Finite Group Databasebeta

← #522086

Subgroups — #522086

Order 512.

2690 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic
2C2×#2331C211yesmaximal, characteristic, complemented
3C2×#2283C211yesmaximal, characteristic
4#14388C214yesmaximal, complemented
5#14364C214yesmaximal
6#14282C211yesmaximal, characteristic, complemented
7#14276C211yesmaximal, characteristic, complemented
8#14132C212yesmaximal, complemented
9#14108C214yesmaximal
10#14090C214yesmaximal, complemented
11#13994C212yesmaximal
12#13974C212yesmaximal
13#13818C212yesmaximal, complemented
14#13779C211yesmaximal, characteristic, complemented
15#13753C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C23×C42C2211yescharacteristic
19#2857C4116yescomplemented
20#2331—28no—
21C2×#391C2214yes—
22C2×#391C2212yes—
23C2×#389C2212yes—
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×#388C2214yes—
28C2×#388C2214yes—
29C2×#387C2214yes—
30C2×#384C2211yescharacteristic
31#2283—28no—
32C2×#383C2212yes—
33C2×#382C2211yescharacteristic
34C2×#382C2211yescharacteristic
35C2×#381C2212yes—
36C2×#380C2212yes—
37C2×#380C2212yes—
38C2×#380C2212yes—
39C2×C4×C4⋊C4C2212yes—
40C2×C4×C4⋊C4C2212yes—
41C2×C4×C4⋊C4C2212yes—
42C2×C4×C22⋊C4C2212yes—
43C2×C4×C22⋊C4C2211yescharacteristic
44C2×C4×C22⋊C4C2211yescharacteristic
45C2×#376C2212yes—
46C2×#376C2211yescharacteristic
47C2×#376C2211yescharacteristic
48#1446C2214yes—
49#1445C2214yes—
50#1445C2214yes—
51#1443C2214yes—
52#1442C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1440C2214yes—
57#1440C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64C4⋊#95C2214yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69#1434C2212yes—
70#1434C2214yes—
71#1434C2214yes—
72#1434C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C4×#95C2214yes—
76C4×#95C2211yescharacteristic
77C4×#95C2211yescharacteristic
78C23×Q8C2312yes—
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C2×C4⋊Q8—24no—
83C2×C4⋊Q8—24no—
84C2×C4⋊Q8C2×C418yes—
85C2×C22⋊Q8—48no—
86C2×C22⋊Q8C2×C4116yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C42—24no—
102C22×C42—24no—
103C22×C42C2×C418yes—
104C22×C42C2311yescharacteristic
105C22×C42C2311yescharacteristic
106C22×C42C2314yes—
107C22×C42C2311yescharacteristic
108C22×C42C2311yescharacteristic
109C22×C42C2311yescharacteristic
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112C22×C42C2314yes—
113C22×C42C2314yes—
114C22×C42C2311yescharacteristic
115#391—216no—
116#391C2×C4116yes—
117#389—28no—
118#389—28no—
119#389C2×C4116yes—
120#389C2×C4116yes—
121#388—48no—
122#388—48no—
123#387—216no—
124#384C2×C418yes—
125#383—28no—
126#382—24no—
127#382D418yes—
128#381—28no—
129#380—28no—
130#380C2×C4116yes—
131#380C2×C4116yes—
132C4×C4⋊C4—28no—
133C4×C4⋊C4—28no—
134C4×C4⋊C4—28no—
135C4×C22⋊C4—24no—
136C4×C22⋊C4—24no—
137C4×C22⋊C4—28no—
138#376—24no—
139#376—24no—
140#376—28no—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C25C2411yescharacteristic
172C22×Q8—24no—
173C22×Q8—24no—
174C22×Q8—44no—
175C22×Q8—44no—
176C22×Q8C22×C418yes—
177C23×C4—24no—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—24no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—42no—
190C23×C4—42no—
191C23×C4C22×C414yes—
192C23×C4C22×C414yes—
193C23×C4C22×C414yes—
194C23×C4C2414yes—
195C23×C4C2412yes—
196C23×C4C2412yes—
197C23×C4C2414yes—
198C23×C4C2414yes—
199C23×C4C2411yescharacteristic
200C23×C4C2411yescharacteristic
201C23×C4C2414yes—
202C23×C4C2412yes—
203C23×C4C2412yes—
204C23×C4C2411yescharacteristic
205C23×C4C2411yescharacteristic
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208C4⋊Q8—232no—
209C22⋊Q8—432no—
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—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—28no—
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⋊C4C22×C418yes—
232C2×C4⋊C4C22×C418yes—
233C2×C4⋊C4C22×C418yes—
234C2×C4⋊C4C22×C418yes—
235C2×C4⋊C4C22×C4116yes—
236C2×C4⋊C4C22×C418yes—
237C2×C4⋊C4C2×D418yes—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4C22×C4116yes—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—232no—
261C2×C42—28no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—28no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—28no—
270C2×C42—28no—
271C2×C42—44no—
272C2×C42—44no—
273C2×C42—44no—
274C2×C42—44no—
275C2×C42—44no—
276C2×C42C22×C414yes—
277C2×C42C22×C414yes—
278C2×C42C22×C414yes—
279C2×C42C22×C414yes—
280C2×C42C2×D414yes—
281C2×C42#4114yes—
282C2×C42#4114yes—
283C2×C42#4114yes—
284#95—28no—
285#95—24no—
286#95—24no—
287#95—24no—
288#95—24no—
289#95—28no—
290#95—28no—
291#95—28no—
292#95—24no—
293#95—24no—
294#95—24no—
295#95—24no—
296#95—28no—
297#95—28no—
298#95—24no—
299#95—24no—
300#95—42no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—42no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95C22×C4116yes—
311#95C22×C4116yes—
312#95C2×D418yes—
313#95C2×D418yes—
314C24—21no—
315C24—21no—
316C24—21no—
317C24—21no—
318C24—22no—
319C24—21no—
320C24—21no—
321C24—22no—
322C24—22no—
323C24C23×C412yes—
324C24H(5,𝔽2)14yes—
325C24C2511yescharacteristic
326C2×Q8—216no—
327C2×Q8—48no—
328C2×Q8—48no—
329C2×Q8—88no—
330C2×Q8—88no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—28no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—28no—
380C22×C4—28no—
381C22×C4—28no—
382C22×C4—28no—
383C22×C4—28no—
384C22×C4—28no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4—48no—
409C22×C4—48no—
410C22×C4—48no—
411C22×C4—48no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4C23×C412yes—
417C22×C4C23×C418yes—
418C22×C4C23×C418yes—
419C22×C4C23×C412yes—
420C22×C4C23×C412yes—
421C22×C4C23×C412yes—
422C22×C4C23×C412yes—
423C22×C4C23×C412yes—
424C22×C4C22×D418yes—
425C22×C4C22×D412yes—
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C2×#4114yes—
429C22×C4C2×#4114yes—
430C22×C4C2×#4112yes—
431C22×C4C2×#4112yes—
432C22×C4C2×#4112yes—
433C22×C4C2×#4112yes—
434C22×C4C2×#4112yes—
435C22×C42−1+414yes—
436C22×C42−1+412yes—
437C22×C42−1+412yes—
438C4⋊C4—216no—
439C4⋊C4—216no—
440C4⋊C4—216no—
441C4⋊C4—216no—
442C4⋊C4—48no—
443C4⋊C4—48no—
444C4⋊C4—416no—
445C4⋊C4—48no—
446C4⋊C4—88no—
447C4⋊C4—88no—
448C22⋊C4—232no—
449C22⋊C4—416no—
450C22⋊C4—48no—
451C22⋊C4—48no—
452C42—28no—
453C42—28no—
454C42—28no—
455C42—28no—
456C42—28no—
457C42—416no—
458C42—44no—
459C42—44no—
460C42—88no—
461C42—88no—
462C42C4×D4116yes—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—42no—
476C23—42no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23#51814yes—
486C23C24×C411yescharacteristic
487C23C23×D411yescharacteristic
488C23C22×#4111yescharacteristic
489C23C22×#4111yescharacteristic
490C23C22×#4111yescharacteristic
491C23C2×H(5,𝔽2)12yes—
492C23C2×2−1+412yes—
493C23C2×2−1+411yescharacteristic
494C23C2×2−1+411yescharacteristic
495C23#58512yes—
496C23#58512yes—
497Q8—816no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—28no—
513C2×C4—28no—
514C2×C4—22no—
515C2×C4—28no—
516C2×C4—28no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—28no—
520C2×C4—22no—
521C2×C4—28no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—28no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—28no—
528C2×C4—28no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—416no—
532C2×C4—416no—
533C2×C4—44no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—416no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4C2×C4×D414yes—
582C2×C4C2×C4×D414yes—
583C2×C4C2×C4×D414yes—
584C2×C4C4×#4114yes—
585C2×C4#51914yes—
586C2×C4#51914yes—
587C2×C4#54714yes—
588C2×C4#54814yes—
589C2×C4#55214yes—
590C2×C4#55614yes—
591C22—24no—
592C22—24no—
593C22—24no—
594C22—42no—
595C22—44no—
596C22—44no—
597C22—42no—
598C22—42no—
599C22—44no—
600C22—44no—
601C22—42no—
602C22C22×C4×D411yescharacteristic
603C22C2×C4×#4111yescharacteristic
604C22C2×#51811yescharacteristic
605C22C2×#51911yescharacteristic
606C22C2×#51911yescharacteristic
607C22#342811yescharacteristic
608C22#342811yescharacteristic
609C22#346112yes—
610C22#346112yes—
611C22C2×#54711yescharacteristic
612C22C2×#54811yescharacteristic
613C22C2×#55211yescharacteristic
614C22C2×#55611yescharacteristic
615C22#348812yes—
616C22#348912yes—
617C22#349412yes—
618C22#349412yes—
619C22#349412yes—
620C22#349512yes—
621C22#354412yes—
622C22#354412yes—
623C22#356512yes—
624C22#356512yes—
625C4—24no—
626C4—24no—
627C4—24no—
628C4—44no—
629C4—42no—
630C4—44no—
631C4—48no—
632C4—42no—
633C4—48no—
634C4—48no—
635C4—82no—
636C4—84no—
637C4—82no—
638C4#236218yes—
639C2—44no—
640C2#3408511yescharacteristic
641C2C2×#236211yescharacteristic
642C2#3420311yescharacteristic
643C2#3420411yescharacteristic
644C2#3422211yescharacteristic
645C2#3436511yescharacteristic
646C2#3440311yescharacteristic
647C2#3775812yes—
648C2#3776612yes—
649C2#3793912yes—
650C2#3796312yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC25C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#2331C2semidirect
4. #14388C2semidirect
6. #14282C2semidirect
7. #14276C2semidirect
8. #14132C2semidirect
10. #14090C2semidirect
13. #13818C2semidirect
14. #13779C2semidirect
19. #2857C4semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
C2×#23312
C2×#22832
#143882
#143642
#142822
#142762
#141322
#141082
#140902
#139942
#139742
#138182
#137792
#137532