Finite Group Databasebeta

← #892166

Subgroups — #892166

Order 512.

2882 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2804C212yesmaximal, complemented
2#17080C211yesmaximal, characteristic, complemented
3#17068C211yesmaximal, characteristic
4#16652C212yesmaximal, complemented
5#15440C214yesmaximal, complemented
6#15314C214yesmaximal, complemented
7#15004C214yesmaximal, complemented
8#14883C212yesmaximal, complemented
9#14708C212yesmaximal, complemented
10#14535C212yesmaximal, complemented
11#14189C214yesmaximal, complemented
12#14176C212yesmaximal, complemented
13C4×#391C211yesmaximal, characteristic, complemented
14C22×#124C2212yes—
15C22×C4×Q8C2211yescharacteristic
16C22×C4×D4C2212yes—
17#2804—216no—
18C2×#400C2214yes—
19C2×#394C2212yes—
20C2×#394C2212yes—
21C2×#394C2212yes—
22C2×#393C2214yes—
23C2×#393C2214yes—
24C2×#392C2214yes—
25C2×#391C2211yescharacteristic
26C2×#391C2212yes—
27C2×#391C2212yes—
28C2×#391C2211yescharacteristic
29C2×#390C2212yes—
30C2×#390C2212yes—
31C2×#390C2212yes—
32C2×#388C2214yes—
33C2×#386C2214yes—
34C2×#386C2214yes—
35C2×#385C2214yes—
36C2×#385C2214yes—
37C2×#385C2214yes—
38C2×#382C2212yes—
39C2×C4×C4⋊C4C2211yescharacteristic
40C2×C43C2211yescharacteristic
41#1446C2214yes—
42#1444C2212yes—
43#1444C2214yes—
44#1443C2214yes—
45#1442C2214yes—
46#1441C2212yes—
47#1441C2212yes—
48#1441C2212yes—
49#1440C2214yes—
50#1440C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1438C2214yes—
55#1438C2212yes—
56#1437C2214yes—
57#1437C2214yes—
58#1437C2214yes—
59#1437C2214yes—
60#1437C2214yes—
61#1436C2214yes—
62C4⋊#95C2211yescharacteristic
63C4⋊#95C2211yescharacteristic
64#1434C2212yes—
65#1434C2212yes—
66C22⋊#95C2214yes—
67C4×#95C2214yes—
68C23×Q8C2312yes—
69C23×Q8C2311yescharacteristic
70C23×D4C2312yes—
71C23×D4C2312yes—
72C23×D4C2312yes—
73C24×C4C2314yes—
74C2×#124—28no—
75C2×#124—44no—
76C2×#124—44no—
77C2×C4×Q8—24no—
78C2×C4×Q8—42no—
79C2×C4×Q8—42no—
80C2×C4×D4—44no—
81C2×C4×D4—48no—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2311yescharacteristic
91C22×C22⋊C4C2314yes—
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C42C2312yes—
103C22×C42C2311yescharacteristic
104C22×C42C2312yes—
105C22×C42C2312yes—
106C22×C42C2312yes—
107C22×C42C2311yescharacteristic
108C22×C42C2311yescharacteristic
109#400—216no—
110#394—28no—
111#394—28no—
112#394—44no—
113#393—216no—
114#393—216no—
115#392—216no—
116#391—24no—
117#391—28no—
118#391—44no—
119#391D418yescomplemented
120#390—28no—
121#390—44no—
122#390—44no—
123#388—48no—
124#386—216no—
125#386—48no—
126#385—216no—
127#385—216no—
128#385—48no—
129#382—44no—
130C4×C4⋊C4—42no—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2312yes—
137C2×#95C2312yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C43—42no—
152C25C2414yes—
153C22×Q8—24no—
154C22×Q8—22no—
155C22×Q8—22no—
156C22×Q8—24no—
157C22×Q8—24no—
158C22×Q8—41no—
159C22×Q8—41no—
160C22×Q8—41no—
161C22×Q8—41no—
162C22×Q8—42no—
163C22×Q8—42no—
164C22×Q8—42no—
165C22×Q8—42no—
166C22×Q8C2×D414yes—
167C22×D4—28no—
168C22×D4—28no—
169C22×D4—24no—
170C22×D4—24no—
171C22×D4—24no—
172C22×D4—24no—
173C22×D4—44no—
174C22×D4—42no—
175C22×D4—42no—
176C22×D4—42no—
177C22×D4—42no—
178C22×D4—42no—
179C22×D4—44no—
180C22×D4—42no—
181C22×D4—42no—
182C22×D4—42no—
183C22×D4#4118yes—
184C23×C4—28no—
185C23×C4—44no—
186C23×C4—44no—
187C23×C4—44no—
188C23×C4—44no—
189C23×C4—44no—
190C23×C4—44no—
191C23×C4C2414yes—
192C23×C4C2414yes—
193C23×C4C2414yes—
194C23×C4C2412yes—
195C23×C4C2412yes—
196C23×C4C2412yes—
197C23×C4C2412yes—
198C23×C4C2411yescharacteristic
199C23×C4C2412yes—
200C23×C4C2411yescharacteristic
201C23×C4C2412yes—
202C23×C4C2411yescharacteristic
203#124—432no—
204C4×Q8—416no—
205C4×D4—432no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4C2×D414yes—
229C2×C4⋊C4#4114yes—
230C2×C4⋊C4#4114yes—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
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×C42—22no—
265C2×C42—24no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—24no—
269C2×C42—24no—
270C2×C42—24no—
271C2×C42—22no—
272C2×C42—24no—
273C2×C42—28no—
274C2×C42—24no—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42—42no—
278C2×C42—41no—
279C2×C42—42no—
280C2×C42—41no—
281C2×C42—41no—
282C2×C42—41no—
283C2×C42C2×D414yes—
284#95—28no—
285#95—28no—
286#95—28no—
287#95—28no—
288#95—28no—
289#95—28no—
290#95—28no—
291#95—24no—
292#95—24no—
293#95—28no—
294#95—28no—
295#95—28no—
296#95—42no—
297#95—42no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—42no—
303#95C2×D4116yes—
304C24—24no—
305C24—24no—
306C24—24no—
307C24—24no—
308C24—24no—
309C24—24no—
310C24—24no—
311C24—24no—
312C24—24no—
313C24—24no—
314C24—24no—
315C24—24no—
316C24—24no—
317C24—24no—
318C24C2×#4118yes—
319C24C2511yescharacteristic
320C2×Q8—28no—
321C2×Q8—44no—
322C2×Q8—44no—
323C2×Q8—44no—
324C2×Q8—44no—
325C2×Q8—44no—
326C2×Q8—44no—
327C2×Q8—48no—
328C2×Q8—48no—
329C2×Q8—48no—
330C2×Q8—84no—
331C2×Q8—84no—
332C2×Q8—84no—
333C2×Q8—84no—
334C2×D4—216no—
335C2×D4—416no—
336C2×D4—48no—
337C2×D4—48no—
338C2×D4—48no—
339C2×D4—416no—
340C2×D4—48no—
341C2×D4—416no—
342C2×D4—48no—
343C2×D4—48no—
344C2×D4—84no—
345C2×D4—84no—
346C2×D4—84no—
347C2×D4—84no—
348C2×D4—84no—
349C2×D4—88no—
350C2×D4—84no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
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—416no—
412C22×C4—88no—
413C22×C4—88no—
414C22×C4—88no—
415C22×C4—88no—
416C22×C4—88no—
417C22×C4—88no—
418C22×C4C22×D414yes—
419C22×C4C22×D414yes—
420C22×C4C22×D412yes—
421C22×C4C22×D414yes—
422C22×C4C2×#4114yes—
423C22×C4C2×#4114yes—
424C22×C4C2×#4112yes—
425C22×C4C2×#4112yes—
426C22×C4C2×#4114yes—
427C22×C4H(5,𝔽2)14yes—
428C22×C4H(5,𝔽2)12yes—
429C22×C4H(5,𝔽2)14yes—
430C22×C4H(5,𝔽2)12yes—
431C22×C42−1+414yes—
432C4⋊C4—44no—
433C4⋊C4—44no—
434C4⋊C4—44no—
435C4⋊C4—48no—
436C4⋊C4—48no—
437C4⋊C4—84no—
438C4⋊C4—84no—
439C4⋊C4—88no—
440C4⋊C4—84no—
441C22⋊C4—416no—
442C22⋊C4—416no—
443C22⋊C4—416no—
444C22⋊C4—416no—
445C22⋊C4—416no—
446C22⋊C4—416no—
447C22⋊C4—416no—
448C22⋊C4—88no—
449C22⋊C4—88no—
450C22⋊C4—88no—
451C22⋊C4—88no—
452C42—28no—
453C42—216no—
454C42—44no—
455C42—48no—
456C42—48no—
457C42—44no—
458C42—48no—
459C23—28no—
460C23—28no—
461C23—28no—
462C23—28no—
463C23—28no—
464C23—28no—
465C23—28no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—44no—
473C23—44no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23C23×D411yescharacteristic
495C23C22×#4111yescharacteristic
496C23C22×#4111yescharacteristic
497C23C22×#4112yes—
498C23C2×H(5,𝔽2)11yescharacteristic
499C23C2×H(5,𝔽2)12yes—
500C23C2×H(5,𝔽2)11yescharacteristic
501C23C2×H(5,𝔽2)12yes—
502C23C2×2−1+412yes—
503C23#58511yescharacteristic
504C23#58511yescharacteristic
505Q8—416no—
506Q8—816no—
507D4—432no—
508D4—816no—
509D4—816no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—28no—
529C2×C4—28no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—28no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—42no—
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—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—832no—
582C2×C4#53214yes—
583C2×C4#54818yes—
584C2×C4H(3,𝔽4)14yes—
585C22—48no—
586C22—48no—
587C22—48no—
588C22—48no—
589C22—48no—
590C22—48no—
591C22—48no—
592C22—84no—
593C22—84no—
594C22—84no—
595C22—84no—
596C22—84no—
597C22—84no—
598C22—84no—
599C22—84no—
600C22C2×#53211yescharacteristic
601C22#345912yes—
602C22#345911yescharacteristic
603C22#346011yescharacteristic
604C22C2×#54812yes—
605C22#348111yescharacteristic
606C22#348112yes—
607C22#348112yes—
608C22#349011yescharacteristic
609C22#349012yes—
610C22#349311yescharacteristic
611C22#349611yescharacteristic
612C22#349611yescharacteristic
613C22#349812yes—
614C22#351512yes—
615C22#352212yes—
616C22#352212yes—
617C22C2×H(3,𝔽4)11yescharacteristic
618C22#353612yes—
619C22#354411yescharacteristic
620C22#354911yescharacteristic
621C22#355812yes—
622C22#356112yes—
623C4—24no—
624C4—24no—
625C4—28no—
626C4—44no—
627C4—44no—
628C4—42no—
629C4—44no—
630C4—44no—
631C4—44no—
632C4—84no—
633C4—84no—
634C4—84no—
635C2—88no—
636C2#3500311yescharacteristic
637C2#3649612yes—
638C2#3726711yescharacteristic
639C2#3876212yes—
640C2#3876912yes—
641C2#3942012yes—
642C2#4599611yescharacteristic
643C2#4609911yescharacteristic
644C2#5230111yescharacteristic
645C2#5966011yescharacteristic
646C2#6004511yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radical#17068C2
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×#2804C2semidirect
2. #17080C2semidirect
4. #16652C2semidirect
5. #15440C2semidirect
6. #15314C2semidirect
7. #15004C2semidirect
8. #14883C2semidirect
9. #14708C2semidirect
10. #14535C2semidirect
11. #14189C2semidirect
12. #14176C2semidirect
13. C4×#391C2semidirect
119. #391D4semidirect

Maximal subgroups

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

subgroupindex
C2×#28042
#170802
#170682
#166522
#154402
#153142
#150042
#148832
#147082
#145352
#141892
#141762
C4×#3912