Finite Group Databasebeta

← #627732

Subgroups — #627732

Order 512.

2582 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic
2C2×#2851C211yesmaximal, characteristic
3C2×#2642C211yesmaximal, characteristic, complemented
4C2×#2330C211yesmaximal, characteristic, complemented
5#15895C214yesmaximal, complemented
6#15888C214yesmaximal, complemented
7#15679C214yesmaximal
8#15525C214yesmaximal
9#15189C212yesmaximal, complemented
10#15179C212yesmaximal
11#14276C211yesmaximal, characteristic, complemented
12#14274C211yesmaximal, characteristic, complemented
13#14142C212yesmaximal, complemented
14#13997C212yesmaximal
15#13753C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×#125C2211yescharacteristic
18C22×C22⋊Q8C2212yes—
19C23×C42C2211yescharacteristic
20#2857—28no—
21#2851—28no—
22#2642—28no—
23C2×#400C2214yes—
24C2×#400C2214yes—
25#2330—28no—
26C2×#391C2212yes—
27C2×#391C2212yes—
28C2×#391C2212yes—
29C2×#391C2212yes—
30C2×#391C2212yes—
31C2×#389C2212yes—
32C2×#389C2214yes—
33C2×#389C2214yes—
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#387C2214yes—
37C2×#387C2214yes—
38C2×#385C2214yes—
39C2×#384C2211yescharacteristic
40C2×#382C2211yescharacteristic
41C2×#380C2212yes—
42C2×#380C2214yes—
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C22⋊C4C2211yescharacteristic
45C2×C4×C22⋊C4C2211yescharacteristic
46C2×#376C2211yescharacteristic
47C2×#376C2211yescharacteristic
48#1445C2214yes—
49#1445C2214yes—
50#1445C2214yes—
51#1443C2214yes—
52#1443C2214yes—
53#1442C2214yes—
54#1442C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1440C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1437C2214yes—
65#1436C2214yes—
66C4⋊#95C2214yes—
67C4⋊#95C2212yes—
68#1434C2212yes—
69#1434C2212yes—
70#1434C2214yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C4×#95C2211yescharacteristic
74C4×#95C2211yescharacteristic
75C23×Q8C2312yes—
76C24×C4C2311yescharacteristic
77C24×C4C2312yes—
78C2×C4⋊Q8—24no—
79C2×C4⋊Q8—24no—
80C2×C4⋊Q8—24no—
81C2×#125—24no—
82C2×#125—24no—
83C2×#125D418yes—
84C2×C22⋊Q8—28no—
85C2×C22⋊Q8—28no—
86C2×C22⋊Q8—44no—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2314yes—
97C22×C22⋊C4C2311yescharacteristic
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2311yescharacteristic
102C22×C42—24no—
103C22×C42—24no—
104C22×C42—24no—
105C22×C42C2311yescharacteristic
106C22×C42C2312yes—
107C22×C42C2311yescharacteristic
108C22×C42C2311yescharacteristic
109C22×C42C2311yescharacteristic
110C22×C42C2311yescharacteristic
111C22×C42C2314yes—
112C22×C42C2311yescharacteristic
113#400—216no—
114#400—216no—
115#391—28no—
116#391—28no—
117#391—28no—
118#391—28no—
119#391—44no—
120#389—216no—
121#389—28no—
122#389—28no—
123#389—216no—
124#389—44no—
125#387—216no—
126#387—216no—
127#385—216no—
128#384D418yes—
129#382—24no—
130#380—28no—
131#380—216no—
132C4×C4⋊C4—28no—
133C4×C22⋊C4—24no—
134C4×C22⋊C4—24no—
135#376—24no—
136#376—24no—
137C2×#95C2312yes—
138C2×#95C2312yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2311yescharacteristic
149C2×#95C2311yescharacteristic
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C25C2411yescharacteristic
167C22×Q8—24no—
168C22×Q8—24no—
169C22×Q8—24no—
170C22×Q8—24no—
171C22×Q8—24no—
172C22×Q8—24no—
173C22×Q8—24no—
174C23×C4—24no—
175C23×C4—22no—
176C23×C4—24no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—24no—
182C23×C4—22no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—42no—
186C23×C4—42no—
187C23×C4#4114yes—
188C23×C4C2411yescharacteristic
189C23×C4C2414yes—
190C23×C4C2414yes—
191C23×C4C2412yes—
192C23×C4C2411yescharacteristic
193C23×C4C2411yescharacteristic
194C23×C4C2411yescharacteristic
195C23×C4C2414yes—
196C23×C4C2414yes—
197C23×C4C2412yes—
198C23×C4C2411yescharacteristic
199C23×C4C2412yes—
200C23×C4C2412yes—
201C23×C4C2411yescharacteristic
202C4⋊Q8—416no—
203#125—232no—
204C22⋊Q8—432no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4C2×D418yes—
232C2×C4⋊C4C2×D418yes—
233C2×C4⋊C4C2×D4116yes—
234C2×C4⋊C4C2×D4116yes—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—44no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—24no—
254C2×C42—24no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—24no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—28no—
266C2×C42—416no—
267C2×C42—44no—
268C2×C42—44no—
269C2×C42C2×D414yes—
270C2×C42C2×D414yes—
271C2×C42#4114yes—
272C2×C42#4114yes—
273#95—24no—
274#95—24no—
275#95—24no—
276#95—24no—
277#95—28no—
278#95—28no—
279#95—28no—
280#95—28no—
281#95—28no—
282#95—28no—
283#95—28no—
284#95—28no—
285#95—28no—
286#95—28no—
287#95—28no—
288#95—24no—
289#95—22no—
290#95—22no—
291#95—28no—
292#95—28no—
293#95—24no—
294#95—24no—
295#95—42no—
296#95—42no—
297#95—44no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—44no—
302C24—21no—
303C24—21no—
304C24—21no—
305C24—21no—
306C24—21no—
307C24—21no—
308C24—22no—
309C24—22no—
310C24—22no—
311C24C2×#4112yes—
312C242−1+414yes—
313C24C2511yescharacteristic
314C2×Q8—48no—
315C2×Q8—48no—
316C2×Q8—48no—
317C2×Q8—48no—
318C2×Q8—48no—
319C2×Q8—48no—
320C2×Q8—48no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—216no—
326C22×C4—22no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—28no—
357C22×C4—28no—
358C22×C4—28no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—21no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—48no—
401C22×C4—48no—
402C22×C4—48no—
403C22×C4—48no—
404C22×C4—48no—
405C22×C4—48no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4C22×D418yes—
411C22×C4C22×D418yes—
412C22×C4C22×D414yes—
413C22×C4C22×D414yes—
414C22×C4C22×D412yes—
415C22×C4C22×D412yes—
416C22×C4C2×#4114yes—
417C22×C4C2×#4112yes—
418C22×C4C2×#4114yes—
419C22×C4C2×#4112yes—
420C22×C4C2×#4112yes—
421C22×C4C2×#4112yes—
422C22×C4H(5,𝔽2)12yes—
423C22×C42−1+414yes—
424C22×C42−1+412yes—
425C4⋊C4—232no—
426C4⋊C4—216no—
427C4⋊C4—216no—
428C4⋊C4—416no—
429C4⋊C4—416no—
430C4⋊C4—416no—
431C4⋊C4—416no—
432C4⋊C4—48no—
433C4⋊C4—88no—
434C4⋊C4—88no—
435C22⋊C4—416no—
436C22⋊C4—48no—
437C22⋊C4—416no—
438C22⋊C4—44no—
439C22⋊C4—44no—
440C42—28no—
441C42—48no—
442C42—44no—
443C42—44no—
444C42—44no—
445C42—44no—
446C42—88no—
447C42#121116yes—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—24no—
453C23—24no—
454C23—24no—
455C23—24no—
456C23—22no—
457C23—22no—
458C23—24no—
459C23—24no—
460C23—41no—
461C23—41no—
462C23—42no—
463C23—41no—
464C23—41no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23#54114yes—
471C23C23×D411yescharacteristic
472C23C23×D411yescharacteristic
473C23C22×#4111yescharacteristic
474C23C22×#4111yescharacteristic
475C23C22×#4111yescharacteristic
476C23C2×H(5,𝔽2)11yescharacteristic
477C23C2×2−1+412yes—
478C23C2×2−1+411yescharacteristic
479C23C2×2−1+412yes—
480C23#58512yes—
481C23#58512yes—
482Q8—816no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—28no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—28no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—22no—
496C2×C4—28no—
497C2×C4—28no—
498C2×C4—24no—
499C2×C4—28no—
500C2×C4—24no—
501C2×C4—28no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—432no—
505C2×C4—42no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—44no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—416no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—44no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—44no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4C2×#12118yes—
568C2×C4C2×#12114yes—
569C2×C4#52914yes—
570C2×C4#54214yes—
571C2×C4#54614yes—
572C2×C4#54814yes—
573C2×C4#54814yes—
574C2×C4#55014yes—
575C22—24no—
576C22—24no—
577C22—24no—
578C22—44no—
579C22—44no—
580C22—42no—
581C22—42no—
582C22—44no—
583C22—44no—
584C22—42no—
585C22—42no—
586C22C22×#12111yescharacteristic
587C22C2×#52911yescharacteristic
588C22C2×#54111yescharacteristic
589C22C2×#54211yescharacteristic
590C22#345511yescharacteristic
591C22#346112yes—
592C22C2×#54611yescharacteristic
593C22C2×#54811yescharacteristic
594C22C2×#54811yescharacteristic
595C22C2×#55011yescharacteristic
596C22#348512yes—
597C22#348612yes—
598C22#348912yes—
599C22#348912yes—
600C22#349512yes—
601C22#349512yes—
602C22#349512yes—
603C22#350911yescharacteristic
604C22#351211yescharacteristic
605C22#355212yes—
606C22#355512yes—
607C22#356212yes—
608C22#356312yes—
609C4—28no—
610C4—48no—
611C4—44no—
612C4—48no—
613C4—44no—
614C4—42no—
615C4—81no—
616C4—81no—
617C4—81no—
618C4—81no—
619C4—82no—
620C4—84no—
621C4—84no—
622C4#242618yes—
623C2—44no—
624C2C2×#242611yescharacteristic
625C2#3531511yescharacteristic
626C2#3532311yescharacteristic
627C2#3563312yes—
628C2#3738512yes—
629C2#3769511yescharacteristic
630C2#3770011yescharacteristic
631C2#3770511yescharacteristic
632C2#3770911yescharacteristic
633C2#3775712yes—
634C2#3777412yes—
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
3. C2×#2642C2semidirect
4. C2×#2330C2semidirect
5. #15895C2semidirect
6. #15888C2semidirect
9. #15189C2semidirect
11. #14276C2semidirect
12. #14274C2semidirect
13. #14142C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
C2×#28512
C2×#26422
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#158952
#158882
#156792
#155252
#151892
#151792
#142762
#142742
#141422
#139972
#137532