Finite Group Databasebeta

← #628166

Subgroups — #628166

Order 512.

2986 conjugacy classes of subgroups.

The subgroup lattice has 661 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2329C211yesmaximal, characteristic
2C22×#391C211yesmaximal, characteristic
3#16041C214yesmaximal, complemented
4#15382C214yesmaximal, complemented
5#15360C214yesmaximal, complemented
6#15332C214yesmaximal, complemented
7#15026C212yesmaximal
8#14874C212yesmaximal
9#14702C211yesmaximal, characteristic
10#14697C211yesmaximal, characteristic
11#14347C211yesmaximal, characteristic
12#14281C211yesmaximal, characteristic
13#14275C211yesmaximal, characteristic
14#14040C212yesmaximal
15#14039C212yesmaximal
16C24×Q8C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C42C2211yescharacteristic
20C2×#395C2214yes—
21C2×#395C2214yes—
22#2329—28no—
23C2×#391—28no—
24C2×#391C2214yes—
25C2×#391C2214yes—
26C2×#391C2212yes—
27C2×#391C2212yes—
28C2×#389C2212yes—
29C2×#389C2212yes—
30C2×#387C2212yes—
31C2×#387C2212yes—
32C2×#387C2214yes—
33C2×#385C2212yes—
34C2×#385C2212yes—
35C2×#385C2212yes—
36C2×#380C2211yescharacteristic
37C2×#380C2211yescharacteristic
38C2×#380C2211yescharacteristic
39C2×C4×C22⋊C4C2212yes—
40C22×#95C2211yescharacteristic
41C22×#95C2211yescharacteristic
42C22×#95C2211yescharacteristic
43C22×#95C2211yescharacteristic
44#1446C2214yes—
45#1446C2214yes—
46#1446C2212yes—
47#1445C2214yes—
48#1445C2212yes—
49#1444C2212yes—
50#1443C2214yes—
51#1443C2214yes—
52#1443C2214yes—
53#1442C2214yes—
54#1442C2214yes—
55#1442C2214yes—
56#1442C2214yes—
57#1442C2214yes—
58#1442C2212yes—
59#1441C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68#1434C2214yes—
69C22⋊#95C2212yes—
70C22⋊#95C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C23×Q8—24no—
78C23×Q8—24no—
79C23×Q8—24no—
80C23×Q8—44no—
81C23×Q8—44no—
82C23×Q8C2314yes—
83C23×Q8C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C2×C22⋊Q8—24no—
92C2×C22⋊Q8—24no—
93C2×C22⋊Q8Q818yescomplemented
94C22×C4⋊C4—24no—
95C22×C4⋊C4—44no—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C42—24no—
112C22×C42—44no—
113C22×C42C2312yes—
114C22×C42C2312yes—
115C22×C42C2312yes—
116C22×C42C2312yes—
117#395—48no—
118#395—48no—
119#391—28no—
120#391—216no—
121#391—216no—
122#391—44no—
123#389—28no—
124#389—44no—
125#387—28no—
126#387—28no—
127#387—48no—
128#385—28no—
129#385—28no—
130#385—28no—
131#380—24no—
132#380—24no—
133#380—24no—
134C4×C22⋊C4—28no—
135C2×#95—24no—
136C2×#95—24no—
137C2×#95—24no—
138C2×#95D418yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C25C2411yescharacteristic
171C22×Q8—232no—
172C22×Q8—28no—
173C22×Q8—22no—
174C22×Q8—28no—
175C22×Q8—28no—
176C22×Q8—22no—
177C22×Q8—28no—
178C22×Q8—28no—
179C22×Q8—44no—
180C22×Q8—44no—
181C22×Q8—42no—
182C22×Q8—42no—
183C22×Q8—416no—
184C22×Q8—416no—
185C22×Q8—432no—
186C22×Q8—432no—
187C22×Q8C2×Q814yescomplemented
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—24no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4—42no—
209C23×C4—42no—
210C23×C4—42no—
211C23×C4—42no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—42no—
216C23×C4C2×D414yes—
217C23×C4C2×D414yes—
218C23×C4C2×D414yes—
219C23×C4C2×Q814yes—
220C23×C4#4114yes—
221C23×C4#4114yes—
222C23×C4#4114yes—
223C23×C4C2412yes—
224C23×C4C2414yes—
225C23×C4C2414yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C23×C4C2412yes—
237C22⋊Q8—232no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—416no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4C2×Q814yescomplemented
267C2×C4⋊C4C2×Q818yescomplemented
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4C2×Q818yes—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—42no—
287C2×C42—42no—
288C2×C42—416no—
289C2×C42—44no—
290C2×C42—44no—
291#95—28no—
292#95—28no—
293#95—24no—
294#95—24no—
295#95—24no—
296#95—24no—
297#95—24no—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—28no—
303#95—28no—
304#95—28no—
305#95—28no—
306#95—28no—
307#95—28no—
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#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95C2×D418yes—
321#95C2×D418yes—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24C22×D412yes—
327C24C22×Q812yes—
328C24C2×#4112yes—
329C24C2×#4112yes—
330C24C2×#4112yes—
331C242−1+412yes—
332C242−1+412yes—
333C24C2511yescharacteristic
334C2×Q8—232no—
335C2×Q8—28no—
336C2×Q8—48no—
337C2×Q8—416no—
338C2×Q8—416no—
339C2×Q8—416no—
340C2×Q8—416no—
341C2×Q8—4128no—
342C2×Q8—416no—
343C2×Q8—416no—
344C2×Q8—82no—
345C2×Q8—82no—
346C2×Q8—84no—
347C22×C4—28no—
348C22×C4—28no—
349C22×C4—216no—
350C22×C4—28no—
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—216no—
360C22×C4—28no—
361C22×C4—28no—
362C22×C4—28no—
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—24no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—28no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4—44no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—48no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—84no—
441C22×C4—84no—
442C22×C4—84no—
443C22×C4—84no—
444C22×C4C22⋊Q8116yes—
445C22×C4C22×D414yes—
446C22×C4C22×D414yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×Q814yes—
450C22×C4C22×Q814yes—
451C22×C4C22×Q812yes—
452C22×C4C22×Q812yes—
453C22×C4C2×#4114yes—
454C22×C4C2×#4112yes—
455C22×C4C2×#4112yes—
456C22×C4C2×#4114yes—
457C22×C4H(5,𝔽2)14yes—
458C22×C4H(5,𝔽2)14yes—
459C22×C42−1+414yes—
460C4⋊C4—28no—
461C4⋊C4—216no—
462C4⋊C4—416no—
463C4⋊C4—48no—
464C4⋊C4—416no—
465C4⋊C4—82no—
466C4⋊C4—82no—
467C4⋊C4—84no—
468C4⋊C4—84no—
469C4⋊C4—84no—
470C4⋊C4—88no—
471C22⋊C4—216no—
472C22⋊C4—48no—
473C22⋊C4—48no—
474C22⋊C4—48no—
475C42—48no—
476C42—48no—
477C42—84no—
478C42—84no—
479C23—24no—
480C23—24no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—24no—
495C23C2×C22⋊Q814yes—
496C23#52514yes—
497C23#53614yes—
498C23#55614yes—
499C23#55614yes—
500C23#55714yes—
501C23#55714yes—
502C23C23×D411yescharacteristic
503C23C23×Q811yescharacteristic
504C23C22×#4111yescharacteristic
505C23C22×#4111yescharacteristic
506C23C22×#4111yescharacteristic
507C23C2×H(5,𝔽2)12yes—
508C23C2×H(5,𝔽2)12yes—
509C23C2×2−1+411yescharacteristic
510C23C2×2−1+411yescharacteristic
511C23C2×2−1+412yes—
512C23#58512yes—
513Q8—464no—
514Q8—88no—
515C2×C4—232no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—28no—
541C2×C4—28no—
542C2×C4—28no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—416no—
546C2×C4—416no—
547C2×C4—416no—
548C2×C4—416no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—88no—
596C2×C4—88no—
597C2×C4C2×C22⋊Q814yes—
598C2×C4C2×C22⋊Q814yes—
599C2×C4#53818yes—
600C22—24no—
601C22—24no—
602C22—28no—
603C22—24no—
604C22—24no—
605C22—28no—
606C22—24no—
607C22—28no—
608C22—24no—
609C22—28no—
610C22#258118yes—
611C22C22×C22⋊Q811yescharacteristic
612C22C2×#52511yescharacteristic
613C22C2×#53611yescharacteristic
614C22C2×#53812yes—
615C22#346112yes—
616C22C2×#55611yescharacteristic
617C22C2×#55611yescharacteristic
618C22C2×#55711yescharacteristic
619C22C2×#55711yescharacteristic
620C22#348411yescharacteristic
621C22#348411yescharacteristic
622C22#348511yescharacteristic
623C22#348611yescharacteristic
624C22#349012yes—
625C22#350112yes—
626C22#350312yes—
627C22#350512yes—
628C22#350712yes—
629C22#351112yes—
630C22#353812yes—
631C22#353812yes—
632C22#356112yes—
633C22#356112yes—
634C4—24no—
635C4—24no—
636C4—28no—
637C4—44no—
638C4—44no—
639C4—44no—
640C4—44no—
641C4—44no—
642C4—44no—
643C4—82no—
644C4—84no—
645C4—82no—
646C4—84no—
647C4—82no—
648C2—28no—
649C2C2×#258111yescharacteristic
650C2#3498811yescharacteristic
651C2#3509211yescharacteristic
652C2#3534911yescharacteristic
653C2#3557411yescharacteristic
654C2#3670811yescharacteristic
655C2#3696611yescharacteristic
656C2#3964012yes—
657C2#3967612yes—
658C2#3976012yes—
659C2#3979612yes—
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. #16041C2semidirect
4. #15382C2semidirect
5. #15360C2semidirect
6. #15332C2semidirect
93. C2×C22⋊Q8Q8semidirect
187. C22×Q8C2×Q8semidirect
266. C2×C4⋊C4C2×Q8semidirect
267. C2×C4⋊C4C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#23292
C22×#3912
#160412
#153822
#153602
#153322
#150262
#148742
#147022
#146972
#143472
#142812
#142752
#140402
#140392