Finite Group Databasebeta

← #1045723

Subgroups — #1045723

Order 512.

3139 conjugacy classes of subgroups.

The subgroup lattice has 665 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2801C211yesmaximal, characteristic, complemented
2C22×#396C211yesmaximal, characteristic, complemented
3#17213C211yesmaximal, characteristic, complemented
4#17204C211yesmaximal, characteristic, complemented
5#16817C213yesmaximal, complemented
6#16439C213yesmaximal, complemented
7#15810C213yesmaximal, complemented
8#15619C213yesmaximal, complemented
9#14836C211yesmaximal, characteristic, complemented
10C22⋊#401C211yesmaximal, characteristic, complemented
11#14725C213yesmaximal
12#14562C211yesmaximal, characteristic, complemented
13#14019C213yesmaximal, complemented
14#13907C213yesmaximal, complemented
15#13873C213yesmaximal, complemented
16C22×#120C2211yescharacteristic
17C23×C4⋊C4C2213yes—
18C23×C22⋊C4C2213yes—
19#2801—28no—
20C2×#401C2211yescharacteristic, complemented
21C2×#399C2211yescharacteristic, complemented
22C2×#396—28no—
23C2×#396C2213yes—
24C2×#396C2211yescharacteristic
25C2×#396C2213yes—
26C2×#396C2211yescharacteristic
27C2×#394C2213yes—
28C2×#394C2213yes—
29C2×#389C2213yescomplemented
30C2×#389C2213yescomplemented
31C2×#388C2213yes—
32C2×#388C2213yes—
33C2×#388C2213yescomplemented
34C2×#388C2213yescomplemented
35C2×#386C2213yes—
36C2×#386C2213yes—
37C2×#380C2213yes—
38C2×#380C2213yes—
39C2×#379C2213yes—
40C2×C4×C22⋊C4C2213yes—
41C22×#95C2211yescharacteristic
42#1445C2213yescomplemented
43#1444C2211yescharacteristic, complemented
44#1443C2213yescomplemented
45#1443C2211yescharacteristic, complemented
46#1443C2213yescomplemented
47#1442C2213yescomplemented
48#1442C2213yescomplemented
49#1441C2213yescomplemented
50#1441C2213yescomplemented
51#1441C2213yescomplemented
52#1441C2213yescomplemented
53#1440C2213yes—
54#1440C2213yes—
55#1440C2213yes—
56#1440C2213yes—
57#1439C2213yes—
58#1439C2213yes—
59#1439C2213yes—
60#1439C2213yes—
61#1439C2213yescomplemented
62#1439C2213yescomplemented
63#1439C2213yescomplemented
64#1438C2213yes—
65#1438C2213yes—
66#1437C2213yes—
67#1437C2213yes—
68#1436C2213yescomplemented
69#1436C2213yescomplemented
70C4⋊#95C2213yescomplemented
71C4⋊#95C2213yescomplemented
72C22⋊#95C2213yes—
73C22⋊#95C2213yes—
74C22⋊#95C2213yes—
75C26C2311yescharacteristic
76C23×D4C2311yescharacteristic
77C23×D4C2311yescharacteristic
78C23×D4C2311yescharacteristic
79C24×C4C2313yes—
80C24×C4C2313yes—
81C2×#120—46no—
82C22×C4⋊C4—212no—
83C22×C4⋊C4—212no—
84C22×C4⋊C4—46no—
85C22×C4⋊C4C2313yes—
86C22×C4⋊C4C2313yes—
87C22×C4⋊C4C2313yes—
88C22×C4⋊C4C2313yes—
89C22×C4⋊C4C2313yes—
90C22×C4⋊C4C2313yes—
91C22×C4⋊C4C2313yes—
92C22×C4⋊C4C2313yes—
93C22×C22⋊C4—212no—
94C22×C22⋊C4—212no—
95C22×C22⋊C4—46no—
96C22×C22⋊C4C2313yes—
97C22×C22⋊C4C2313yes—
98C22×C22⋊C4C2311yescharacteristic
99C22×C22⋊C4C2313yes—
100C22×C22⋊C4C2313yes—
101C22×C22⋊C4C2313yes—
102C22×C22⋊C4C2313yes—
103C22×C22⋊C4C2313yes—
104C22×C22⋊C4C2313yes—
105C22×C22⋊C4C2313yes—
106C22×C22⋊C4C2313yes—
107C22×C22⋊C4C2313yes—
108C22×C22⋊C4C2313yes—
109C22×C22⋊C4C2311yescharacteristic
110C22×C22⋊C4C2311yescharacteristic
111C22×C22⋊C4C2313yes—
112C22×C22⋊C4C2313yes—
113C22×C42C2313yes—
114C22×C42C2313yes—
115#401—24no—
116#399—24no—
117#396—212no—
118#396—212no—
119#396—24no—
120#396D418yescomplemented
121#394—212no—
122#394—46no—
123#389—212no—
124#389—46no—
125#388—212no—
126#388—212no—
127#388—46no—
128#388—46no—
129#386—212no—
130#386—46no—
131#380—46no—
132#380—46no—
133#379—46no—
134C4×C22⋊C4—46no—
135C2×#95—24no—
136C2×#95C2313yes—
137C2×#95C2313yes—
138C2×#95C2313yes—
139C2×#95C2313yes—
140C2×#95C2313yes—
141C2×#95C2313yes—
142C2×#95C2313yes—
143C2×#95C2313yes—
144C2×#95C2313yes—
145C2×#95C2313yes—
146C2×#95C2313yes—
147C2×#95C2313yes—
148C2×#95C2313yes—
149C2×#95C2313yes—
150C2×#95C2313yes—
151C2×#95C2313yes—
152C2×#95C2313yes—
153C2×#95C2313yes—
154C2×#95C2313yes—
155C2×#95C2313yes—
156C2×#95C2313yes—
157C2×#95C2313yes—
158C2×#95C2313yes—
159C2×#95C2311yescharacteristic
160C25—22no—
161C25—22no—
162C25—22no—
163C25—26no—
164C25—26no—
165C25—43no—
166C25—43no—
167C25C2411yescharacteristic
168C25C2411yescharacteristic
169C25C2411yescharacteristic
170C25C2411yescharacteristic
171C22×D4—26no—
172C22×D4—26no—
173C22×D4—22no—
174C22×D4—22no—
175C22×D4—22no—
176C22×D4—43no—
177C22×D4—43no—
178C22×D4—43no—
179C22×D4—43no—
180C23×C4—26no—
181C23×C4—26no—
182C23×C4—26no—
183C23×C4—26no—
184C23×C4—26no—
185C23×C4—26no—
186C23×C4—26no—
187C23×C4—26no—
188C23×C4—43no—
189C23×C4—43no—
190C23×C4—43no—
191C23×C4—43no—
192C23×C4—43no—
193C23×C4#41112yes—
194C23×C4C2413yes—
195C23×C4C2413yes—
196C23×C4C2413yes—
197C23×C4C2413yes—
198C23×C4C2413yes—
199C23×C4C2411yescharacteristic
200C23×C4C2411yescharacteristic
201C23×C4C2411yescharacteristic
202C23×C4C2413yes—
203C23×C4C2413yes—
204C23×C4C2413yes—
205#120—416no—
206C2×C4⋊C4—26no—
207C2×C4⋊C4—26no—
208C2×C4⋊C4—26no—
209C2×C4⋊C4—26no—
210C2×C4⋊C4—26no—
211C2×C4⋊C4—26no—
212C2×C4⋊C4—26no—
213C2×C4⋊C4—26no—
214C2×C4⋊C4—26no—
215C2×C4⋊C4—26no—
216C2×C4⋊C4—26no—
217C2×C4⋊C4—26no—
218C2×C4⋊C4—448no—
219C2×C4⋊C4—43no—
220C2×C4⋊C4—43no—
221C2×C4⋊C4—43no—
222C2×C4⋊C4—43no—
223C2×C4⋊C4—43no—
224C2×C4⋊C4—43no—
225C2×C4⋊C4—43no—
226C2×C4⋊C4—43no—
227C2×C4⋊C4—43no—
228C2×C4⋊C4—43no—
229C2×C4⋊C4—43no—
230C2×C4⋊C4C2×D4112yes—
231C2×C22⋊C4—26no—
232C2×C22⋊C4—26no—
233C2×C22⋊C4—26no—
234C2×C22⋊C4—26no—
235C2×C22⋊C4—26no—
236C2×C22⋊C4—26no—
237C2×C22⋊C4—26no—
238C2×C22⋊C4—26no—
239C2×C22⋊C4—26no—
240C2×C22⋊C4—26no—
241C2×C22⋊C4—26no—
242C2×C22⋊C4—26no—
243C2×C22⋊C4—26no—
244C2×C22⋊C4—26no—
245C2×C22⋊C4—26no—
246C2×C22⋊C4—26no—
247C2×C22⋊C4—26no—
248C2×C22⋊C4—26no—
249C2×C22⋊C4—26no—
250C2×C22⋊C4—26no—
251C2×C22⋊C4—26no—
252C2×C22⋊C4—26no—
253C2×C22⋊C4—26no—
254C2×C22⋊C4—448no—
255C2×C22⋊C4—43no—
256C2×C22⋊C4—43no—
257C2×C22⋊C4—43no—
258C2×C22⋊C4—43no—
259C2×C22⋊C4—43no—
260C2×C22⋊C4—43no—
261C2×C22⋊C4—43no—
262C2×C22⋊C4—43no—
263C2×C22⋊C4—43no—
264C2×C22⋊C4—43no—
265C2×C22⋊C4—43no—
266C2×C22⋊C4—43no—
267C2×C22⋊C4—43no—
268C2×C22⋊C4—43no—
269C2×C22⋊C4—43no—
270C2×C22⋊C4—43no—
271C2×C22⋊C4—43no—
272C2×C22⋊C4—43no—
273C2×C22⋊C4—43no—
274C2×C22⋊C4—43no—
275C2×C22⋊C4—43no—
276C2×C22⋊C4C2×D4112yes—
277C2×C42—26no—
278C2×C42—26no—
279C2×C42—43no—
280C2×C42—43no—
281C2×C42—43no—
282C2×C42—43no—
283#95—26no—
284#95—26no—
285#95—26no—
286#95—26no—
287#95—26no—
288#95—26no—
289#95—26no—
290#95—43no—
291#95—43no—
292#95—43no—
293#95—43no—
294#95—43no—
295#95—43no—
296#95—43no—
297#95—43no—
298#95—43no—
299#95—43no—
300#95—43no—
301#95—43no—
302#95—43no—
303#95—43no—
304#95—43no—
305#95—43no—
306#95C2×D414yes—
307C24—21no—
308C24—23no—
309C24—21no—
310C24—23no—
311C24—28no—
312C24—21no—
313C24—23no—
314C24—23no—
315C24—23no—
316C24—23no—
317C24—23no—
318C24—23no—
319C24—21no—
320C24—21no—
321C24—21no—
322C24—23no—
323C24—23no—
324C24—21no—
325C24—23no—
326C24—23no—
327C24—23no—
328C24—21no—
329C24—21no—
330C24—23no—
331C24—412no—
332C24—412no—
333C24—412no—
334C24—412no—
335C24—412no—
336C24—412no—
337C24—412no—
338C24—44no—
339C24—86no—
340C24—86no—
341C24—86no—
342C24—86no—
343C24C22×D412yes—
344C24C2×#4116yes—
345C24H(5,𝔽2)16yes—
346C242−1+412yes—
347C242−1+412yes—
348C24C2511yescharacteristic
349C2×D4—44no—
350C2×D4—44no—
351C2×D4—412no—
352C2×D4—412no—
353C2×D4—44no—
354C2×D4—86no—
355C2×D4—86no—
356C2×D4—86no—
357C2×D4—86no—
358C22×C4—23no—
359C22×C4—23no—
360C22×C4—23no—
361C22×C4—23no—
362C22×C4—224no—
363C22×C4—23no—
364C22×C4—23no—
365C22×C4—23no—
366C22×C4—23no—
367C22×C4—23no—
368C22×C4—23no—
369C22×C4—23no—
370C22×C4—23no—
371C22×C4—23no—
372C22×C4—23no—
373C22×C4—23no—
374C22×C4—23no—
375C22×C4—224no—
376C22×C4—23no—
377C22×C4—21no—
378C22×C4—23no—
379C22×C4—23no—
380C22×C4—23no—
381C22×C4—23no—
382C22×C4—23no—
383C22×C4—23no—
384C22×C4—23no—
385C22×C4—23no—
386C22×C4—23no—
387C22×C4—23no—
388C22×C4—23no—
389C22×C4—23no—
390C22×C4—23no—
391C22×C4—23no—
392C22×C4—23no—
393C22×C4—23no—
394C22×C4—23no—
395C22×C4—23no—
396C22×C4—23no—
397C22×C4—23no—
398C22×C4—23no—
399C22×C4—23no—
400C22×C4—23no—
401C22×C4—23no—
402C22×C4—23no—
403C22×C4—23no—
404C22×C4—23no—
405C22×C4—23no—
406C22×C4—23no—
407C22×C4—23no—
408C22×C4—23no—
409C22×C4—23no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—23no—
413C22×C4—23no—
414C22×C4—23no—
415C22×C4—23no—
416C22×C4—23no—
417C22×C4—23no—
418C22×C4—23no—
419C22×C4—23no—
420C22×C4—412no—
421C22×C4—412no—
422C22×C4—412no—
423C22×C4—412no—
424C22×C4—412no—
425C22×C4—412no—
426C22×C4—412no—
427C22×C4—412no—
428C22×C4—412no—
429C22×C4—86no—
430C22×C4—86no—
431C22×C4—86no—
432C22×C4C22×D416yes—
433C22×C4C22×D416yes—
434C22×C4C2×#4116yes—
435C22×C4C2×#4116yes—
436C22×C4H(5,𝔽2)16yes—
437C4⋊C4—224no—
438C4⋊C4—412no—
439C4⋊C4—412no—
440C4⋊C4—412no—
441C4⋊C4—86no—
442C4⋊C4—86no—
443C4⋊C4—86no—
444C4⋊C4—86no—
445C22⋊C4—224no—
446C22⋊C4—412no—
447C22⋊C4—412no—
448C22⋊C4—412no—
449C22⋊C4—44no—
450C22⋊C4—44no—
451C22⋊C4—412no—
452C22⋊C4—412no—
453C22⋊C4—44no—
454C22⋊C4—412no—
455C22⋊C4—412no—
456C22⋊C4—86no—
457C22⋊C4—86no—
458C22⋊C4—86no—
459C22⋊C4—86no—
460C22⋊C4—86no—
461C22⋊C4—86no—
462C42—86no—
463C42—86no—
464C23—26no—
465C23—22no—
466C23—26no—
467C23—26no—
468C23—26no—
469C23—26no—
470C23—26no—
471C23—26no—
472C23—26no—
473C23—26no—
474C23—26no—
475C23—22no—
476C23—22no—
477C23—26no—
478C23—26no—
479C23—26no—
480C23—448no—
481C23—43no—
482C23—43no—
483C23—43no—
484C23—43no—
485C23—43no—
486C23—43no—
487C23—43no—
488C23—43no—
489C23—43no—
490C23—43no—
491C23—43no—
492C23—41no—
493C23—41no—
494C23—43no—
495C23—43no—
496C23—43no—
497C23—43no—
498C23—43no—
499C23—43no—
500C23—43no—
501C23—43no—
502C23—43no—
503C23—43no—
504C23—41no—
505C23—43no—
506C23—43no—
507C23—43no—
508C23—43no—
509C23—41no—
510C23—43no—
511C23—43no—
512C23—43no—
513C23—43no—
514C23—824no—
515C23—88no—
516C23—824no—
517C23—88no—
518C23—824no—
519C23—88no—
520C23#542112yes—
521C23#553112yes—
522C23#56014yes—
523C23C23×D411yescharacteristic
524C23C22×#4113yes—
525C23C2×H(5,𝔽2)13yes—
526C23C2×H(5,𝔽2)13yes—
527C23C2×2−1+411yescharacteristic
528C23C2×2−1+411yescharacteristic
529C23#58513yes—
530D4—88no—
531D4—88no—
532D4—88no—
533C2×C4—26no—
534C2×C4—26no—
535C2×C4—26no—
536C2×C4—26no—
537C2×C4—26no—
538C2×C4—26no—
539C2×C4—26no—
540C2×C4—26no—
541C2×C4—26no—
542C2×C4—26no—
543C2×C4—26no—
544C2×C4—26no—
545C2×C4—448no—
546C2×C4—43no—
547C2×C4—43no—
548C2×C4—43no—
549C2×C4—43no—
550C2×C4—43no—
551C2×C4—43no—
552C2×C4—43no—
553C2×C4—43no—
554C2×C4—43no—
555C2×C4—43no—
556C2×C4—43no—
557C2×C4—43no—
558C2×C4—43no—
559C2×C4—43no—
560C2×C4—43no—
561C2×C4—43no—
562C2×C4—43no—
563C2×C4—43no—
564C2×C4—43no—
565C2×C4—43no—
566C2×C4—43no—
567C2×C4—43no—
568C2×C4—43no—
569C2×C4—43no—
570C2×C4—43no—
571C2×C4—43no—
572C2×C4—43no—
573C2×C4—43no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—43no—
578C2×C4—43no—
579C2×C4—43no—
580C2×C4—43no—
581C2×C4—43no—
582C2×C4—43no—
583C2×C4—43no—
584C2×C4—43no—
585C2×C4—43no—
586C2×C4—43no—
587C2×C4—43no—
588C2×C4—43no—
589C2×C4—43no—
590C2×C4—43no—
591C2×C4—43no—
592C2×C4—43no—
593C2×C4—43no—
594C2×C4—43no—
595C2×C4—43no—
596C2×C4—43no—
597C2×C4—43no—
598C2×C4—824no—
599C2×C4#546112yes—
600C22—212no—
601C22—212no—
602C22—212no—
603C22—24no—
604C22—212no—
605C22—24no—
606C22—46no—
607C22—46no—
608C22—46no—
609C22—42no—
610C22—46no—
611C22—42no—
612C22—832no—
613C22—81no—
614C22—81no—
615C22—83no—
616C22—83no—
617C22—83no—
618C22—81no—
619C22—83no—
620C22—81no—
621C22—83no—
622C22—81no—
623C22—83no—
624C22—83no—
625C22—81no—
626C22—83no—
627C22—81no—
628C22#280118yescomplemented
629C22C2×#54213yes—
630C22C2×#54613yes—
631C22C2×#55313yes—
632C22#348611yescharacteristic
633C22#348813yes—
634C22#349413yes—
635C22#349513yes—
636C22#350513yes—
637C22C2×#56011yescharacteristic
638C22#353613yes—
639C22#353713yes—
640C22#354113yes—
641C22#356013yes—
642C4—212no—
643C4—46no—
644C4—46no—
645C4—83no—
646C4—81no—
647C4—83no—
648C4—83no—
649C4—83no—
650C4—81no—
651C4—83no—
652C4—81no—
653C2—28no—
654C2—82no—
655C2—82no—
656C2—161no—
657C2#3766513yes—
658C2#4056113yes—
659C2#4393613yes—
660C2#4597011yescharacteristic
661C2C2×#280111yescharacteristic
662C2#5247211yescharacteristic
663C2#5901313yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#120C22
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×#2801C2semidirect
2. C22×#396C2semidirect
3. #17213C2semidirect
4. #17204C2semidirect
5. #16817C2semidirect
6. #16439C2semidirect
7. #15810C2semidirect
8. #15619C2semidirect
9. #14836C2semidirect
10. C22⋊#401C2semidirect
12. #14562C2semidirect
13. #14019C2semidirect
14. #13907C2semidirect
15. #13873C2semidirect
20. C2×#401C22semidirect
21. C2×#399C22semidirect
29. C2×#389C22semidirect
30. C2×#389C22semidirect
33. C2×#388C22semidirect
34. C2×#388C22semidirect
42. #1445C22semidirect
43. #1444C22semidirect
44. #1443C22semidirect
45. #1443C22semidirect
46. #1443C22semidirect
47. #1442C22semidirect
48. #1442C22semidirect
49. #1441C22semidirect
50. #1441C22semidirect
51. #1441C22semidirect
52. #1441C22semidirect
61. #1439C22semidirect
62. #1439C22semidirect
63. #1439C22semidirect
68. #1436C22semidirect
69. #1436C22semidirect
70. C4⋊#95C22semidirect
71. C4⋊#95C22semidirect
120. #396D4semidirect
628. C22#2801semidirect

Maximal subgroups

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

subgroupindex
C2×#28012
C22×#3962
#172132
#172042
#168172
#164392
#158102
#156192
#148362
C22⋊#4012
#147252
#145622
#140192
#139072
#138732