Finite Group Databasebeta

← #6797948

Subgroups — #6797948

Order 512.

2216 conjugacy classes of subgroups.

The subgroup lattice has 619 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36179C211yesmaximal, characteristic, complemented
2#36127C211yesmaximal, characteristic, complemented
3#33289C211yesmaximal, characteristic, complemented
4#31930C218yesmaximal, complemented
5#29842C212yesmaximal, complemented
6#28716C211yesmaximal, characteristic, complemented
7#28508C211yesmaximal, characteristic, complemented
8#27950C212yesmaximal, complemented
9#25599C212yesmaximal, complemented
10#24811C214yesmaximal, complemented
11#24116C214yesmaximal, complemented
12#21859C212yesmaximal, complemented
13C4×#493C211yesmaximal, characteristic, complemented
14#21264C211yesmaximal, characteristic, complemented
15C2×#547—21no—
16C2×#547—21no—
17C2×#547—21no—
18C2×#547C2212yes—
19C2×#532C2212yes—
20C2×#493C2211yescharacteristic
21C2×#486C2212yes—
22C2×#474C2214yes—
23C2×C4×D8C2214yes—
24C2×#431C2212yes—
25#2868C2211yescharacteristic
26#2868C2211yescharacteristic
27#2857C2212yes—
28#2856C2212yes—
29#2547—21no—
30#2542—21no—
31#2542—21no—
32#2542C2212yes—
33#2537—21no—
34#2537C2212yes—
35#2501—22no—
36#2501—22no—
37#2501C2214yes—
38#2432—22no—
39#2422—22no—
40#2422C2214yes—
41D4×C4⋊C4—21no—
42D4×C4⋊C4C2212yes—
43C4×#127C2211yescharacteristic
44C4×#127C2211yescharacteristic
45C2×#384C2212yes—
46C42×D4C2211yescharacteristic
47#2093C2214yes—
48#2083C2218yes—
49#2071C2214yes—
50#2067C2218yes—
51#2033C2218yes—
52#1999C2218yes—
53#1990C2212yes—
54#1990C2214yes—
55#1990C2212yes—
56#1990C2212yes—
57#1970C2214yes—
58#1957C2212yes—
59#1957C2211yescharacteristic
60#1957C2211yescharacteristic
61#1954C2212yes—
62#1939C2212yes—
63#1923C2214yes—
64#1879C2218yes—
65#1874C2218yes—
66#1852C2218yes—
67#1845C2211yescharacteristic
68#1845C2212yes—
69#1834C2214yes—
70#1802C2214yes—
71#1802C2218yes—
72C4×C4⋊C8C2211yescharacteristic
73C4×C22⋊C8C2212yes—
74C42×C8C2211yescharacteristic
75C22×#41—22no—
76C22×#41—22no—
77C22×D8C2314yes—
78#547—48no—
79#547—48no—
80#547—48no—
81#547—48no—
82#532—28no—
83#532—44no—
84C2×C4⋊Q8—21no—
85C2×C4⋊Q8—21no—
86C2×C4⋊Q8—21no—
87C2×C4⋊Q8—21no—
88C2×C4⋊Q8C2312yes—
89C2×C4⋊Q8C2312yes—
90C2×#127C2311yescharacteristic
91C2×#127C2311yescharacteristic
92C2×#124—22no—
93C2×#123—24no—
94C2×#123—24no—
95C2×#123—24no—
96C2×#123—24no—
97C2×C22⋊Q8—22no—
98C2×C22⋊Q8—22no—
99C2×C22⋊Q8—22no—
100C2×C22⋊Q8—22no—
101C2×#121—22no—
102C2×#121—22no—
103C2×#121—22no—
104C2×#121—22no—
105C2×#121C2314yes—
106C2×#121C2314yes—
107C2×C4×D4—21no—
108C2×C4×D4—21no—
109C2×C4×D4—21no—
110C2×C4×D4—21no—
111C2×C4×D4—22no—
112C2×C4×D4D411yescharacteristic
113C2×C4×D4D411yescharacteristic
114C2×C4×D4C2312yes—
115C2×C4×D4C2312yes—
116C2×C4×D4C2314yes—
117C2×C4×D4C2311yescharacteristic
118C22×C4⋊C4—22no—
119C22×C4⋊C4—22no—
120C22×C4⋊C4—22no—
121C22×C4⋊C4C2314yes—
122C22×C42C2312yes—
123#493—24no—
124#486—44no—
125#474—48no—
126C4×D8—48no—
127#431—44no—
128C2×#107C2318yes—
129C2×#107C2314yes—
130C2×C4⋊C8C2312yes—
131C2×C4⋊C8C2314yes—
132C2×#102C2314yes—
133C2×#102C2318yes—
134C2×#102C2318yes—
135C2×#102C2314yes—
136C2×C22⋊C8C2314yes—
137C2×C22⋊C8C2314yes—
138C2×C4×C8C2312yes—
139C2×C4×C8C2312yes—
140C2×C4×C8C2311yescharacteristic
141C2×C4×C8C2311yescharacteristic
142C2×C4×C8C2312yes—
143#399—24no—
144#399—24no—
145#399—24no—
146#397—24no—
147#394—24no—
148#394—24no—
149#391—22no—
150#390—21no—
151#390—21no—
152#390—21no—
153#390—21no—
154#390—22no—
155#390—22no—
156#390C2312yes—
157#390C2312yes—
158#390C2314yes—
159#389—22no—
160#389C2314yes—
161#385—24no—
162#385—24no—
163#384—21no—
164#384—21no—
165#384—21no—
166#384—21no—
167#384—22no—
168#384—22no—
169#384C2311yescharacteristic
170#384C2311yescharacteristic
171#384C2314yes—
172#384C2312yes—
173#384C2312yes—
174#384C2312yes—
175#384C2312yes—
176#380—22no—
177#380—22no—
178#380—22no—
179#380C2314yes—
180C4×C4⋊C4D411yescharacteristic
181C4×C4⋊C4D411yescharacteristic
182C4×C4⋊C4C2311yescharacteristic
183C4×C22⋊C4D412yes—
184C4×C22⋊C4D412yes—
185C4×C22⋊C4C2312yes—
186C43C2311yescharacteristic
187#340C2318yes—
188#340C2318yes—
189#340C2318yes—
190#336C2312yes—
191#336C2314yes—
192C2×#41—48no—
193C2×#41—48no—
194C2×#41—48no—
195C2×#41—48no—
196C2×#41—48no—
197C2×#41—84no—
198C22×Q8—22no—
199C22×Q8—22no—
200C22×D4—24no—
201C22×D4—24no—
202C22×D4—21no—
203C22×D4—21no—
204C22×D4—21no—
205C22×D4—21no—
206C22×D4C2×D411yescharacteristic
207C22×D4C2412yes—
208C22×D4C2412yes—
209C23×C4—24no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—24no—
213C23×C4C2×D412yes—
214C23×C4C2×D412yes—
215C23×C4C2412yes—
216C2×D8—28no—
217C2×D8—28no—
218C2×D8—44no—
219C22×C8C2414yes—
220C22×C8C2412yes—
221C22×C8C2412yes—
222C4⋊Q8—24no—
223C4⋊Q8—24no—
224C4⋊Q8—28no—
225C4⋊Q8—44no—
226C4⋊Q8—44no—
227C4⋊Q8—44no—
228C4⋊Q8—82no—
229#127—22no—
230#127—22no—
231#127—22no—
232#127#4114yescomplemented
233#124—48no—
234#123—416no—
235#123—416no—
236#123—416no—
237#123—416no—
238C22⋊Q8—48no—
239C22⋊Q8—48no—
240C22⋊Q8—48no—
241C22⋊Q8—48no—
242#121—28no—
243#121—44no—
244#121—48no—
245#121—48no—
246#121—48no—
247#121—48no—
248#121—48no—
249C4×D4—28no—
250C4×D4—24no—
251C4×D4—24no—
252C4×D4—28no—
253C4×D4—28no—
254C4×D4—44no—
255C4×D4—48no—
256C4×D4—44no—
257C4×D4—44no—
258C4×D4—44no—
259C4×D4—44no—
260C4×D4D818yescomplemented
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—48no—
280C2×C4⋊C4—48no—
281C2×C4⋊C4—48no—
282C2×C4⋊C4—48no—
283C2×C4⋊C4C2×D411yescharacteristic
284C2×C4⋊C4C2×D411yescharacteristic
285C2×C4⋊C4C2×D411yescharacteristic
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2414yes—
288C2×C4⋊C4C2414yes—
289C2×C4⋊C4C2412yes—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4C2×D412yes—
303C2×C22⋊C4C2×D412yes—
304C2×C22⋊C4C2×D412yes—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42C2×D412yes—
308C2×C42C2×D412yes—
309C2×C42C2×D411yescharacteristic
310C2×C42C2×D411yescharacteristic
311C2×C42C2×D411yescharacteristic
312C2×C42C2×D411yescharacteristic
313C2×C42C2412yes—
314C2×C42C2411yescharacteristic
315C2×C42C2411yescharacteristic
316C2×C42C2411yescharacteristic
317#107—44no—
318#107—48no—
319C4⋊C8—24no—
320C4⋊C8—44no—
321#102—44no—
322#102—44no—
323#102—48no—
324#102—48no—
325C22⋊C8—28no—
326C22⋊C8—44no—
327C4×C8—24no—
328C4×C8—22no—
329C4×C8—22no—
330C4×C8—24no—
331C4×C8—24no—
332#95—24no—
333#95—24no—
334#95C2×D412yes—
335#95C2×D412yes—
336#95C2×D412yes—
337C24—22no—
338C24—22no—
339C24C22×D412yes—
340#41—832no—
341#41—832no—
342C2×Q8—44no—
343C2×Q8—44no—
344C2×Q8—44no—
345C2×Q8—44no—
346C2×Q8—44no—
347C2×Q8—44no—
348C2×D4—22no—
349C2×D4—24no—
350C2×D4—24no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—24no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—24no—
357C2×D4—24no—
358C2×D4—24no—
359C2×D4—48no—
360C2×D4—48no—
361C2×D4—48no—
362C2×D4—48no—
363C2×D4—48no—
364C2×D4—42no—
365C2×D4—42no—
366C2×D4—42no—
367C2×D4—42no—
368C2×D4—42no—
369C2×D4—42no—
370C2×D4—42no—
371C2×D4—42no—
372C2×D4—42no—
373C2×D4—42no—
374C2×D4—84no—
375C2×D4C2×D814yes—
376C2×D4C2×#4114yes—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—48no—
391C22×C4—48no—
392C22×C4—48no—
393C22×C4—48no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—48no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—44no—
401C22×C4—84no—
402C22×C4C2×D818yes—
403C22×C4C22×D412yes—
404C22×C4C22×D412yes—
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D412yes—
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4C2511yescharacteristic
413D8—416no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—24no—
418C2×C8—22no—
419C2×C8—24no—
420C2×C8—22no—
421C2×C8—24no—
422C2×C82−1+414yes—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—28no—
430C4⋊C4—48no—
431C4⋊C4—48no—
432C4⋊C4—44no—
433C4⋊C4—44no—
434C4⋊C4—42no—
435C4⋊C4—44no—
436C4⋊C4—44no—
437C4⋊C4—48no—
438C4⋊C4—48no—
439C4⋊C4—42no—
440C4⋊C4—44no—
441C4⋊C4—44no—
442C4⋊C4—42no—
443C4⋊C4—44no—
444C4⋊C4—42no—
445C4⋊C4—44no—
446C4⋊C4—82no—
447C4⋊C4—82no—
448C4⋊C4C2×D814yes—
449C22⋊C4—24no—
450C22⋊C4—24no—
451C22⋊C4—48no—
452C22⋊C4—48no—
453C22⋊C4—48no—
454C22⋊C4—48no—
455C22⋊C4—48no—
456C22⋊C4—44no—
457C22⋊C4—44no—
458C22⋊C4—44no—
459C22⋊C4—44no—
460C22⋊C4—48no—
461C22⋊C4—44no—
462C22⋊C4—44no—
463C22⋊C4C2×D818yes—
464C42—24no—
465C42—24no—
466C42—22no—
467C42—22no—
468C42—22no—
469C42—22no—
470C42—22no—
471C42C2×D814yes—
472C42AGL(1,ℤ/8)14yes—
473C42C2×#4112yes—
474C42H(5,𝔽2)14yes—
475C42H(5,𝔽2)12yes—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—24no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23C22×D814yes—
497C23C23×D411yescharacteristic
498Q8—88no—
499Q8—88no—
500D4—28no—
501D4—28no—
502D4—48no—
503D4—44no—
504D4—44no—
505D4—44no—
506D4—48no—
507D4—816no—
508D4—816no—
509D4—84no—
510D4—84no—
511C2×C4—22no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—216no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—48no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—816no—
542C2×C4—816no—
543C2×C4—88no—
544C2×C4—88no—
545C2×C4#53512yes—
546C2×C4#53512yes—
547C2×C4#54612yes—
548C2×C4#54612yes—
549C2×C4C22×D812yes—
550C2×C4C22×D814yes—
551C2×C4C22×D812yes—
552C2×C4C22×D812yes—
553C2×C4C2×AGL(1,ℤ/8)12yes—
554C2×C4C2×AGL(1,ℤ/8)12yes—
555C2×C4C2×AGL(1,ℤ/8)12yes—
556C2×C4#57712yes—
557C2×C4#57812yes—
558C2×C4C22×#4111yescharacteristic
559C2×C4C2×H(5,𝔽2)12yes—
560C2×C4C2×H(5,𝔽2)11yescharacteristic
561C2×C4C2×2−1+412yes—
562C8—24no—
563C8—42no—
564C8—44no—
565C22—24no—
566C22—24no—
567C22—24no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—44no—
573C22—44no—
574C22—44no—
575C22—82no—
576C22—82no—
577C22—82no—
578C22—82no—
579C22—82no—
580C22—82no—
581C22—82no—
582C22—82no—
583C22—82no—
584C22—82no—
585C22—82no—
586C22C2×#53511yescharacteristic
587C22C2×#54611yescharacteristic
588C22#349911yescharacteristic
589C22C23×D811yescharacteristic
590C22C22×AGL(1,ℤ/8)11yescharacteristic
591C22C2×#57711yescharacteristic
592C22C2×#57811yescharacteristic
593C4—22no—
594C4—22no—
595C4—22no—
596C4—24no—
597C4—42no—
598C4—42no—
599C4—82no—
600C4—82no—
601C4—84no—
602C4—84no—
603C4#320514yes—
604C4#329414yes—
605C4#333014yes—
606C4#352012yes—
607C4#354812yes—
608C2—44no—
609C2—84no—
610C2—162no—
611C2#3895111yescharacteristic
612C2C2×#320511yescharacteristic
613C2#6082411yescharacteristic
614C2C2×#329411yescharacteristic
615C2C2×#333011yescharacteristic
616C2#6247111yescharacteristic
617C2#6247711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
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. #36179C2semidirect
2. #36127C2semidirect
3. #33289C2semidirect
4. #31930C2semidirect
5. #29842C2semidirect
6. #28716C2semidirect
7. #28508C2semidirect
8. #27950C2semidirect
9. #25599C2semidirect
10. #24811C2semidirect
11. #24116C2semidirect
12. #21859C2semidirect
13. C4×#493C2semidirect
14. #21264C2semidirect
232. #127#41semidirect
260. C4×D4D8semidirect

Maximal subgroups

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

subgroupindex
#361792
#361272
#332892
#319302
#298422
#287162
#285082
#279502
#255992
#248112
#241162
#218592
C4×#4932
#212642