Finite Group Databasebeta

← #4453091

Subgroups — #4453091

Order 512.

2296 conjugacy classes of subgroups.

The subgroup lattice has 633 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2640C211yesmaximal, characteristic
2C2×#2399C211yesmaximal, characteristic, complemented
3#16787C214yesmaximal, complemented
4#16758C214yesmaximal, complemented
5#16378C214yesmaximal, complemented
6#16375C214yesmaximal, complemented
7#15201C212yesmaximal, complemented
8#15192C212yesmaximal, complemented
9#15157C212yesmaximal
10#15147C212yesmaximal
11#14161C211yesmaximal, characteristic, complemented
12#14155C211yesmaximal, characteristic, complemented
13#14153C211yesmaximal, characteristic
14#14153C211yesmaximal, characteristic
15#13797C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×#124C2211yescharacteristic
18C22×C4×Q8C2212yes—
19#2640—28no—
20#2399—28no—
21C2×#398C2212yes—
22C2×#398C2212yes—
23C2×#397C2214yes—
24C2×#397C2214yes—
25C2×#395C2212yes—
26C2×#395C2212yes—
27C2×#393C2214yes—
28C2×#393C2214yes—
29C2×#391C2211yescharacteristic
30C2×#391C2211yescharacteristic
31C2×#391C2212yes—
32C2×#391C2212yes—
33C2×#391C2214yes—
34C2×#390C2211yescharacteristic
35C2×#390C2211yescharacteristic
36C2×#390C2212yes—
37C2×#390C2212yes—
38C2×C4×C4⋊C4C2212yes—
39C2×C4×C4⋊C4C2211yescharacteristic
40C2×C4×C4⋊C4C2211yescharacteristic
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×#376C2211yescharacteristic
44#1446C2212yes—
45#1446C2212yes—
46#1446C2214yes—
47#1445C2212yes—
48#1445C2212yes—
49#1445C2214yes—
50#1443C2214yes—
51#1443C2214yes—
52#1442C2214yes—
53#1442C2214yes—
54#1442C2214yes—
55#1442C2214yes—
56#1440C2214yes—
57#1440C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1437C2212yes—
65#1437C2212yes—
66#1437C2212yes—
67#1437C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4⋊#95C2214yes—
75#1434C2214yes—
76#1434C2214yes—
77C23×Q8C2311yescharacteristic
78C23×Q8C2312yes—
79C23×Q8C2312yes—
80C23×D4C2311yescharacteristic
81C2×C4⋊Q8—24no—
82C2×C4⋊Q8—24no—
83C2×C4⋊Q8—24no—
84C2×#124—24no—
85C2×#124—24no—
86C2×#124—24no—
87C2×C4×Q8—28no—
88C2×C4×Q8—44no—
89C2×C4×Q8—44no—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C42C2312yes—
111C22×C42C2312yes—
112C22×C42C2311yescharacteristic
113C22×C42C2311yescharacteristic
114C22×C42C2312yes—
115C22×C42C2312yes—
116C22×C42C2311yescharacteristic
117#398—28no—
118#398—28no—
119#397—216no—
120#397—48no—
121#395—28no—
122#395—28no—
123#393—216no—
124#393—48no—
125#391—24no—
126#391—24no—
127#391—48no—
128#391—44no—
129#391—44no—
130#390—24no—
131#390—24no—
132#390—44no—
133#390—44no—
134C4×C4⋊C4—28no—
135C4×C4⋊C4—24no—
136C4×C4⋊C4—24no—
137C4×C4⋊C4—24no—
138C4×C4⋊C4—24no—
139#376D418yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2311yescharacteristic
144C2×#95C2311yescharacteristic
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2311yescharacteristic
148C2×#95C2311yescharacteristic
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C25C2412yes—
169C22×Q8—24no—
170C22×Q8—22no—
171C22×Q8—24no—
172C22×Q8—22no—
173C22×Q8—22no—
174C22×Q8—24no—
175C22×Q8—24no—
176C22×Q8—24no—
177C22×Q8—24no—
178C22×Q8—24no—
179C22×Q8—24no—
180C22×Q8—42no—
181C22×Q8—42no—
182C22×Q8—42no—
183C22×Q8—42no—
184C22×Q8—44no—
185C22×Q8—44no—
186C22×D4—22no—
187C22×D4—22no—
188C22×D4—22no—
189C22×D4—24no—
190C22×D4—24no—
191C23×C4C2412yes—
192C23×C4C2412yes—
193C23×C4C2412yes—
194C23×C4C2412yes—
195C23×C4C2414yes—
196C23×C4C2414yes—
197C23×C4C2412yes—
198C23×C4C2412yes—
199C23×C4C2412yes—
200C23×C4C2412yes—
201C23×C4C2412yes—
202C23×C4C2411yescharacteristic
203C23×C4C2411yescharacteristic
204C23×C4C2411yescharacteristic
205C4⋊Q8—416no—
206#124—416no—
207C4×Q8—432no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
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—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⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C42—24no—
269C2×C42—24no—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—24no—
273C2×C42—24no—
274C2×C42—24no—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42—24no—
278C2×C42—44no—
279C2×C42—44no—
280C2×C42C2×D414yes—
281C2×C42C2×D414yes—
282C2×C42C2×D414yes—
283C2×C42#4114yes—
284C2×C42#4114yes—
285#95—24no—
286#95—22no—
287#95—22no—
288#95—24no—
289#95—22no—
290#95—22no—
291#95—28no—
292#95—28no—
293#95—24no—
294#95—24no—
295#95—28no—
296#95—28no—
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—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95C2×D418yes—
312#95C2×D418yes—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—24no—
317C24—24no—
318C24—24no—
319C24—24no—
320C24—24no—
321C24—24no—
322C24C2511yescharacteristic
323C2×Q8—48no—
324C2×Q8—48no—
325C2×Q8—48no—
326C2×Q8—48no—
327C2×Q8—48no—
328C2×Q8—48no—
329C2×Q8—416no—
330C2×Q8—48no—
331C2×Q8—44no—
332C2×Q8—48no—
333C2×Q8—44no—
334C2×Q8—44no—
335C2×Q8—48no—
336C2×Q8—48no—
337C2×Q8—88no—
338C2×Q8—84no—
339C2×Q8—84no—
340C2×D4—44no—
341C2×D4—44no—
342C2×D4—44no—
343C2×D4—48no—
344C2×D4—48no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4C22×D414yes—
417C22×C4C22×D414yes—
418C22×C4C22×D412yes—
419C22×C4C22×D412yes—
420C22×C4C22×D412yes—
421C22×C4C2×#4112yes—
422C22×C4C2×#4112yes—
423C22×C4C2×#4112yes—
424C22×C4C2×#4112yes—
425C22×C4C2×#4112yes—
426C22×C4C2×#4112yes—
427C22×C4H(5,𝔽2)14yes—
428C22×C4H(5,𝔽2)14yes—
429C22×C42−1+414yes—
430C22×C42−1+414yes—
431C4⋊C4—48no—
432C4⋊C4—48no—
433C4⋊C4—48no—
434C4⋊C4—48no—
435C4⋊C4—48no—
436C4⋊C4—48no—
437C4⋊C4—48no—
438C4⋊C4—48no—
439C4⋊C4—44no—
440C4⋊C4—44no—
441C4⋊C4—48no—
442C4⋊C4—48no—
443C22⋊C4—48no—
444C22⋊C4—48no—
445C22⋊C4—88no—
446C22⋊C4—88no—
447C22⋊C4—88no—
448C22⋊C4—88no—
449C22⋊C4—88no—
450C22⋊C4—88no—
451C42—28no—
452C42—28no—
453C42—48no—
454C42—48no—
455C42—84no—
456C42—84no—
457C42#124116yes—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—44no—
462C23—44no—
463C23—44no—
464C23—44no—
465C23—44no—
466C23—44no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—44no—
472C23—44no—
473C23—44no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23C23×D411yescharacteristic
480C23C22×#4111yescharacteristic
481C23C22×#4111yescharacteristic
482C23C2×H(5,𝔽2)12yes—
483C23C2×H(5,𝔽2)12yes—
484C23C2×2−1+412yes—
485C23C2×2−1+412yes—
486C23#58512yes—
487C23#58512yes—
488Q8—432no—
489Q8—816no—
490Q8—88no—
491D4—88no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—42no—
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—44no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4C2×#12414yes—
572C2×C4C2×#12414yes—
573C2×C4C2×#12414yes—
574C2×C4#53514yes—
575C2×C4#53614yes—
576C2×C4#54218yes—
577C2×C4#54218yes—
578C22—84no—
579C22—84no—
580C22—82no—
581C22—82no—
582C22—84no—
583C22—84no—
584C22—84no—
585C22—84no—
586C22—82no—
587C22C22×#12411yescharacteristic
588C22C2×#53511yescharacteristic
589C22C2×#53611yescharacteristic
590C22C2×#54212yes—
591C22C2×#54212yes—
592C22#348812yes—
593C22#348912yes—
594C22#350612yes—
595C22#350712yes—
596C22#350811yescharacteristic
597C22#350911yescharacteristic
598C22#351011yescharacteristic
599C22#351011yescharacteristic
600C22#353612yes—
601C22#354012yes—
602C22#354112yes—
603C22#354212yes—
604C22#355012yes—
605C22#355412yes—
606C22#355512yes—
607C22#355712yes—
608C4—24no—
609C4—24no—
610C4—44no—
611C4—44no—
612C4—44no—
613C4—44no—
614C4—84no—
615C4—82no—
616C4—82no—
617C4—82no—
618C4—84no—
619C4—82no—
620C4—82no—
621C4#264018yes—
622C2—162no—
623C2C2×#264011yescharacteristic
624C2#3555611yescharacteristic
625C2#3562811yescharacteristic
626C2#3727112yes—
627C2#3727512yes—
628C2#4252312yes—
629C2#4252612yes—
630C2#4269912yes—
631C2#4270012yes—
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. C2×#2399C2semidirect
3. #16787C2semidirect
4. #16758C2semidirect
5. #16378C2semidirect
6. #16375C2semidirect
7. #15201C2semidirect
8. #15192C2semidirect
11. #14161C2semidirect
12. #14155C2semidirect
15. #13797C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26402
C2×#23992
#167872
#167582
#163782
#163752
#152012
#151922
#151572
#151472
#141612
#141552
#141532
#137972