Finite Group Databasebeta

← #2737117

Subgroups — #2737117

Order 512.

2120 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×#2551C211yesmaximal, characteristic, complemented
2C2×#2494C211yesmaximal, characteristic
3#17155C214yesmaximal, complemented
4#17127C214yesmaximal
5#17040C212yesmaximal
6#16602C212yesmaximal
7#16560C212yesmaximal, complemented
8#15242C211yesmaximal, characteristic, complemented
9#15230C212yesmaximal, complemented
10#15223C212yesmaximal
11#15195C211yesmaximal, characteristic
12#15149C211yesmaximal, characteristic, complemented
13#15139C212yesmaximal, complemented
14#15125C212yesmaximal
15#14631C212yesmaximal, complemented
16#13936C211yesmaximal, characteristic, complemented
17C4×#397C211yesmaximal, characteristic
18C22×C4⋊Q8C2212yes—
19C22×C4⋊Q8C2212yes—
20#2551—28no—
21#2494—28no—
22C2×#401C2212yes—
23C2×#400C2214yes—
24C2×#398C2212yes—
25C2×#397C2211yescharacteristic
26C2×#397C2212yes—
27C2×#395C2214yes—
28C2×#393C2214yes—
29C2×#389C2211yescharacteristic
30C2×#389C2212yes—
31C2×#389C2212yes—
32C2×#389C2211yescharacteristic
33C2×#389C2212yes—
34C2×#388C2212yes—
35C2×#387C2212yes—
36C2×#384C2211yescharacteristic
37C2×#382C2211yescharacteristic
38C2×#382C2212yes—
39C2×#380C2211yescharacteristic
40C2×#380C2211yescharacteristic
41C2×#380C2212yes—
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C22⋊C4C2211yescharacteristic
45C2×C4×C22⋊C4C2212yes—
46#1446C2214yes—
47#1446C2214yes—
48#1446C2214yes—
49#1445C2214yes—
50#1445C2214yes—
51#1445C2214yes—
52#1445C2211yescharacteristic
53#1443C2212yes—
54#1443C2214yes—
55#1443C2214yes—
56#1442C2212yes—
57#1442C2214yes—
58#1442C2214yes—
59#1442C2214yes—
60#1442C2214yes—
61#1442C2211yescharacteristic
62#1442C2212yes—
63#1441C2214yes—
64#1441C2214yes—
65#1441C2212yes—
66#1441C2212yes—
67#1441C2212yes—
68#1440C2212yes—
69#1439C2214yes—
70#1439C2211yescharacteristic
71#1439C2212yes—
72#1439C2212yes—
73#1439C2214yes—
74#1439C2214yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2211yescharacteristic
81#1434C2212yes—
82#1434C2212yes—
83C4×#95C2212yes—
84C4×#95C2211yescharacteristic
85C23×Q8C2314yes—
86C24×C4C2311yescharacteristic
87C2×C4⋊Q8—28no—
88C2×C4⋊Q8—28no—
89C2×C4⋊Q8—44no—
90C2×C4⋊Q8—44no—
91C2×C4⋊Q8—48no—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2312yes—
107C22×C22⋊C4C2311yescharacteristic
108C22×C22⋊C4C2311yescharacteristic
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C42C2312yes—
115C22×C42C2312yes—
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2312yes—
119#401—28no—
120#400—48no—
121#398—28no—
122#397—24no—
123#397—28no—
124#395—216no—
125#393—48no—
126#389—24no—
127#389—28no—
128#389—24no—
129#389—28no—
130#389—44no—
131#388—28no—
132#387—28no—
133#384—24no—
134#382—24no—
135#382—44no—
136#380—24no—
137#380—24no—
138#380—44no—
139C4×C4⋊C4—24no—
140C4×C4⋊C4D418yes—
141C4×C22⋊C4—24no—
142C4×C22⋊C4—44no—
143C2×#95C2314yes—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2311yescharacteristic
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2311yescharacteristic
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2311yescharacteristic
159C2×#95C2311yescharacteristic
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C25C2411yescharacteristic
180C22×Q8—28no—
181C22×Q8—28no—
182C22×Q8—44no—
183C22×Q8—44no—
184C22×Q8—44no—
185C22×Q8—44no—
186C22×Q8—44no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4C2414yes—
193C23×C4C2412yes—
194C23×C4C2412yes—
195C23×C4C2414yes—
196C23×C4C2411yescharacteristic
197C23×C4C2411yescharacteristic
198C23×C4C2412yes—
199C23×C4C2412yes—
200C23×C4C2411yescharacteristic
201C23×C4C2412yes—
202C23×C4C2412yes—
203C23×C4C2411yescharacteristic
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208C4⋊Q8—432no—
209C4⋊Q8—432no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4C2×D414yes—
247C2×C4⋊C4C2×D414yes—
248C2×C4⋊C4#4114yes—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—41no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C42—24no—
270C2×C42—24no—
271C2×C42—22no—
272C2×C42—24no—
273C2×C42—24no—
274C2×C42—24no—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42—24no—
278C2×C42—44no—
279C2×C42C2×D414yes—
280C2×C42C2×D418yes—
281#95—22no—
282#95—22no—
283#95—24no—
284#95—24no—
285#95—24no—
286#95—24no—
287#95—24no—
288#95—24no—
289#95—28no—
290#95—24no—
291#95—24no—
292#95—22no—
293#95—28no—
294#95—24no—
295#95—28no—
296#95—24no—
297#95—22no—
298#95—44no—
299#95—42no—
300#95—42no—
301#95—42no—
302#95—42no—
303#95—44no—
304#95—44no—
305#95—42no—
306#95—42no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95C2×D418yes—
317C24—21no—
318C24—21no—
319C24—22no—
320C24—21no—
321C24—21no—
322C24—21no—
323C24—22no—
324C24—22no—
325C24—21no—
326C24—22no—
327C242−1+412yes—
328C24C2511yescharacteristic
329C2×Q8—416no—
330C2×Q8—416no—
331C2×Q8—88no—
332C2×Q8—88no—
333C2×Q8—88no—
334C2×Q8—88no—
335C2×Q8—88no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4C22×D412yes—
423C22×C4C22×D414yes—
424C22×C4C22×D414yes—
425C22×C4C22×D412yes—
426C22×C4C22×D412yes—
427C22×C4C2×#4114yes—
428C22×C4C2×#4112yes—
429C22×C4H(5,𝔽2)14yes—
430C22×C4H(5,𝔽2)12yes—
431C22×C4H(5,𝔽2)12yes—
432C22×C42−1+412yes—
433C22×C42−1+414yes—
434C22×C42−1+414yes—
435C4⋊C4—28no—
436C4⋊C4—28no—
437C4⋊C4—28no—
438C4⋊C4—28no—
439C4⋊C4—416no—
440C4⋊C4—48no—
441C4⋊C4—416no—
442C4⋊C4—44no—
443C4⋊C4—48no—
444C4⋊C4—44no—
445C4⋊C4—48no—
446C4⋊C4—48no—
447C4⋊C4—84no—
448C4⋊C4—84no—
449C4⋊C4—88no—
450C22⋊C4—44no—
451C22⋊C4—48no—
452C22⋊C4—48no—
453C22⋊C4—48no—
454C22⋊C4—44no—
455C22⋊C4—84no—
456C22⋊C4—88no—
457C42—28no—
458C42—48no—
459C42—48no—
460C42—44no—
461C42—48no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—24no—
466C23—24no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—42no—
474C23—41no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—41no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23C23×D411yescharacteristic
486C23C22×#4111yescharacteristic
487C23C2×H(5,𝔽2)11yescharacteristic
488C23C2×H(5,𝔽2)11yescharacteristic
489C23C2×H(5,𝔽2)12yes—
490C23C2×2−1+412yes—
491C23C2×2−1+412yes—
492C23C2×2−1+411yescharacteristic
493C23C2×2−1+411yescharacteristic
494C23#58511yescharacteristic
495C23#58512yes—
496Q8—832no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—44no—
547C2×C4—41no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—88no—
586C2×C4#53514yes—
587C2×C4#53614yes—
588C2×C4#53614yes—
589C2×C4#54814yes—
590C2×C4#56318yes—
591C2×C4#56318yes—
592C22—44no—
593C22—44no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—81no—
598C22—81no—
599C22—82no—
600C22—81no—
601C22—81no—
602C22—82no—
603C22C2×#53511yescharacteristic
604C22C2×#53611yescharacteristic
605C22C2×#53611yescharacteristic
606C22C2×#54811yescharacteristic
607C22#348411yescharacteristic
608C22#348911yescharacteristic
609C22#350011yescharacteristic
610C22#351112yes—
611C22#351411yescharacteristic
612C22#351812yes—
613C22#352011yescharacteristic
614C22C2×#56312yes—
615C22C2×#56312yes—
616C22#354212yes—
617C22#354711yescharacteristic
618C22#354712yes—
619C22#355211yescharacteristic
620C22#355212yes—
621C22#355212yes—
622C22#355512yes—
623C22#355812yes—
624C22#356312yes—
625C22#356512yes—
626C4—24no—
627C4—44no—
628C4—44no—
629C4—44no—
630C4—42no—
631C4—44no—
632C4—81no—
633C4—82no—
634C4—82no—
635C4—82no—
636C4—81no—
637C4—84no—
638C4—82no—
639C4—84no—
640C4—81no—
641C4#282118yes—
642C2—82no—
643C2C2×#282111yescharacteristic
644C2#3912411yescharacteristic
645C2#3917211yescharacteristic
646C2#3972011yescharacteristic
647C2#4242611yescharacteristic
648C2#4248511yescharacteristic
649C2#4297811yescharacteristic
650C2#4562712yes—
651C2#4606212yes—
652C2#5300212yes—
653C2#5304812yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
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×#2551C2semidirect
3. #17155C2semidirect
7. #16560C2semidirect
8. #15242C2semidirect
9. #15230C2semidirect
12. #15149C2semidirect
13. #15139C2semidirect
15. #14631C2semidirect
16. #13936C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25512
C2×#24942
#171552
#171272
#170402
#166022
#165602
#152422
#152302
#152232
#151952
#151492
#151392
#151252
#146312
#139362
C4×#3972