Finite Group Databasebeta

← #6816672

Subgroups — #6816672

Order 512.

1876 conjugacy classes of subgroups.

The subgroup lattice has 567 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3234C212yesmaximal, complemented
2#36149C211yesmaximal, characteristic
3D4×#125C211yesmaximal, characteristic, complemented
4#33317C211yesmaximal, characteristic, complemented
5#31088C214yesmaximal, complemented
6#31080C214yesmaximal, complemented
7#29884C212yesmaximal, complemented
8#28724C211yesmaximal, characteristic, complemented
9#28695C211yesmaximal, characteristic, complemented
10#26059C214yesmaximal, complemented
11#25819C212yesmaximal, complemented
12#25676C212yesmaximal, complemented
13#25022C214yesmaximal, complemented
14#21656C211yesmaximal, characteristic, complemented
15#21271C211yesmaximal, characteristic, complemented
16C2×#532C2212yes—
17C22×#125C2212yes—
18#3234—216no—
19C2×#501C2212yes—
20C2×#501C2212yes—
21C2×#501C2212yes—
22C2×#490C2212yes—
23C2×#490C2211yescharacteristic
24C2×#481C2214yes—
25C2×#480C2214yes—
26C2×#478C2214yes—
27C2×#478C2214yes—
28C2×#432C2212yes—
29#2862C2211yescharacteristic
30#2859C2211yescharacteristic
31#2857C2212yes—
32#2851C2212yes—
33#2552C2212yes—
34#2535C2212yes—
35#2506—22no—
36#2473—22no—
37#2473—22no—
38#2447—22no—
39#2442C2212yes—
40#2442C2212yes—
41#2427—22no—
42#2420C2214yes—
43#2407—22no—
44#2349C2212yes—
45D4×C4⋊C4—22no—
46D4×C4⋊C4C2212yes—
47#2339—22no—
48C4×#127C2211yescharacteristic
49C4×#125C2211yescharacteristic
50C42×D4C2211yescharacteristic
51#2088C2214yes—
52#2086C2214yes—
53#2085C2214yes—
54#2077C2214yes—
55#2076C2214yes—
56#2073C2212yes—
57#2073C2212yes—
58#2059C2214yes—
59#2059C2214yes—
60#2034C2214yes—
61#2033C2214yes—
62#1993C2212yes—
63#1992C2212yes—
64#1992C2211yescharacteristic
65#1961C2211yescharacteristic
66#1955C2212yes—
67#1949C2214yes—
68#1940C2212yes—
69#1934C2214yes—
70#1923C2214yes—
71#1864C2214yes—
72#1862C2214yes—
73#1853C2214yes—
74#1852C2214yes—
75#1847C2214yes—
76#1846C2212yes—
77#1846C2212yes—
78#1836C2211yescharacteristic
79#1802C2214yes—
80#1767C2211yescharacteristic
81#1762C2214yes—
82#1749C2212yes—
83C4×#97C2211yescharacteristic
84C22×#41—22no—
85#532—28no—
86#532—44no—
87C2×C4⋊Q8C2312yes—
88C2×#127C2311yescharacteristic
89C2×#125—28no—
90C2×#125—22no—
91C2×#125—22no—
92C2×#125—44no—
93C2×#125C2312yes—
94C2×#125C2312yes—
95C2×#125C2312yes—
96C2×#125C2311yescharacteristic
97C2×#124—22no—
98C2×C22⋊Q8—22no—
99C2×#121—22no—
100C2×#121C2314yes—
101C2×C4×D4—22no—
102C2×C4×D4—22no—
103C2×C4×D4—22no—
104C2×C4×D4D412yes—
105C2×C4×D4C2312yes—
106C2×C4×D4C2314yes—
107C2×C4×D4C2311yescharacteristic
108C22×C4⋊C4—22no—
109C22×C4⋊C4—22no—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C42C2312yes—
113#501—28no—
114#501—28no—
115#501—44no—
116#490—28no—
117#490D418yescomplemented
118#481—216no—
119#480—216no—
120#478—216no—
121#478—216no—
122#432—28no—
123C2×#107C2314yes—
124C2×#107C2314yes—
125C2×#107C2314yes—
126C2×#106C2314yes—
127C2×#106C2314yes—
128C2×#106C2314yes—
129C2×C4⋊C8C2314yes—
130C2×C4⋊C8C2312yes—
131C2×#102C2314yes—
132C2×#102C2314yes—
133C2×#102C2314yes—
134C2×#102C2314yes—
135C2×C22⋊C8C2314yes—
136C2×C22⋊C8C2314yes—
137C2×#97C2312yes—
138C2×#97C2312yes—
139C2×#97C2311yescharacteristic
140C2×#97C2311yescharacteristic
141C2×C4×C8C2312yes—
142#400—22no—
143#400—22no—
144#398—22no—
145#398—22no—
146#398—22no—
147#398—22no—
148#397—22no—
149#397—24no—
150#394—24no—
151#392—22no—
152#391—22no—
153#391—22no—
154#391—22no—
155#390—22no—
156#390C2312yes—
157#389—22no—
158#389—22no—
159#389—22no—
160#389—22no—
161#389—22no—
162#389C2312yes—
163#389C2314yes—
164#389C2312yes—
165#389C2312yes—
166#387—22no—
167#387—22no—
168#387—22no—
169#386—24no—
170#386—24no—
171#385—22no—
172#385—22no—
173#385—24no—
174#385—22no—
175#385—22no—
176#385—22no—
177#385—22no—
178#384—22no—
179#384C2311yescharacteristic
180#384C2312yes—
181#382C2311yescharacteristic
182#382C2311yescharacteristic
183#382C2311yescharacteristic
184#382C2312yes—
185#382C2312yes—
186#381—24no—
187#380—22no—
188#380—22no—
189#380—22no—
190#380—22no—
191#380C2312yes—
192#380C2314yes—
193#380C2312yes—
194C4×C4⋊C4—22no—
195C4×C4⋊C4D412yes—
196C4×C4⋊C4C2312yes—
197C4×C4⋊C4C2311yescharacteristic
198C4×C22⋊C4D412yes—
199C4×C22⋊C4D412yes—
200C4×C22⋊C4C2312yes—
201C2×#95—22no—
202C43C2311yescharacteristic
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#336C2314yes—
210#336C2312yes—
211C2×#41—48no—
212C2×#41—48no—
213C2×#41—84no—
214C22×Q8—22no—
215C22×D4—24no—
216C22×D4—22no—
217C22×D4C2×D411yescharacteristic
218C22×D4C2412yes—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—24no—
222C23×C4—22no—
223C23×C4C2×D412yes—
224C23×C4C2×D412yes—
225C23×C4C2412yes—
226C22×C8C2414yes—
227C22×C8C2412yes—
228C22×C8C2412yes—
229C4⋊Q8—24no—
230C4⋊Q8—24no—
231#127—22no—
232#127C2×D414yes—
233#125—24no—
234#125—24no—
235#125—24no—
236#125—22no—
237#125—24no—
238#125—42no—
239#125—42no—
240#125—48no—
241#125—48no—
242#125C2×D414yes—
243#124—48no—
244C22⋊Q8—84no—
245#121—28no—
246#121—44no—
247#121—48no—
248C4×D4—28no—
249C4×D4—24no—
250C4×D4—28no—
251C4×D4—44no—
252C4×D4—48no—
253C4×D4—44no—
254C4×D4—48no—
255C4×D4—84no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4—48no—
278C2×C4⋊C4—48no—
279C2×C4⋊C4C2×D411yescharacteristic
280C2×C4⋊C4C2×D412yes—
281C2×C4⋊C4C2412yes—
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2414yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2412yes—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4C2×D412yes—
295C2×C22⋊C4C2×D412yes—
296C2×C22⋊C4C2×D412yes—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—22no—
300C2×C42C2×D412yes—
301C2×C42C2×D412yes—
302C2×C42C2×D412yes—
303C2×C42C2×D412yes—
304C2×C42C2412yes—
305C2×C42C2411yescharacteristic
306C2×C42C2411yescharacteristic
307C2×C42C2411yescharacteristic
308#107—28no—
309#107—28no—
310#107—44no—
311#106—28no—
312#106—28no—
313#106—44no—
314C4⋊C8—28no—
315C4⋊C8—24no—
316#102—28no—
317#102—28no—
318#102—28no—
319#102C2×D4116yes—
320C22⋊C8—28no—
321C22⋊C8—28no—
322#97—24no—
323#97—22no—
324#97—24no—
325#97C2×D414yes—
326C4×C8—24no—
327#95—24no—
328#95—22no—
329#95—22no—
330#95—22no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—24no—
337#95—22no—
338#95—22no—
339#95—24no—
340#95—22no—
341#95C2×D411yescharacteristic
342#95C2×D411yescharacteristic
343#95C2×D412yes—
344#95C2×D412yes—
345C24—22no—
346C24C22×D412yes—
347#41—832no—
348C2×Q8—44no—
349C2×Q8—44no—
350C2×Q8—44no—
351C2×D4—24no—
352C2×D4—22no—
353C2×D4—28no—
354C2×D4—22no—
355C2×D4—24no—
356C2×D4—24no—
357C2×D4—44no—
358C2×D4—44no—
359C2×D4—48no—
360C2×D4—48no—
361C2×D4—84no—
362C2×D4C22×D414yes—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—48no—
382C22×C4—48no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—84no—
387C22×C4—84no—
388C22×C4—82no—
389C22×C4—82no—
390C22×C4C22×D412yes—
391C22×C4C22×D412yes—
392C22×C4C22×D412yes—
393C22×C4C22×D412yes—
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D412yes—
397C22×C4C2511yescharacteristic
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—24no—
405C2×C8C22×D414yes—
406C2×C82−1+414yes—
407C4⋊C4—22no—
408C4⋊C4—28no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—24no—
413C4⋊C4—22no—
414C4⋊C4—24no—
415C4⋊C4—22no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—42no—
419C4⋊C4—44no—
420C4⋊C4—44no—
421C4⋊C4—44no—
422C4⋊C4—44no—
423C4⋊C4—44no—
424C4⋊C4—44no—
425C4⋊C4—44no—
426C4⋊C4—44no—
427C4⋊C4—44no—
428C4⋊C4—44no—
429C4⋊C4—44no—
430C4⋊C4—82no—
431C4⋊C4—82no—
432C4⋊C4—82no—
433C4⋊C4—82no—
434C4⋊C4—84no—
435C4⋊C4C22×D414yes—
436C22⋊C4—24no—
437C22⋊C4—24no—
438C22⋊C4—42no—
439C22⋊C4—48no—
440C22⋊C4—48no—
441C22⋊C4—44no—
442C22⋊C4—44no—
443C22⋊C4—44no—
444C22⋊C4—84no—
445C22⋊C4—84no—
446C42—24no—
447C42—24no—
448C42—22no—
449C42—22no—
450C42—22no—
451C42—42no—
452C42—42no—
453C42—44no—
454C42C22×D412yes—
455C42H(5,𝔽2)12yes—
456C422−1+414yes—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—24no—
464C23—44no—
465C23—42no—
466C23—42no—
467C23—44no—
468C23#57514yes—
469C23C23×D411yescharacteristic
470Q8—88no—
471D4—28no—
472D4—28no—
473D4—416no—
474D4—44no—
475D4—816no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—48no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—84no—
513C2×C4—88no—
514C2×C4—88no—
515C2×C4—816no—
516C2×C4—164no—
517C2×C4#53612yes—
518C2×C4#53612yes—
519C2×C4D4214yes—
520C2×C4C23×D411yescharacteristic
521C2×C4C2×H(5,𝔽2)11yescharacteristic
522C2×C4C2×2−1+412yes—
523C2×C4C2×2−1+412yes—
524C8—24no—
525C8—24no—
526C8—44no—
527C22—24no—
528C22—24no—
529C22—24no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—48no—
535C22—44no—
536C22—84no—
537C22—82no—
538C22—82no—
539C22C2×#53611yescharacteristic
540C22C2×D4211yescharacteristic
541C22#349911yescharacteristic
542C22C2×#57512yes—
543C22#358512yes—
544C4—24no—
545C4—22no—
546C4—22no—
547C4—42no—
548C4—42no—
549C4—42no—
550C4—44no—
551C4—42no—
552C4—82no—
553C4—82no—
554C4—82no—
555C4—84no—
556C4—82no—
557C4#349912yes—
558C4#355212yes—
559C2—44no—
560C2—84no—
561C2#3936011yescharacteristic
562C2#6086111yescharacteristic
563C2#6088311yescharacteristic
564C2#6106612yes—
565C2#6250612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36149C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3234C2semidirect
3. D4×#125C2semidirect
4. #33317C2semidirect
5. #31088C2semidirect
6. #31080C2semidirect
7. #29884C2semidirect
8. #28724C2semidirect
9. #28695C2semidirect
10. #26059C2semidirect
11. #25819C2semidirect
12. #25676C2semidirect
13. #25022C2semidirect
14. #21656C2semidirect
15. #21271C2semidirect
117. #490D4semidirect

Maximal subgroups

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

subgroupindex
C2×#32342
#361492
D4×#1252
#333172
#310882
#310802
#298842
#287242
#286952
#260592
#258192
#256762
#250222
#216562
#212712