Finite Group Databasebeta

← #539109

Subgroups — #539109

Order 512.

3114 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2354C212yesmaximal
2C2×#2349C211yesmaximal, characteristic
3#14382C214yesmaximal, complemented
4#14369C214yesmaximal, complemented
5#14072C214yesmaximal, complemented
6#14041C212yesmaximal, complemented
7#13981C212yesmaximal, complemented
8#13916C212yesmaximal
9#13910C211yesmaximal, characteristic, complemented
10#13906C211yesmaximal, characteristic
11#13878C214yesmaximal, complemented
12#13848C211yesmaximal, characteristic, complemented
13#13804C211yesmaximal, characteristic, complemented
14#13785C211yesmaximal, characteristic
15#13779C211yesmaximal, characteristic, complemented
16C22×C4×D4C2212yes—
17C22×C4×D4C2211yescharacteristic
18C22×#117C2214yes—
19#2354—216no—
20#2349C4116yescomplemented
21C2×#390C2212yes—
22C2×#390C2211yescharacteristic
23C2×#390C2211yescharacteristic
24C2×#389C2214yes—
25C2×#389C2212yes—
26C2×#388C2214yes—
27C2×#388C2214yes—
28C2×#387C2214yes—
29C2×#385C2214yes—
30C2×#384C2212yes—
31C2×#384C2211yescharacteristic
32C2×#382C2211yescharacteristic
33C2×#381C2214yes—
34C2×#380C2214yes—
35C2×#380C2214yes—
36C2×#380C2212yes—
37C2×C4×C4⋊C4C2212yes—
38C2×C4×C4⋊C4C2211yescharacteristic
39C2×C4×C4⋊C4C2212yes—
40C2×C4×C4⋊C4C2212yes—
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×#376C2211yescharacteristic
44C22×#95C2212yes—
45#1445C2214yes—
46#1444C2212yes—
47#1444C2212yes—
48#1444C2212yes—
49#1442C2212yes—
50#1442C2214yes—
51#1442C2214yes—
52#1441C2212yes—
53#1441C2212yes—
54#1441C2214yes—
55#1440C2214yes—
56#1440C2214yes—
57#1439C2212yes—
58#1439C2212yes—
59#1437C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2211yescharacteristic
71C4⋊#95C2211yescharacteristic
72#1434C2212yes—
73C22⋊#95C2214yes—
74C22⋊#95C2214yes—
75C4×#95C2212yes—
76C4×#95C2212yes—
77C23×D4C2311yescharacteristic
78C24×C4C2314yes—
79C24×C4C2312yes—
80C2×C4×D4—28no—
81C2×C4×D4—28no—
82C2×C4×D4—24no—
83C2×C4×D4—24no—
84C2×C4×D4C2×C4116yes—
85C2×C4×D4C2×C418yes—
86C2×#117—216no—
87C2×#117—48no—
88C2×#117—48no—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C42C2311yescharacteristic
110C22×C42C2312yes—
111C22×C42C2314yes—
112C22×C42C2312yes—
113C22×C42C2311yescharacteristic
114C22×C42C2314yes—
115C22×C42C2312yes—
116C22×C42C2311yescharacteristic
117#390—28no—
118#390—24no—
119#390D418yes—
120#389—48no—
121#389C2×C4116yes—
122#388—216no—
123#388—216no—
124#387—48no—
125#385C2×C4132yes—
126#384—28no—
127#384—24no—
128#382C2×C418yes—
129#381—48no—
130#380—216no—
131#380—48no—
132#380C2×C4116yes—
133C4×C4⋊C4—28no—
134C4×C4⋊C4—28no—
135C4×C4⋊C4—28no—
136C4×C4⋊C4—24no—
137C4×C4⋊C4—24no—
138C4×C4⋊C4C2×C418yes—
139#376—24no—
140C2×#95C2×C4116yes—
141C2×#95C2311yescharacteristic
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2311yescharacteristic
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2311yescharacteristic
168C2×#95C2311yescharacteristic
169C25C2412yes—
170C22×D4—22no—
171C22×D4—22no—
172C22×D4—22no—
173C22×D4—22no—
174C22×D4C22×C414yes—
175C22×D4C2×D414yes—
176C22×D4#4114yes—
177C23×C4—28no—
178C23×C4—28no—
179C23×C4—28no—
180C23×C4—28no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—44no—
184C23×C4—44no—
185C23×C4—44no—
186C23×C4—44no—
187C23×C4C22×C4116yes—
188C23×C4C22×C418yes—
189C23×C4C2414yes—
190C23×C4C2412yes—
191C23×C4C2412yes—
192C23×C4C2412yes—
193C23×C4C2412yes—
194C23×C4C2412yes—
195C23×C4C2414yes—
196C23×C4C2412yes—
197C23×C4C2412yes—
198C23×C4C2412yes—
199C23×C4C2411yescharacteristic
200C23×C4C2411yescharacteristic
201C23×C4C2412yes—
202C23×C4C2411yescharacteristic
203C4×D4—264no—
204C4×D4—232no—
205#117—464no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4C22×C418yes—
242C2×C4⋊C4C22×C414yes—
243C2×C4⋊C4C22×C418yes—
244C2×C4⋊C4C22×C4116yes—
245C2×C4⋊C4C2×D414yes—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4C22×C4116yes—
262C2×C22⋊C4C22×C418yes—
263C2×C22⋊C4C2×D4116yes—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—28no—
269C2×C42—28no—
270C2×C42—24no—
271C2×C42—22no—
272C2×C42—24no—
273C2×C42—22no—
274C2×C42—28no—
275C2×C42—24no—
276C2×C42—22no—
277C2×C42—24no—
278C2×C42—44no—
279C2×C42—44no—
280C2×C42—44no—
281C2×C42—44no—
282C2×C42C22×C418yes—
283C2×C42C22×C414yes—
284C2×C42C2×D414yes—
285C2×C42#4114yes—
286C2×C42#4114yes—
287#95—28no—
288#95—28no—
289#95—24no—
290#95—24no—
291#95—24no—
292#95—24no—
293#95—22no—
294#95—24no—
295#95—22no—
296#95—28no—
297#95—24no—
298#95—28no—
299#95—24no—
300#95—24no—
301#95—44no—
302#95—44no—
303#95—42no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95C22×C414yes—
311#95C22×C418yes—
312#95C22×C4116yes—
313#95C22×C414yes—
314#95C22×C418yes—
315C24—22no—
316C24—22no—
317C24—24no—
318C24—24no—
319C24—24no—
320C24—22no—
321C24—22no—
322C24—24no—
323C24C23×C414yes—
324C24C22×D414yes—
325C24C2×#4114yes—
326C24C2511yescharacteristic
327C2×D4—28no—
328C2×D4—28no—
329C2×D4—28no—
330C2×D4—28no—
331C2×D4—28no—
332C2×D4—28no—
333C2×D4C4×D4116yes—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—232no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—232no—
376C22×C4—216no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—416no—
404C22×C4—416no—
405C22×C4—416no—
406C22×C4—416no—
407C22×C4—416no—
408C22×C4—416no—
409C22×C4—88no—
410C22×C4—84no—
411C22×C4—84no—
412C22×C4C23×C418yes—
413C22×C4C23×C414yes—
414C22×C4C23×C414yes—
415C22×C4C23×C412yes—
416C22×C4C23×C412yes—
417C22×C4C23×C414yes—
418C22×C4C23×C412yes—
419C22×C4C22×D414yes—
420C22×C4C22×D414yes—
421C22×C4C22×D412yes—
422C22×C4C2×#4112yes—
423C22×C4C2×#4112yes—
424C22×C4C2×#4112yes—
425C22×C4C2×#4112yes—
426C22×C4C2×#4112yes—
427C22×C4C2×#4112yes—
428C22×C4C2×#4112yes—
429C22×C4H(5,𝔽2)14yes—
430C22×C4H(5,𝔽2)14yes—
431C22×C42−1+414yes—
432C22×C42−1+412yes—
433C22×C42−1+412yes—
434C4⋊C4—216no—
435C4⋊C4—28no—
436C4⋊C4—216no—
437C4⋊C4—216no—
438C4⋊C4—28no—
439C4⋊C4—416no—
440C4⋊C4—48no—
441C4⋊C4—48no—
442C4⋊C4—416no—
443C4⋊C4—48no—
444C4⋊C4—44no—
445C4⋊C4—84no—
446C4⋊C4—84no—
447C4⋊C4—88no—
448C22⋊C4—232no—
449C22⋊C4—216no—
450C22⋊C4—416no—
451C22⋊C4—416no—
452C22⋊C4—88no—
453C22⋊C4—88no—
454C42—216no—
455C42—28no—
456C42—416no—
457C42—416no—
458C42—48no—
459C42—44no—
460C42—84no—
461C42#117116yescomplemented
462C23—24no—
463C23—24no—
464C23—28no—
465C23—28no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—28no—
471C23—28no—
472C23—28no—
473C23—28no—
474C23—44no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23C2×C4×D418yes—
485C23C24×C411yescharacteristic
486C23C23×D411yescharacteristic
487C23C22×#4111yescharacteristic
488C23C22×#4111yescharacteristic
489C23C22×#4111yescharacteristic
490C23C2×H(5,𝔽2)12yes—
491C23C2×H(5,𝔽2)12yes—
492C23C2×2−1+411yescharacteristic
493C23C2×2−1+411yescharacteristic
494C23C2×2−1+412yes—
495C23#58512yes—
496D4—232no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—28no—
514C2×C4—24no—
515C2×C4—28no—
516C2×C4—24no—
517C2×C4—28no—
518C2×C4—28no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—28no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—464no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—816no—
577C2×C4C2×#11714yes—
578C2×C4C2×#11714yes—
579C2×C4C2×#11714yes—
580C2×C4C2×C4×D414yes—
581C2×C4#51914yes—
582C2×C4#52018yes—
583C2×C4#54714yes—
584C2×C4#54714yes—
585C2×C4#55214yes—
586C2×C4#55214yes—
587C22—28no—
588C22—216no—
589C22—44no—
590C22—44no—
591C22—44no—
592C22—48no—
593C22—44no—
594C22—44no—
595C22—48no—
596C22—44no—
597C22—48no—
598C22C22×#11711yescharacteristic
599C22C22×C4×D411yescharacteristic
600C22C2×#51911yescharacteristic
601C22C2×#52012yes—
602C22C4×H(5,𝔽2)12yes—
603C22C2×#54711yescharacteristic
604C22C2×#54711yescharacteristic
605C22C2×#55211yescharacteristic
606C22C2×#55211yescharacteristic
607C22#348412yes—
608C22#348912yes—
609C22#349412yes—
610C22#350811yescharacteristic
611C22#350811yescharacteristic
612C22#350911yescharacteristic
613C22#351011yescharacteristic
614C22#351112yes—
615C22#351312yes—
616C22#352012yes—
617C22#354412yes—
618C22#354712yes—
619C22#354812yes—
620C22#355512yes—
621C4—24no—
622C4—24no—
623C4—28no—
624C4—44no—
625C4—44no—
626C4—44no—
627C4—44no—
628C4—44no—
629C4—48no—
630C4—82no—
631C4—82no—
632C4—84no—
633C4—82no—
634C4#234418yes—
635C2—48no—
636C2C2×#234411yescharacteristic
637C2#3418811yescharacteristic
638C2#3425511yescharacteristic
639C2#3428812yes—
640C2#3448412yes—
641C2#4492812yes—
642C2#4493111yescharacteristic
643C2#4505012yes—
644C2#4505611yescharacteristic
645C2#4508311yescharacteristic
646C2#4542911yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #14382C2semidirect
4. #14369C2semidirect
5. #14072C2semidirect
6. #14041C2semidirect
7. #13981C2semidirect
9. #13910C2semidirect
11. #13878C2semidirect
12. #13848C2semidirect
13. #13804C2semidirect
15. #13779C2semidirect
20. #2349C4semidirect
461. C42#117semidirect

Maximal subgroups

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

subgroupindex
C2×#23542
C2×#23492
#143822
#143692
#140722
#140412
#139812
#139162
#139102
#139062
#138782
#138482
#138042
#137852
#137792