Finite Group Databasebeta

← #4308091

Subgroups — #4308091

Order 512.

2662 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×#2647C211yesmaximal, characteristic, complemented
2C22×#376C211yesmaximal, characteristic
3#14904C214yesmaximal, complemented
4#14895C214yesmaximal, complemented
5#14767C214yesmaximal
6#14736C214yesmaximal
7#14697C212yesmaximal
8#14697C212yesmaximal
9#14686C211yesmaximal, characteristic
10#14682C211yesmaximal, characteristic
11#14348C211yesmaximal, characteristic, complemented
12#14277C211yesmaximal, characteristic, complemented
13#14276C211yesmaximal, characteristic, complemented
14#14266C212yesmaximal, complemented
15#14266C212yesmaximal, complemented
16C23×C42C2211yescharacteristic
17C23×C42C2211yescharacteristic
18C23×C42C2211yescharacteristic
19#2647—28no—
20C2×#400C2214yes—
21C2×#398C2214yes—
22C2×#389C2212yes—
23C2×#389C2212yes—
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#385C2214yes—
27C2×#385C2214yes—
28C2×#384C2211yescharacteristic
29C2×#384C2211yescharacteristic
30C2×#382C2211yescharacteristic
31C2×#382C2211yescharacteristic
32C2×#380C2212yes—
33C2×#380C2212yes—
34C2×#380C2214yes—
35C2×#376—28no—
36C2×#376C2212yes—
37C2×#376C2212yes—
38C2×#376C2211yescharacteristic
39C2×#376C2211yescharacteristic
40C2×#376C2211yescharacteristic
41C2×#376C2211yescharacteristic
42C22×#95C2212yes—
43C22×#95C2212yes—
44#1445C2214yes—
45#1445C2214yes—
46#1444C2214yes—
47#1443C2214yes—
48#1443C2214yes—
49#1442C2214yes—
50#1442C2214yes—
51#1442C2214yes—
52#1442C2214yes—
53#1442C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2214yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69#1434C2214yes—
70C22⋊#95C2214yes—
71C22⋊#95C2214yes—
72C22⋊#95C2214yes—
73C24×C4C2311yescharacteristic
74C24×C4C2312yes—
75C24×C4C2311yescharacteristic
76C24×C4C2312yes—
77C24×C4C2311yescharacteristic
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2314yes—
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2314yes—
88C22×C22⋊C4C2314yes—
89C22×C22⋊C4C2314yes—
90C22×C42—24no—
91C22×C42—24no—
92C22×C42—24no—
93C22×C42—24no—
94C22×C42—24no—
95C22×C42—24no—
96C22×C42—24no—
97C22×C42—24no—
98C22×C42Q818yes—
99C22×C42C2311yescharacteristic
100C22×C42C2312yes—
101C22×C42C2311yescharacteristic
102C22×C42C2311yescharacteristic
103C22×C42C2311yescharacteristic
104C22×C42C2312yes—
105C22×C42C2311yescharacteristic
106C22×C42C2311yescharacteristic
107C22×C42C2311yescharacteristic
108C22×C42C2311yescharacteristic
109#400—216no—
110#398—216no—
111#389—28no—
112#389—28no—
113#389—28no—
114#389—28no—
115#385—48no—
116#385—48no—
117#384—24no—
118#384—24no—
119#382—24no—
120#382—24no—
121#380—28no—
122#380—28no—
123#380—48no—
124#376—24no—
125#376—24no—
126#376—24no—
127#376—28no—
128#376—28no—
129#376D418yes—
130C2×#95—28no—
131C2×#95—28no—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C25C2411yescharacteristic
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—24no—
168C23×C4—24no—
169C23×C4—24no—
170C23×C4—22no—
171C23×C4—24no—
172C23×C4—22no—
173C23×C4—24no—
174C23×C4—22no—
175C23×C4—24no—
176C23×C4—22no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—42no—
183C23×C4—42no—
184C23×C4—42no—
185C23×C4—42no—
186C23×C4—42no—
187C23×C4—42no—
188C23×C4C2×Q818yes—
189C23×C4C2×Q814yes—
190C23×C4#4114yes—
191C23×C4#4114yes—
192C23×C4C2414yes—
193C23×C4C2414yes—
194C23×C4C2414yes—
195C23×C4C2414yes—
196C23×C4C2411yescharacteristic
197C23×C4C2412yes—
198C23×C4C2411yescharacteristic
199C23×C4C2412yes—
200C23×C4C2411yescharacteristic
201C23×C4C2412yes—
202C23×C4C2411yescharacteristic
203C23×C4C2412yes—
204C23×C4C2411yescharacteristic
205C23×C4C2411yescharacteristic
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—28no—
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⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
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×C42—232no—
243C2×C42—232no—
244C2×C42—22no—
245C2×C42—24no—
246C2×C42—22no—
247C2×C42—24no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—24no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—24no—
257C2×C42—24no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—232no—
266C2×C42C2×D414yes—
267C2×C42C2×D414yes—
268C2×C42C2×D414yes—
269C2×C42C2×Q814yes—
270C2×C42C2×Q818yes—
271C2×C42C2×Q814yes—
272C2×C42#4114yes—
273C2×C42#4114yes—
274C2×C42#4114yes—
275#95—24no—
276#95—24no—
277#95—24no—
278#95—24no—
279#95—24no—
280#95—24no—
281#95—28no—
282#95—28no—
283#95—28no—
284#95—28no—
285#95—28no—
286#95—28no—
287#95—28no—
288#95—28no—
289#95—44no—
290#95—44no—
291#95—44no—
292#95—44no—
293#95—44no—
294#95—44no—
295#95—44no—
296#95—44no—
297#95—44no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95C2×D418yes—
302#95C2×D418yes—
303C24—21no—
304C24—21no—
305C24—21no—
306C24—21no—
307C24—22no—
308C24—22no—
309C24C22×Q812yes—
310C24C2×#4112yes—
311C24C2×#4112yes—
312C24H(5,𝔽2)14yes—
313C242−1+414yes—
314C24C2511yescharacteristic
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—28no—
324C22×C4—28no—
325C22×C4—28no—
326C22×C4—28no—
327C22×C4—28no—
328C22×C4—28no—
329C22×C4—216no—
330C22×C4—28no—
331C22×C4—28no—
332C22×C4—216no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
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—216no—
368C22×C4—24no—
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—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—48no—
397C22×C4—48no—
398C22×C4—48no—
399C22×C4—48no—
400C22×C4—48no—
401C22×C4—48no—
402C22×C4—48no—
403C22×C4—48no—
404C22×C4—48no—
405C22×C4—48no—
406C22×C4—48no—
407C22×C4—48no—
408C22×C4—48no—
409C22×C4—48no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—48no—
415C22×C4#125116yes—
416C22×C4C22×D412yes—
417C22×C4C22×D414yes—
418C22×C4C22×D412yes—
419C22×C4C22×D414yes—
420C22×C4C22×D412yes—
421C22×C4C22×Q814yes—
422C22×C4C22×Q814yes—
423C22×C4C22×Q812yes—
424C22×C4C22×Q812yes—
425C22×C4C2×#4112yes—
426C22×C4C2×#4114yes—
427C22×C4C2×#4112yes—
428C22×C4C2×#4114yes—
429C22×C4C2×#4112yes—
430C22×C4C2×#4112yes—
431C22×C4C2×#4112yes—
432C22×C4C2×#4112yes—
433C22×C4C2×#4112yes—
434C22×C4C2×#4112yes—
435C22×C4C2×#4112yes—
436C4⋊C4—48no—
437C4⋊C4—48no—
438C4⋊C4—48no—
439C4⋊C4—48no—
440C4⋊C4—88no—
441C4⋊C4—88no—
442C4⋊C4—88no—
443C4⋊C4—88no—
444C4⋊C4—88no—
445C4⋊C4—88no—
446C22⋊C4—416no—
447C22⋊C4—416no—
448C42—216no—
449C42—28no—
450C42—28no—
451C42—216no—
452C42—28no—
453C42—28no—
454C42—28no—
455C42—28no—
456C42—28no—
457C42C22⋊Q8116yes—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—24no—
465C23—22no—
466C23—22no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23C2×#12514yes—
477C23#54014yes—
478C23#54114yes—
479C23#54218yes—
480C23#54418yes—
481C23C23×D411yescharacteristic
482C23C23×Q811yescharacteristic
483C23C22×#4111yescharacteristic
484C23C22×#4111yescharacteristic
485C23C22×#4111yescharacteristic
486C23C22×#4111yescharacteristic
487C23C22×#4111yescharacteristic
488C23C2×H(5,𝔽2)12yes—
489C23C2×2−1+412yes—
490C23#58512yes—
491C23#58512yes—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—232no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—432no—
523C2×C4—432no—
524C2×C4—416no—
525C2×C4—416no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
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×C4C2×C22⋊Q818yes—
579C2×C4C2×C22⋊Q814yes—
580C2×C4#52514yes—
581C2×C4#52514yes—
582C2×C4C2×#12514yes—
583C2×C4C2×#12514yes—
584C2×C4#52914yes—
585C2×C4#52914yes—
586C2×C4#53314yes—
587C22—24no—
588C22—24no—
589C22—28no—
590C22—28no—
591C22—24no—
592C22—28no—
593C22—24no—
594C22—28no—
595C22—24no—
596C22—24no—
597C22#264718yescomplemented
598C22C22×C22⋊Q811yescharacteristic
599C22C2×#52511yescharacteristic
600C22C2×#52511yescharacteristic
601C22C22×#12511yescharacteristic
602C22C2×#52911yescharacteristic
603C22C2×#52911yescharacteristic
604C22C2×#53311yescharacteristic
605C22C2×#54011yescharacteristic
606C22C2×#54111yescharacteristic
607C22C2×#54212yes—
608C22#345512yes—
609C22#345511yescharacteristic
610C22#345511yescharacteristic
611C22C2×#54412yes—
612C22#345812yes—
613C22#348812yes—
614C22#348912yes—
615C22#349412yes—
616C22#349412yes—
617C22#349412yes—
618C22#349512yes—
619C22#349512yes—
620C22#349512yes—
621C4—24no—
622C4—28no—
623C4—44no—
624C4—44no—
625C4—44no—
626C4—42no—
627C4—42no—
628C4—42no—
629C4—42no—
630C4—84no—
631C4—84no—
632C4—84no—
633C4—84no—
634C4#285118yes—
635C2—28no—
636C2C2×#264711yescharacteristic
637C2#3463811yescharacteristic
638C2#3525312yes—
639C2#3526012yes—
640C2#3526111yescharacteristic
641C2#3527012yes—
642C2#3527111yescharacteristic
643C2#3530412yes—
644C2#3531011yescharacteristic
645C2#3531511yescharacteristic
646C2C2×#285111yescharacteristic
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×#2647C2semidirect
3. #14904C2semidirect
4. #14895C2semidirect
11. #14348C2semidirect
12. #14277C2semidirect
13. #14276C2semidirect
14. #14266C2semidirect
15. #14266C2semidirect
597. C22#2647semidirect

Maximal subgroups

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

subgroupindex
C2×#26472
C22×#3762
#149042
#148952
#147672
#147362
#146972
#146862
#146822
#143482
#142772
#142762
#142662