteristic
14C25×C4C2211yescharacteristic
15C22×#117C2212yes—
16#2842—28no—
17C2×#399C2214yes—
18C2×#398C2214yescomplemented
19C2×#396C2214yes—
20C2×#396C2214yes—
21C2×#387C2214yescomplemented
22#2280—28no—
23C2×#381C2212yes—
24C2×#381C2212yes—
25C2×#381C2212yes—
26C2×#381C2212yes—
27C2×#381C2212yes—
28C2×#380C2212yes—
29C2×#380C2212yes—
30C2×#379C2211yescharacteristic
31C2×#379C2211yescharacteristic
32C2×#379C2211yescharacteristic
33C2×#379C2211yescharacteristic
34C2×#379C2211yescharacteristic
35C2×#379C2211yescharacteristic
36C2×C4×C22⋊C4C2212yes—
37C2×C4×C22⋊C4C2212yes—
38#1445C2214yescomplemented
39#1444C2214yescomplemented
40#1444C2214yescomplemented
41#1443C2214yescomplemented
42#1442C2214yescomplemented
43#1442C2214yescomplemented
44#1442C2214yescomplemented
45#1441C2214yescomplemented
46#1441C2214yescomplemented
47#1441C2214yescomplemented
48#1441C2214yescomplemented
49#1439C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1436C2214yescomplemented
59#1436C2214yescomplemented
60C4⋊#95C2214yescomplemented
61C22⋊#95C2212yes—
62C22⋊#95C2212yes—
63C22⋊#95C2212yes—
64C22⋊#95C2212yes—
65C22⋊#95C2212yes—
66C22⋊#95C2212yes—
67C22⋊#95C2212yes—
68C22⋊#95C2212yes—
69C26C2311yescharacteristic
70C24×C4—24no—
71C24×C4—24no—
72C24×C4—44no—
73C24×C4—44no—
74C24×C4Q818yes—
75C24×C4C2311yescharacteristic
76C24×C4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C2×#117—28no—
82C2×#117—44no—
83C2×#117—44no—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2314yes—
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×C22⋊C4C2312yes—
109C22×C42C2314yes—
110#399—216no—
111#398—48no—
112#396—216no—
113#396—216no—
114#387—48no—
115#381—28no—
116#381—44no—
117#381—44no—
118#381—44no—
119#381—44no—
120#380—28no—
121#380—28no—
122#379—24no—
123#379—24no—
124#379—24no—
125#379—24no—
126#379—24no—
127#379D418yes—
128C4×C22⋊C4—28no—
129C4×C22⋊C4—28no—
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—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C25—42no—
151C25—42no—
152C25—42no—
153C25—42no—
154C25—42no—
155C25—42no—
156C25C2×D414yes—
157C25C2×Q814yes—
158C25#4114yes—
159C25C2411yescharacteristic
160C25C2411yescharacteristic
161C25C2411yescharacteristic
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—24no—
165C23×C4—24no—
166C23×C4—232no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—416no—
182C23×C4—416no—
183C23×C4—42no—
184C23×C4—42no—
185C23×C4—42no—
186C23×C4—42no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4—432no—
190C23×C4—432no—
191C23×C4C2×Q814yes—
192C23×C4C2×Q814yes—
193C23×C4C2×Q814yes—
194C23×C4C2×Q814yes—
195C23×C4C2×Q814yes—
196C23×C4C2×Q814yes—
197C23×C4C2414yes—
198C23×C4C2414yes—
199C23×C4C2414yes—
200C23×C4C2414yes—
201C23×C4C2414yes—
202C23×C4C2414yes—
203C23×C4C2411yescharacteristic
204C23×C4C2411yescharacteristic
205C23×C4C2411yescharacteristic
206C23×C4C2411yescharacteristic
207#117—432no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
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—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
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—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
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—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⋊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⋊C4—42no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4C2×D418yes—
281C2×C22⋊C4C2×D418yes—
282C2×C22⋊C4C2×D418yes—
283C2×C42—28no—
284C2×C42—28no—
285C2×C42—44no—
286#95—28no—
287#95—28no—
288#95—28no—
289#95—28no—
290#95—28no—
291#95—28no—
292#95—28no—
293#95—44no—
294#95—44no—
295#95—44no—
296#95—44no—
297#95—44no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—44no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—48no—
325C24—44no—
326C24—44no—
327C24—48no—
328C24—48no—
329C24—48no—
330C24—44no—
331C24—44no—
332C24—44no—
333C24—44no—
334C24—48no—
335C24—48no—
336C24—416no—
337C24—416no—
338C24—416no—
339C24—416no—
340C24C22⋊Q8116yes—
341C24C22×D412yes—
342C24C22×D412yes—
343C24C22×D412yes—
344C24C22×Q812yes—
345C24C22×Q812yes—
346C24C22×Q812yes—
347C24C2×#4112yes—
348C24C2×#4112yes—
349C24C2×#4112yes—
350C24C2511yescharacteristic
351C22×C4—216no—
352C22×C4—216no—
353C22×C4—28no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—28no—
361C22×C4—28no—
362C22×C4—28no—
363C22×C4—28no—
364C22×C4—28no—
365C22×C4—28no—
366C22×C4—28no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—28no—
382C22×C4—28no—
383C22×C4—28no—
384C22×C4—28no—
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—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
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—24no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—44no—
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—4128no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4C22×D418yes—
438C22×C4C22×Q812yes—
439C22×C4C22×Q812yes—
440C22×C4C22×Q812yes—
441C22×C4C22×Q812yes—
442C22×C4H(5,𝔽2)14yes—
443C22×C4H(5,𝔽2)14yes—
444C4⋊C4—416no—
445C4⋊C4—416no—
446C4⋊C4—88no—
447C4⋊C4—88no—
448C4⋊C4—88no—
449C4⋊C4—88no—
450C4⋊C4—88no—
451C22⋊C4—48no—
452C22⋊C4—48no—
453C22⋊C4—48no—
454C22⋊C4—48no—
455C22⋊C4—48no—
456C22⋊C4—48no—
457C22⋊C4—84no—
458C22⋊C4—84no—
459C22⋊C4—84no—
460C22⋊C4—84no—
461C22⋊C4—84no—
462C22⋊C4—84no—
463C22⋊C4—84no—
464C22⋊C4—84no—
465C22⋊C4—84no—
466C22⋊C4—84no—
467C22⋊C4—84no—
468C22⋊C4—84no—
469C42—416no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—24no—
506C23—416no—
507C23—416no—
508C23—416no—
509C23—416no—
510C23—416no—
511C23—416no—
512C23—416no—
513C23—464no—
514C23—464no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23C2×C22⋊Q814yes—
528C23C2×C22⋊Q814yes—
529C23C2×C22⋊Q814yes—
530C23C23×D411yescharacteristic
531C23C23×Q811yescharacteristic
532C23C22×#4111yescharacteristic
533C23C2×H(5,𝔽2)12yes—
534C23C2×H(5,𝔽2)12yes—
535C23#58512yes—
536C23#58512yes—
537C23#58512yes—
538C23#58512yes—
539C2×C4—232no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—28no—
555C2×C4—28no—
556C2×C4—28no—
557C2×C4—416no—
558C2×C4—416no—
559C2×C4—416no—
560C2×C4—416no—
561C2×C4—416no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
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×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4#54314yes—
606C2×C4#54314yes—
607C22—24no—
608C22—24no—
609C22—24no—
610C22—24no—
611C22—24no—
612C22—24no—
613C22—24no—
614C22—24no—
615C22—24no—
616C22—464no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—48no—
630C22—48no—
631C22—48no—
632C22—48no—
633C22—48no—
634C22—48no—
635C22C22×C22⋊Q811yescharacteristic
636C22C2×#54311yescharacteristic
637C22C2×#54311yescharacteristic
638C22#345811yescharacteristic
639C22#345811yescharacteristic
640C22#345811yescharacteristic
641C22#345811yescharacteristic
642C22#348412yes—
643C22#348512yes—
644C22#348512yes—
645C22#349012yes—
646C22#349112yes—
647C22#349112yes—
648C22#352914yes—
649C22#353914yes—
650C22#353914yes—
651C22#354014yes—
652C4—24no—
653C4—24no—
654C4—48no—
655C4—42no—
656C4—42no—
657C4—84no—
658C4—84no—
659C4—84no—
660C4—84no—
661C4—84no—
662C2—44no—
663C2—44no—
664C2—44no—
665C2#3498211yescharacteristic
666C2#3509011yescharacteristic
667C2#3509011yescharacteristic
668C2#3667411yescharacteristic
669C2#3748011yescharacteristic
670C2#3748111yescharacteristic
671C2#3748111yescharacteristic
672C2#4544014yes—
673C2#4545914yes—
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. #16667C2semidirect
4. #16302C2semidirect
5. #16300C2semidirect
6. #14891C2semidirect
7. #14773C2semidirect
8. #14772C2semidirect
18. C2×#398C22semidirect
21. C2×#387C22semidirect
38. #1445C22semidirect
39. #1444C22semidirect
40. #1444C22semidirect
41. #1443C22semidirect
42. #1442C22semidirect
43. #1442C22semidirect
44. #1442C22semidirect
45. #1441C22semidirect
46. #1441C22semidirect
47. #1441C22semidirect
48. #1441C22semidirect
58. #1436C22semidirect
59. #1436C22semidirect
60. C4⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28422
C2×#22802
#166672
#163022
#163002
#148912
#147732
#147722
#146812
#146732