Finite Group Databasebeta

← #6797946

Subgroups — #6797946

Order 512.

1864 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36188C211yesmaximal, characteristic, complemented
2#36127C211yesmaximal, characteristic
3#33289C211yesmaximal, characteristic, complemented
4#31937C218yesmaximal, complemented
5#29844C212yesmaximal, complemented
6#28717C211yesmaximal, characteristic, complemented
7#28508C211yesmaximal, characteristic, complemented
8#27949C212yesmaximal, complemented
9#25597C212yesmaximal, complemented
10#24816C214yesmaximal, complemented
11#24119C214yesmaximal, complemented
12#21863C212yesmaximal, complemented
13C4×#492C211yesmaximal, characteristic, complemented
14#21264C211yesmaximal, characteristic, complemented
15C2×#556C2212yes—
16C2×#547—21no—
17C2×#532C2212yes—
18C2×#492C2211yescharacteristic
19C2×#487C2212yes—
20C2×#474C2214yes—
21C2×C4×SD16C2214yes—
22C2×#431C2212yes—
23#2868C2211yescharacteristic
24#2868C2211yescharacteristic
25#2857C2212yes—
26#2857C2212yes—
27#2568C2212yes—
28#2547—21no—
29#2547—21no—
30#2547—21no—
31#2537—21no—
32#2537C2212yes—
33#2499C2214yes—
34#2496—22no—
35#2496—22no—
36#2432—22no—
37#2422—22no—
38#2422C2214yes—
39#2365—21no—
40#2365—21no—
41D4×C4⋊C4—21no—
42D4×C4⋊C4C2212yes—
43C4×C4⋊Q8C2211yescharacteristic
44C4×#127C2211yescharacteristic
45#2285C2212yes—
46C42×D4C2211yescharacteristic
47#2093C2214yes—
48#2085C2218yes—
49#2072C2214yes—
50#2068C2218yes—
51#2036C2218yes—
52#2000C2218yes—
53#1990C2212yes—
54#1990C2214yes—
55#1990C2212yes—
56#1990C2212yes—
57#1968C2214yes—
58#1957C2212yes—
59#1957C2211yescharacteristic
60#1956C2211yescharacteristic
61#1954C2212yes—
62#1939C2212yes—
63#1923C2214yes—
64#1877C2218yes—
65#1875C2218yes—
66#1854C2218yes—
67#1845C2212yes—
68#1844C2211yescharacteristic
69#1834C2214yes—
70#1804C2218yes—
71#1803C2214yes—
72C4×C4⋊C8C2211yescharacteristic
73C4×C22⋊C8C2212yes—
74C42×C8C2211yescharacteristic
75C22×#41—22no—
76C22×SD16C2314yes—
77#556—48no—
78#547—48no—
79#532—28no—
80#532—44no—
81C2×C4⋊Q8—21no—
82C2×C4⋊Q8—21no—
83C2×C4⋊Q8—21no—
84C2×C4⋊Q8—21no—
85C2×C4⋊Q8C2311yescharacteristic
86C2×C4⋊Q8C2312yes—
87C2×C4⋊Q8C2312yes—
88C2×#127C2311yescharacteristic
89C2×#126—24no—
90C2×#124—22no—
91C2×#124—22no—
92C2×#123—24no—
93C2×C22⋊Q8—22no—
94C2×C22⋊Q8C2314yes—
95C2×#121—22no—
96C2×#121—22no—
97C2×#121C2314yes—
98C2×C4×Q8—22no—
99C2×C4×Q8C2312yes—
100C2×C4×D4—21no—
101C2×C4×D4—21no—
102C2×C4×D4—22no—
103C2×C4×D4D411yescharacteristic
104C2×C4×D4D411yescharacteristic
105C2×C4×D4C2312yes—
106C2×C4×D4C2314yes—
107C2×C4×D4C2311yescharacteristic
108C2×#117C2314yes—
109C22×C4⋊C4—22no—
110C22×C42C2312yes—
111#492—24no—
112#487—44no—
113#474—48no—
114C4×SD16—48no—
115#431—44no—
116C2×#107C2318yes—
117C2×#106C2314yes—
118C2×C4⋊C8C2312yes—
119C2×C4⋊C8C2314yes—
120C2×#103C2318yes—
121C2×#103C2314yes—
122C2×#102C2314yes—
123C2×#102C2318yes—
124C2×C22⋊C8C2314yes—
125C2×C22⋊C8C2314yes—
126C2×C4×C8C2312yes—
127C2×C4×C8C2311yescharacteristic
128C2×C4×C8C2312yes—
129C2×C4×C8C2311yescharacteristic
130C2×C4×C8C2312yes—
131#400—24no—
132#397—24no—
133#397—24no—
134#397—24no—
135#394—24no—
136#394—24no—
137#391—22no—
138#391—22no—
139#391—22no—
140#391C2312yes—
141#391C2314yes—
142#390—21no—
143#390—21no—
144#390C2312yes—
145#389—21no—
146#389—21no—
147#389—22no—
148#389—22no—
149#389—22no—
150#389C2314yes—
151#387—24no—
152#387—24no—
153#385—24no—
154#385—24no—
155#384—21no—
156#384—21no—
157#384—21no—
158#384—21no—
159#384C2311yescharacteristic
160#384C2311yescharacteristic
161#384C2312yes—
162#384C2312yes—
163#384C2312yes—
164#384C2312yes—
165#382C2314yes—
166#380—22no—
167#380—22no—
168#380—22no—
169#380—22no—
170#380—22no—
171#380—22no—
172#380—22no—
173#380C2314yes—
174C4×C4⋊C4—21no—
175C4×C4⋊C4—21no—
176C4×C4⋊C4—22no—
177C4×C4⋊C4—22no—
178C4×C4⋊C4D411yescharacteristic
179C4×C4⋊C4D411yescharacteristic
180C4×C4⋊C4C2311yescharacteristic
181C4×C22⋊C4D412yes—
182C4×C22⋊C4D412yes—
183C4×C22⋊C4C2312yes—
184C43C2311yescharacteristic
185#340C2318yes—
186#340C2318yes—
187#340C2318yes—
188#336C2312yes—
189#336C2314yes—
190C2×#41—48no—
191C2×#41—48no—
192C2×#41—84no—
193C22×Q8—22no—
194C22×Q8—22no—
195C22×Q8C2412yes—
196C22×D4—24no—
197C22×D4—21no—
198C22×D4—21no—
199C22×D4C2×D411yescharacteristic
200C22×D4C2412yes—
201C23×C4—22no—
202C23×C4—24no—
203C23×C4C2×D412yes—
204C23×C4C2×D412yes—
205C23×C4C2412yes—
206C2×SD16—28no—
207C2×SD16—28no—
208C2×SD16—44no—
209C22×C8C2414yes—
210C22×C8C2412yes—
211C22×C8C2412yes—
212C4⋊Q8—24no—
213C4⋊Q8—22no—
214C4⋊Q8—24no—
215C4⋊Q8—28no—
216C4⋊Q8—22no—
217C4⋊Q8—44no—
218C4⋊Q8—44no—
219C4⋊Q8—44no—
220C4⋊Q8—82no—
221#127—22no—
222#127#4114yes—
223#126—416no—
224#124—48no—
225#124—48no—
226#123—416no—
227C22⋊Q8—48no—
228C22⋊Q8—48no—
229#121—28no—
230#121—44no—
231#121—48no—
232#121—48no—
233C4×Q8—28no—
234C4×Q8—48no—
235C4×D4—28no—
236C4×D4—24no—
237C4×D4—24no—
238C4×D4—24no—
239C4×D4—28no—
240C4×D4—44no—
241C4×D4—48no—
242C4×D4—44no—
243C4×D4—44no—
244#117—48no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—48no—
265C2×C4⋊C4C2×D411yescharacteristic
266C2×C4⋊C4C2×D411yescharacteristic
267C2×C4⋊C4C2×D411yescharacteristic
268C2×C4⋊C4C2412yes—
269C2×C4⋊C4C2414yes—
270C2×C4⋊C4C2414yes—
271C2×C4⋊C4C2412yes—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4C2×D412yes—
281C2×C22⋊C4C2×D412yes—
282C2×C22⋊C4C2×D412yes—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—22no—
287C2×C42C2×D412yes—
288C2×C42C2×D412yes—
289C2×C42C2×D411yescharacteristic
290C2×C42C2×D411yescharacteristic
291C2×C42C2×D411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C2×C42C2412yes—
294C2×C42C2411yescharacteristic
295C2×C42C2411yescharacteristic
296C2×C42C2411yescharacteristic
297#107—48no—
298#106—44no—
299C4⋊C8—24no—
300C4⋊C8—44no—
301#103—44no—
302#103—48no—
303#102—44no—
304#102—48no—
305C22⋊C8—28no—
306C22⋊C8—44no—
307C4×C8—24no—
308C4×C8—22no—
309C4×C8—22no—
310C4×C8—24no—
311C4×C8—24no—
312#95—24no—
313#95—24no—
314#95—22no—
315#95—22no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95C2×D412yes—
321#95C2×D412yes—
322#95C2×D412yes—
323C24—22no—
324C24C22×D412yes—
325#41—832no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—24no—
329C2×Q8—24no—
330C2×Q8—44no—
331C2×Q8—44no—
332C2×Q8—44no—
333C2×Q8—44no—
334C2×Q8—44no—
335C2×Q8—44no—
336C2×D4—22no—
337C2×D4—24no—
338C2×D4—24no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—24no—
342C2×D4—22no—
343C2×D4—24no—
344C2×D4—48no—
345C2×D4—48no—
346C2×D4—42no—
347C2×D4—42no—
348C2×D4—42no—
349C2×D4—42no—
350C2×D4—84no—
351C2×D4C2×#4114yes—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—48no—
368C22×C4—48no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4—84no—
373C22×C4C22×D412yes—
374C22×C4C22×D412yes—
375C22×C4C22×D412yes—
376C22×C4C22×D411yescharacteristic
377C22×C4C22×D411yescharacteristic
378C22×C4C22×D411yescharacteristic
379C22×C4C22×D411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C22×C4C22×D411yescharacteristic
382C22×C4C2511yescharacteristic
383SD16—416no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—24no—
388C2×C8—22no—
389C2×C8—24no—
390C2×C8—22no—
391C2×C8—24no—
392C2×C82−1+414yes—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—24no—
397C4⋊C4—24no—
398C4⋊C4—28no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—48no—
402C4⋊C4—48no—
403C4⋊C4—44no—
404C4⋊C4—44no—
405C4⋊C4—44no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—44no—
410C4⋊C4—44no—
411C4⋊C4—44no—
412C4⋊C4—42no—
413C4⋊C4—44no—
414C4⋊C4—44no—
415C4⋊C4—44no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—82no—
419C4⋊C4—82no—
420C22⋊C4—24no—
421C22⋊C4—24no—
422C22⋊C4—24no—
423C22⋊C4—48no—
424C22⋊C4—48no—
425C22⋊C4—48no—
426C22⋊C4—48no—
427C22⋊C4—48no—
428C22⋊C4—44no—
429C22⋊C4—44no—
430C22⋊C4—44no—
431C42—24no—
432C42—24no—
433C42—22no—
434C42—22no—
435C42—22no—
436C42—22no—
437C42—22no—
438C42—48no—
439C42—44no—
440C42#13514yes—
441C42AGL(1,ℤ/8)14yes—
442C42C2×#4112yes—
443C42H(5,𝔽2)14yes—
444C42H(5,𝔽2)12yes—
445C23—22no—
446C23—24no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23C23×D411yescharacteristic
460Q8—44no—
461Q8—44no—
462Q8—88no—
463Q8—88no—
464D4—28no—
465D4—44no—
466D4—44no—
467D4—48no—
468D4—48no—
469D4—816no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
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—44no—
486C2×C4—44no—
487C2×C4—48no—
488C2×C4—48no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—816no—
505C2×C4—88no—
506C2×C4#53512yes—
507C2×C4#53512yes—
508C2×C4#54612yes—
509C2×C4#54612yes—
510C2×C4C2×#13512yes—
511C2×C4C2×#13512yes—
512C2×C4C2×#13512yes—
513C2×C4C2×AGL(1,ℤ/8)12yes—
514C2×C4C2×AGL(1,ℤ/8)12yes—
515C2×C4C2×AGL(1,ℤ/8)12yes—
516C2×C4#57712yes—
517C2×C4#57812yes—
518C2×C4C22×#4111yescharacteristic
519C2×C4C2×H(5,𝔽2)12yes—
520C2×C4C2×H(5,𝔽2)11yescharacteristic
521C2×C4C2×2−1+412yes—
522C8—24no—
523C8—42no—
524C8—44no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22—44no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—44no—
533C22—42no—
534C22—44no—
535C22—82no—
536C22—82no—
537C22—82no—
538C22—82no—
539C22C2×#53511yescharacteristic
540C22C2×#54611yescharacteristic
541C22#349911yescharacteristic
542C22C22×#13511yescharacteristic
543C22C22×AGL(1,ℤ/8)11yescharacteristic
544C22C2×#57711yescharacteristic
545C22C2×#57811yescharacteristic
546C4—22no—
547C4—22no—
548C4—22no—
549C4—42no—
550C4—42no—
551C4—42no—
552C4—82no—
553C4—82no—
554C4—84no—
555C4—82no—
556C4—84no—
557C4#320714yes—
558C4#330214yes—
559C4#332414yes—
560C4#352012yes—
561C4#354812yes—
562C2—82no—
563C2—84no—
564C2#3895111yescharacteristic
565C2C2×#320711yescharacteristic
566C2#6082411yescharacteristic
567C2C2×#330211yescharacteristic
568C2C2×#332411yescharacteristic
569C2#6247011yescharacteristic
570C2#6247911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36127C2
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. #36188C2semidirect
3. #33289C2semidirect
4. #31937C2semidirect
5. #29844C2semidirect
6. #28717C2semidirect
7. #28508C2semidirect
8. #27949C2semidirect
9. #25597C2semidirect
10. #24816C2semidirect
11. #24119C2semidirect
12. #21863C2semidirect
13. C4×#492C2semidirect
14. #21264C2semidirect

Maximal subgroups

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

subgroupindex
#361882
#361272
#332892
#319372
#298442
#287172
#285082
#279492
#255972
#248162
#241192
#218632
C4×#4922
#212642