Finite Group Databasebeta

← #1040868

Subgroups — #1040868

Order 512.

2514 conjugacy classes of subgroups.

The subgroup lattice has 655 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2782C212yesmaximal
2#17222C212yesmaximal, complemented
3#16689C212yesmaximal, complemented
4#16307C214yesmaximal, complemented
5#15966C212yesmaximal
6#15357C214yesmaximal, complemented
7#15153C212yesmaximal, complemented
8#15152C211yesmaximal, characteristic, complemented
9#15132C212yesmaximal, complemented
10#13994C212yesmaximal, complemented
11#13911C211yesmaximal, characteristic, complemented
12#13880C212yesmaximal, complemented
13#13880C212yesmaximal, complemented
14#13872C212yesmaximal
15C4×#394C211yesmaximal, characteristic
16C22×C4×D4C2211yescharacteristic
17C23×C4⋊C4C2212yes—
18#2782—216no—
19C2×#400C2212yes—
20C2×#400C2214yes—
21C2×#400C2214yes—
22C2×#397C2214yes—
23C2×#394C2211yescharacteristic
24C2×#389C2212yes—
25C2×#388C2214yes—
26C2×#386C2214yes—
27C2×#385C2214yes—
28C2×#385C2214yes—
29C2×#385C2214yes—
30C2×#385C2214yes—
31C2×#384C2211yescharacteristic
32C2×#384C2212yes—
33C2×#384C2211yescharacteristic
34C2×#382C2212yes—
35C2×#380C2212yes—
36C2×#380C2212yes—
37C2×#380C2212yes—
38C2×C4×C4⋊C4C2211yescharacteristic
39C2×C4×C4⋊C4C2211yescharacteristic
40C2×C4×C22⋊C4C2212yes—
41C2×C4×C22⋊C4C2212yes—
42#1445C2212yes—
43#1445C2214yes—
44#1445C2214yes—
45#1444C2214yes—
46#1444C2214yes—
47#1444C2214yes—
48#1444C2214yes—
49#1443C2214yes—
50#1442C2214yes—
51#1442C2211yescharacteristic
52#1442C2212yes—
53#1442C2212yes—
54#1442C2214yes—
55#1442C2212yes—
56#1441C2211yescharacteristic
57#1441C2214yes—
58#1441C2212yes—
59#1440C2214yes—
60#1440C2212yes—
61#1440C2214yes—
62#1439C2212yes—
63#1439C2214yes—
64#1438C2211yescharacteristic
65C4⋊#95C2211yescharacteristic
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73#1434C2214yes—
74C22⋊#95C2214yes—
75C4×#95C2211yescharacteristic
76C23×D4C2311yescharacteristic
77C24×C4C2312yes—
78C24×C4C2314yes—
79C2×C4×D4—28no—
80C2×C4×D4—42no—
81C22×C4⋊C4—44no—
82C22×C4⋊C4—44no—
83C22×C4⋊C4—44no—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C42C2311yescharacteristic
108C22×C42C2312yes—
109C22×C42C2312yes—
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112#400—28no—
113#400—216no—
114#400—48no—
115#397—216no—
116#394D418yes—
117#389—44no—
118#388—216no—
119#386—216no—
120#385—216no—
121#385—216no—
122#385—48no—
123#385—48no—
124#384—42no—
125#384—42no—
126#384—44no—
127#382—28no—
128#380—28no—
129#380—44no—
130#380—44no—
131C4×C4⋊C4—24no—
132C4×C4⋊C4—42no—
133C4×C22⋊C4—28no—
134C4×C22⋊C4—28no—
135C2×#95C2312yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2312yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2311yescharacteristic
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2311yescharacteristic
161C2×#95C2311yescharacteristic
162C2×#95C2311yescharacteristic
163C25C2412yes—
164C22×D4—24no—
165C22×D4—24no—
166C22×D4—41no—
167C22×D4—41no—
168C22×D4C2×D414yes—
169C23×C4—24no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—28no—
176C23×C4—44no—
177C23×C4—44no—
178C23×C4—44no—
179C23×C4—44no—
180C23×C4—44no—
181C23×C4—44no—
182C23×C4#4118yes—
183C23×C4C2414yes—
184C23×C4C2414yes—
185C23×C4C2411yescharacteristic
186C23×C4C2414yes—
187C23×C4C2414yes—
188C23×C4C2412yes—
189C23×C4C2411yescharacteristic
190C23×C4C2411yescharacteristic
191C23×C4C2411yescharacteristic
192C23×C4C2412yes—
193C23×C4C2412yes—
194C23×C4C2411yescharacteristic
195C23×C4C2411yescharacteristic
196C23×C4C2411yescharacteristic
197C4×D4—416no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—432no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—42no—
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⋊C4C2×D414yes—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4C2×D418yes—
264C2×C22⋊C4C2×D418yes—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—24no—
268C2×C42—24no—
269C2×C42—24no—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—24no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42#4114yes—
278#95—28no—
279#95—28no—
280#95—28no—
281#95—28no—
282#95—28no—
283#95—28no—
284#95—24no—
285#95—24no—
286#95—24no—
287#95—24no—
288#95—24no—
289#95—24no—
290#95—28no—
291#95—28no—
292#95—28no—
293#95—24no—
294#95—24no—
295#95—41no—
296#95—41no—
297#95—41no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95C2×D414yes—
306C24—22no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24C22×D414yes—
319C24C2×#4114yes—
320C242−1+414yes—
321C24C2511yescharacteristic
322C2×D4—216no—
323C2×D4—44no—
324C2×D4—44no—
325C2×D4—48no—
326C2×D4—44no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—216no—
366C22×C4—216no—
367C22×C4—216no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—21no—
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—21no—
398C22×C4—21no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—48no—
402C22×C4—48no—
403C22×C4—48no—
404C22×C4—48no—
405C22×C4—416no—
406C22×C4—88no—
407C22×C4—88no—
408C22×C4—88no—
409C22×C4—88no—
410C22×C4—88no—
411C22×C4—88no—
412C22×C4C22×D412yes—
413C22×C4C22×D412yes—
414C22×C4C22×D414yes—
415C22×C4C22×D412yes—
416C22×C4C2×#4112yes—
417C22×C4C2×#4114yes—
418C22×C4C2×#4112yes—
419C22×C4C2×#4114yes—
420C22×C4C2×#4112yes—
421C22×C4H(5,𝔽2)14yes—
422C22×C4H(5,𝔽2)12yes—
423C22×C4H(5,𝔽2)14yes—
424C22×C42−1+412yes—
425C22×C42−1+412yes—
426C22×C42−1+412yes—
427C4⋊C4—44no—
428C4⋊C4—44no—
429C4⋊C4—44no—
430C4⋊C4—416no—
431C4⋊C4—48no—
432C4⋊C4—48no—
433C4⋊C4—48no—
434C4⋊C4—48no—
435C4⋊C4—84no—
436C4⋊C4—84no—
437C4⋊C4—88no—
438C4⋊C4—88no—
439C4⋊C4—88no—
440C4⋊C4—84no—
441C4⋊C4—84no—
442C22⋊C4—216no—
443C22⋊C4—216no—
444C22⋊C4—216no—
445C22⋊C4—216no—
446C22⋊C4—48no—
447C22⋊C4—416no—
448C22⋊C4—88no—
449C22⋊C4—88no—
450C42—28no—
451C42—48no—
452C42—44no—
453C42—44no—
454C42—48no—
455C23—24no—
456C23—24no—
457C23—24no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—24no—
462C23—24no—
463C23—24no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23#54718yes—
490C23C23×D411yescharacteristic
491C23C22×#4112yes—
492C23C22×#4111yescharacteristic
493C23C2×H(5,𝔽2)11yescharacteristic
494C23C2×H(5,𝔽2)12yes—
495C23C2×H(5,𝔽2)12yes—
496C23C2×2−1+411yescharacteristic
497C23C2×2−1+412yes—
498C23C2×2−1+411yescharacteristic
499C23C2×2−1+411yescharacteristic
500C23#58511yescharacteristic
501D4—416no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—42no—
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—42no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—432no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—41no—
555C2×C4—44no—
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—42no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—41no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—41no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—832no—
584C2×C4#53914yes—
585C2×C4#53918yes—
586C2×C4#54714yes—
587C2×C4#55114yes—
588C2×C4#55318yes—
589C2×C4H(3,𝔽4)14yes—
590C22—28no—
591C22—28no—
592C22—28no—
593C22—44no—
594C22—44no—
595C22—44no—
596C22—44no—
597C22—44no—
598C22—44no—
599C22—44no—
600C22—44no—
601C22—44no—
602C22—44no—
603C22—44no—
604C22—44no—
605C22C2×#53911yescharacteristic
606C22C2×#53912yes—
607C22C2×#54712yes—
608C22C2×#54711yescharacteristic
609C22C2×#55111yescharacteristic
610C22C2×#55312yes—
611C22#349212yes—
612C22#349212yes—
613C22#349512yes—
614C22#349611yescharacteristic
615C22#349812yes—
616C22#349911yescharacteristic
617C22#350711yescharacteristic
618C22#351112yes—
619C22C2×H(3,𝔽4)11yescharacteristic
620C22#353811yescharacteristic
621C22#353811yescharacteristic
622C22#354911yescharacteristic
623C22#355212yes—
624C22#355212yes—
625C22#355212yes—
626C22#355812yes—
627C22#356311yescharacteristic
628C4—24no—
629C4—42no—
630C4—42no—
631C4—42no—
632C4—44no—
633C4—42no—
634C4—44no—
635C4—44no—
636C4—81no—
637C4—84no—
638C4—84no—
639C4—84no—
640C4—84no—
641C4#278218yescomplemented
642C2—48no—
643C2#3934812yes—
644C2#3937611yescharacteristic
645C2#4250211yescharacteristic
646C2#4292112yes—
647C2#4491411yescharacteristic
648C2#4512012yes—
649C2#4599611yescharacteristic
650C2C2×#278211yescharacteristic
651C2#5303211yescharacteristic
652C2#5919811yescharacteristic
653C2#5930412yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17222C2semidirect
3. #16689C2semidirect
4. #16307C2semidirect
6. #15357C2semidirect
7. #15153C2semidirect
8. #15152C2semidirect
9. #15132C2semidirect
10. #13994C2semidirect
11. #13911C2semidirect
12. #13880C2semidirect
13. #13880C2semidirect
641. C4#2782semidirect

Maximal subgroups

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

subgroupindex
C2×#27822
#172222
#166892
#163072
#159662
#153572
#151532
#151522
#151322
#139942
#139112
#138802
#138722
C4×#3942