Finite Group Databasebeta

← #627743

Subgroups — #627743

Order 512.

2520 conjugacy classes of subgroups.

The subgroup lattice has 625 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2868C211yesmaximal, characteristic, complemented
2C2×#2314C211yesmaximal, characteristic
3#16095C214yesmaximal, complemented
4#16093C214yesmaximal, complemented
5#15396C214yesmaximal
6#15322C214yesmaximal
7#15131C212yesmaximal
8#15125C212yesmaximal
9#14707C211yesmaximal, characteristic, complemented
10#14348C211yesmaximal, characteristic, complemented
11#14340C211yesmaximal, characteristic
12#14276C211yesmaximal, characteristic, complemented
13#14053C212yesmaximal, complemented
14#13981C212yesmaximal, complemented
15#13761C211yesmaximal, characteristic
16C22×C4⋊Q8C2212yes—
17C22×C4⋊Q8C2214yes—
18C22×#117C2211yescharacteristic
19#2868—28no—
20C2×#396C2214yes—
21C2×#393C2214yes—
22#2314—28no—
23C2×#391C2214yes—
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#388C2214yes—
27C2×#387C2214yes—
28C2×#387C2214yes—
29C2×#384C2211yescharacteristic
30C2×#384C2212yes—
31C2×#384C2214yes—
32C2×#384C2211yescharacteristic
33C2×#382C2211yescharacteristic
34C2×#382C2211yescharacteristic
35C2×#380C2212yes—
36C2×#380C2212yes—
37C2×C4×C4⋊C4C2212yes—
38C2×C4×C22⋊C4C2211yescharacteristic
39C2×C4×C22⋊C4C2211yescharacteristic
40C2×#376C2211yescharacteristic
41C2×C43C2211yescharacteristic
42#1446C2214yes—
43#1446C2214yes—
44#1442C2214yes—
45#1442C2214yes—
46#1442C2214yes—
47#1442C2214yes—
48#1441C2214yes—
49#1441C2214yes—
50#1441C2214yes—
51#1441C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1437C2212yes—
59#1437C2212yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2212yes—
65#1436C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4×#95C2211yescharacteristic
72C4×#95C2211yescharacteristic
73C23×Q8C2314yes—
74C24×C4C2311yescharacteristic
75C2×C4⋊Q8—28no—
76C2×C4⋊Q8—216no—
77C2×C4⋊Q8—28no—
78C2×C4⋊Q8—44no—
79C2×C4⋊Q8—48no—
80C2×C4⋊Q8—48no—
81C2×#117—24no—
82C2×#117—24no—
83C2×#117Q818yescomplemented
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C22⋊C4C2311yescharacteristic
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2311yescharacteristic
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C42C2312yes—
103C22×C42C2311yescharacteristic
104C22×C42C2311yescharacteristic
105C22×C42C2312yes—
106C22×C42C2311yescharacteristic
107C22×C42C2311yescharacteristic
108C22×C42C2311yescharacteristic
109C22×C42C2314yes—
110C22×C42C2311yescharacteristic
111#396—48no—
112#393—48no—
113#391—48no—
114#389—28no—
115#389—44no—
116#388—216no—
117#387—216no—
118#387—48no—
119#384—28no—
120#384—216no—
121#384—24no—
122#384D418yes—
123#382—24no—
124#382—24no—
125#380—28no—
126#380—44no—
127C4×C4⋊C4—28no—
128C4×C22⋊C4—24no—
129C4×C22⋊C4—24no—
130#376—24no—
131C2×#95C2314yes—
132C2×#95C2312yes—
133C2×#95C2312yes—
134C2×#95C2312yes—
135C2×#95C2312yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2312yes—
140C2×#95C2311yescharacteristic
141C2×#95C2314yes—
142C2×#95C2311yescharacteristic
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C43—24no—
158C25C2411yescharacteristic
159C22×Q8—28no—
160C22×Q8—28no—
161C22×Q8—28no—
162C22×Q8—28no—
163C22×Q8—44no—
164C22×Q8—44no—
165C22×Q8—44no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4C2×Q814yes—
173C23×C4C2414yes—
174C23×C4C2414yes—
175C23×C4C2414yes—
176C23×C4C2412yes—
177C23×C4C2411yescharacteristic
178C23×C4C2411yescharacteristic
179C23×C4C2414yes—
180C23×C4C2412yes—
181C23×C4C2411yescharacteristic
182C23×C4C2412yes—
183C23×C4C2411yescharacteristic
184C23×C4C2412yes—
185C23×C4C2411yescharacteristic
186C23×C4C2411yescharacteristic
187C4⋊Q8—432no—
188C4⋊Q8—464no—
189#117—232no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4C2×D418yes—
224C2×C4⋊C4C2×D4116yes—
225C2×C4⋊C4C2×Q818yes—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4C2×Q818yes—
242C2×C42—22no—
243C2×C42—24no—
244C2×C42—22no—
245C2×C42—24no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—24no—
249C2×C42—24no—
250C2×C42—22no—
251C2×C42—24no—
252C2×C42—24no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—28no—
261C2×C42—44no—
262C2×C42—44no—
263C2×C42C2×D414yes—
264C2×C42C2×Q814yes—
265C2×C42C2×Q814yes—
266C2×C42#4114yes—
267C2×C42#4114yes—
268C2×C42#4114yes—
269#95—28no—
270#95—28no—
271#95—28no—
272#95—24no—
273#95—24no—
274#95—22no—
275#95—22no—
276#95—24no—
277#95—28no—
278#95—28no—
279#95—24no—
280#95—24no—
281#95—28no—
282#95—24no—
283#95—24no—
284#95—28no—
285#95—44no—
286#95—44no—
287#95—44no—
288#95—44no—
289#95—44no—
290#95—44no—
291#95—44no—
292#95—44no—
293#95—44no—
294#95—44no—
295C24—21no—
296C24—21no—
297C24—21no—
298C24—21no—
299C24—22no—
300C24—22no—
301C24—21no—
302C24—22no—
303C24—22no—
304C24—21no—
305C24C22×Q812yes—
306C24C2511yescharacteristic
307C2×Q8—416no—
308C2×Q8—416no—
309C2×Q8—416no—
310C2×Q8—416no—
311C2×Q8—88no—
312C2×Q8—88no—
313C2×Q8—88no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—21no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—28no—
347C22×C4—28no—
348C22×C4—28no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—24no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4C22×D418yes—
392C22×C4C22×D414yes—
393C22×C4C22×D412yes—
394C22×C4C22×Q814yes—
395C22×C4C22×Q812yes—
396C22×C4C22×Q814yes—
397C22×C4C22×Q812yes—
398C22×C4C2×#4114yes—
399C22×C4C2×#4112yes—
400C22×C4C2×#4112yes—
401C22×C4C2×#4114yes—
402C22×C4C2×#4112yes—
403C22×C4C2×#4112yes—
404C22×C4C2×#4112yes—
405C22×C4H(5,𝔽2)14yes—
406C22×C4H(5,𝔽2)14yes—
407C22×C4H(5,𝔽2)12yes—
408C22×C42−1+414yes—
409C22×C42−1+414yes—
410C22×C42−1+412yes—
411C4⋊C4—216no—
412C4⋊C4—216no—
413C4⋊C4—416no—
414C4⋊C4—416no—
415C4⋊C4—416no—
416C4⋊C4—416no—
417C4⋊C4—416no—
418C4⋊C4—416no—
419C4⋊C4—48no—
420C4⋊C4—48no—
421C4⋊C4—48no—
422C4⋊C4—88no—
423C22⋊C4—216no—
424C22⋊C4—44no—
425C22⋊C4—44no—
426C22⋊C4—48no—
427C22⋊C4—88no—
428C22⋊C4—88no—
429C42—216no—
430C42—216no—
431C42—28no—
432C42—28no—
433C42—28no—
434C42—44no—
435C42—44no—
436C42—88no—
437C42C22⋊Q8116yes—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—24no—
442C23—24no—
443C23—41no—
444C23—41no—
445C23—42no—
446C23—41no—
447C23—41no—
448C23—41no—
449C23—41no—
450C23—42no—
451C23—42no—
452C23—41no—
453C23—41no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23C23×D411yescharacteristic
462C23C23×Q811yescharacteristic
463C23C22×#4111yescharacteristic
464C23C22×#4111yescharacteristic
465C23C22×#4111yescharacteristic
466C23C2×H(5,𝔽2)12yes—
467C23C2×H(5,𝔽2)12yes—
468C23C2×H(5,𝔽2)11yescharacteristic
469C23C2×2−1+412yes—
470C23C2×2−1+412yes—
471C23C2×2−1+411yescharacteristic
472Q8—832no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—28no—
498C2×C4—28no—
499C2×C4—28no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—44no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—416no—
515C2×C4—41no—
516C2×C4—44no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—41no—
532C2×C4—44no—
533C2×C4—41no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4C2×C22⋊Q818yes—
554C2×C4C2×C22⋊Q814yes—
555C2×C4#53314yes—
556C2×C4#54014yes—
557C2×C4#54114yes—
558C2×C4#54218yes—
559C2×C4#54414yes—
560C2×C4#54614yes—
561C2×C4#54814yes—
562C2×C4#55514yes—
563C2×C4#55614yes—
564C22—42no—
565C22—44no—
566C22—42no—
567C22—42no—
568C22—44no—
569C22—81no—
570C22—81no—
571C22—82no—
572C22—82no—
573C22—81no—
574C22—81no—
575C22C22×C22⋊Q811yescharacteristic
576C22C2×#53311yescharacteristic
577C22C2×#54011yescharacteristic
578C22C2×#54111yescharacteristic
579C22C2×#54212yes—
580C22C2×#54411yescharacteristic
581C22C2×#54611yescharacteristic
582C22C2×#54811yescharacteristic
583C22C2×#55511yescharacteristic
584C22C2×#55611yescharacteristic
585C22#348412yes—
586C22#348612yes—
587C22#350312yes—
588C22#350312yes—
589C22#350811yescharacteristic
590C22#350911yescharacteristic
591C22#351112yes—
592C22#351112yes—
593C22#351112yes—
594C22#354312yes—
595C22#354412yes—
596C22#355212yes—
597C22#355212yes—
598C4—24no—
599C4—24no—
600C4—28no—
601C4—28no—
602C4—48no—
603C4—42no—
604C4—44no—
605C4—84no—
606C4—84no—
607C4—82no—
608C4—84no—
609C4—81no—
610C4—81no—
611C4#243918yes—
612C2—82no—
613C2C2×#243911yescharacteristic
614C2#3528311yescharacteristic
615C2#3528411yescharacteristic
616C2#3557412yes—
617C2#3731212yes—
618C2#3769411yescharacteristic
619C2#3769511yescharacteristic
620C2#3770412yes—
621C2#3950512yes—
622C2#3951311yescharacteristic
623C2#3951411yescharacteristic
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×#2868C2semidirect
3. #16095C2semidirect
4. #16093C2semidirect
9. #14707C2semidirect
10. #14348C2semidirect
12. #14276C2semidirect
13. #14053C2semidirect
14. #13981C2semidirect
83. C2×#117Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#28682
C2×#23142
#160952
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