Finite Group Databasebeta

← #241391

Subgroups — #241391

Order 512.

786 conjugacy classes of subgroups.

The subgroup lattice has 579 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35192C211yesmaximal, characteristic
2#21984C211yesmaximal, characteristic
3#21962C211yesmaximal, characteristic
4#21362C211yesmaximal, characteristic
5#10578C211yesmaximal, characteristic, complemented
6#10578C211yesmaximal, characteristic, complemented
7#10577C211yesmaximal, characteristic, complemented
8#10576C211yesmaximal, characteristic, complemented
9#10575C211yesmaximal, characteristic, complemented
10#10575C211yesmaximal, characteristic, complemented
11#10574C211yesmaximal, characteristic, complemented
12#10573C211yesmaximal, characteristic, complemented
13#8039C211yesmaximal, characteristic
14#7894C211yesmaximal, characteristic
15#7891C211yesmaximal, characteristic
16C2×#431C2211yescharacteristic
17C2×#431C2211yescharacteristic
18C22×C4⋊C8C2211yescharacteristic
19#2864—21no—
20#2862—21no—
21#2665—22no—
22#2605—22no—
23#2571C2211yescharacteristic
24#2571C2211yescharacteristic
25#2551C2211yescharacteristic
26#2551C2211yescharacteristic
27#2439—21no—
28#2439—21no—
29#2439—21no—
30#2439—21no—
31#2331—21no—
32#2331—21no—
33#2285C2211yescharacteristic
34#2285C2211yescharacteristic
35#2283C2211yescharacteristic
36C4×#117C2211yescharacteristic
37#1949—21no—
38#1949—21no—
39#1949—22no—
40#1948C2211yescharacteristic
41#1948C2211yescharacteristic
42#1948C2211yescharacteristic
43#1948C2211yescharacteristic
44#1947C2211yescharacteristic
45#1947C2211yescharacteristic
46#1947C2211yescharacteristic
47#1947C2211yescharacteristic
48#1854—22no—
49#1854—21no—
50#1854—21no—
51#1845C2211yescharacteristic
52#1844C2211yescharacteristic
53#1834C2211yescharacteristic
54#1834C2211yescharacteristic
55#1808—21no—
56#1807—21no—
57C4×#107—21no—
58C4×#106—21no—
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1279C2211yescharacteristic
68#1279C2211yescharacteristic
69#1279C2211yescharacteristic
70#1279C2211yescharacteristic
71C23×C8—21no—
72C2×C4⋊Q8—21no—
73C2×C4⋊Q8—21no—
74C2×C4⋊Q8—21no—
75C2×C4⋊Q8—21no—
76C2×#117—21no—
77C2×#117—21no—
78C2×#117D411yescharacteristic
79C2×#117D411yescharacteristic
80C22×C42C2311yescharacteristic
81#431—22no—
82#431—22no—
83#431Q814yescomplemented
84#431Q814yescomplemented
85C2×#107—21no—
86C2×#107—21no—
87C2×#107—21no—
88C2×#107—21no—
89C2×#107—22no—
90C2×#106—21no—
91C2×#106—21no—
92C2×#106—21no—
93C2×#106—21no—
94C2×#106—22no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—22no—
98C2×C4⋊C8Q814yescomplemented
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C22⋊C8—21no—
104C2×C22⋊C8—21no—
105C2×C4×C8—21no—
106C2×C4×C8C2311yescharacteristic
107C2×C4×C8C2311yescharacteristic
108#400—22no—
109#400—22no—
110#396—22no—
111#396—22no—
112#395—22no—
113#395—22no—
114#393—22no—
115#393—22no—
116#391—22no—
117#391—22no—
118#391—22no—
119#391—22no—
120#391—22no—
121#391—22no—
122#389—21no—
123#389—21no—
124#389—21no—
125#389—21no—
126#389—21no—
127#389—21no—
128#389—21no—
129#389—21no—
130#387—22no—
131#387—22no—
132#387—22no—
133#387—22no—
134#387—22no—
135#387—22no—
136#384C2311yescharacteristic
137#384C2311yescharacteristic
138#384C2311yescharacteristic
139#384C2311yescharacteristic
140#383—21no—
141#382—21no—
142#382—21no—
143#382—21no—
144#382—21no—
145#382—21no—
146#382—21no—
147#382D411yescharacteristic
148#382D411yescharacteristic
149#381—21no—
150#380—21no—
151#380—21no—
152#380—21no—
153#380D411yescharacteristic
154#380D411yescharacteristic
155#380D412yes—
156C4×C4⋊C4—21no—
157C4×C4⋊C4—21no—
158C4×C4⋊C4—21no—
159C4×C4⋊C4D411yescharacteristic
160C4×C4⋊C4D411yescharacteristic
161C4×C4⋊C4D412yes—
162C4×C4⋊C4C2311yescharacteristic
163C4×C4⋊C4C2311yescharacteristic
164C4×C4⋊C4C2311yescharacteristic
165C4×C4⋊C4C2311yescharacteristic
166C4×C22⋊C4—21no—
167C4×C22⋊C4—21no—
168C4×C22⋊C4—21no—
169#376—21no—
170#376—21no—
171C43—21no—
172#340—21no—
173#340—21no—
174#340—21no—
175#340—21no—
176#340—22no—
177#340—22no—
178#340—21no—
179#340—21no—
180#340—21no—
181#340—21no—
182C22×Q8—22no—
183C22×Q8—22no—
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—24no—
194C22×C8—81no—
195C22×C8—81no—
196C4⋊Q8—42no—
197C4⋊Q8—42no—
198C4⋊Q8—42no—
199C4⋊Q8—42no—
200C4⋊Q8—81no—
201C4⋊Q8—81no—
202C4⋊Q8—81no—
203C4⋊Q8—81no—
204#117—22no—
205#117—22no—
206#117—41no—
207#117—41no—
208#117—81no—
209#117—81no—
210#117—82no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4C2×D411yescharacteristic
228C2×C4⋊C4C2×D411yescharacteristic
229C2×C4⋊C4C2×D411yescharacteristic
230C2×C4⋊C4C2×D411yescharacteristic
231C2×C4⋊C4C2×D411yescharacteristic
232C2×C4⋊C4C2×D411yescharacteristic
233C2×C4⋊C4C2×D411yescharacteristic
234C2×C4⋊C4C2×D411yescharacteristic
235C2×C4⋊C4C2×D412yes—
236C2×C4⋊C4C2×D412yes—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—41no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—22no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—41no—
253C2×C42C2×D411yescharacteristic
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C2×C42C2×Q812yes—
257C2×C42C2×Q812yes—
258C2×C42C2×Q812yes—
259C2×C42#4112yes—
260C2×C42C2411yescharacteristic
261#107—42no—
262#107—42no—
263#107—81no—
264#107—81no—
265#107—82no—
266#106—42no—
267#106—42no—
268#106—81no—
269#106—81no—
270#106—82no—
271C4⋊C8—24no—
272C4⋊C8—24no—
273C4⋊C8—21no—
274C4⋊C8—21no—
275C4⋊C8—21no—
276C4⋊C8—21no—
277C4⋊C8C2×Q812yes—
278C4⋊C8C2×Q812yes—
279C4⋊C8C2×Q812yes—
280C4⋊C8C2×Q812yes—
281C22⋊C8—24no—
282C22⋊C8—24no—
283C4×C8—24no—
284C4×C8—21no—
285C4×C8—21no—
286C4×C8C2×Q812yes—
287C4×C8C2×Q812yes—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—22no—
296#95—21no—
297#95—21no—
298#95—21no—
299#95—21no—
300#95—21no—
301#95—21no—
302#95—21no—
303#95—21no—
304#95—22no—
305#95—22no—
306#95—21no—
307#95—21no—
308#95—21no—
309#95—21no—
310#95—41no—
311#95—41no—
312C24#12011yescharacteristic
313C2×Q8—44no—
314C2×Q8—44no—
315C2×Q8—82no—
316C2×Q8—82no—
317C2×Q8—82no—
318C2×Q8—82no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4#12011yescharacteristic
342C22×C4#12011yescharacteristic
343C22×C4#12011yescharacteristic
344C22×C4C22⋊Q812yes—
345C22×C4C22⋊Q812yes—
346C22×C4C22⋊Q812yes—
347C22×C4C4⋊Q812yes—
348C22×C4C4⋊Q812yes—
349C22×C4C4⋊Q812yes—
350C22×C4C22×D411yescharacteristic
351C22×C4C22×D411yescharacteristic
352C22×C4C22×D411yescharacteristic
353C2×C8—22no—
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—22no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—84no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—44no—
389C4⋊C4—44no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—82no—
409C4⋊C4—82no—
410C4⋊C4—82no—
411C4⋊C4—82no—
412C4⋊C4#13512yes—
413C4⋊C4#13512yes—
414C22⋊C4—42no—
415C22⋊C4—42no—
416C22⋊C4—81no—
417C22⋊C4—81no—
418C22⋊C4—82no—
419C22⋊C4—82no—
420C42—42no—
421C42—42no—
422C42—41no—
423C42—41no—
424C42—41no—
425C42—41no—
426C42—41no—
427C42—41no—
428C42—41no—
429C42—41no—
430C42—81no—
431C42—81no—
432C42—82no—
433C42—82no—
434C42C22⋊Q812yes—
435C42C22⋊Q812yes—
436C42C22⋊Q812yes—
437C42C4⋊Q812yes—
438C42C4⋊Q812yes—
439C42C4⋊Q812yes—
440C42C22×Q811yescharacteristic
441C42C22×Q811yescharacteristic
442C42C22×Q811yescharacteristic
443C42C2×#4111yescharacteristic
444C23—21no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23#39512yes—
451C23C2×#12011yescharacteristic
452Q8—164no—
453Q8—164no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4—81no—
506C2×C4—81no—
507C2×C4—81no—
508C2×C4#39512yes—
509C2×C4#39512yes—
510C2×C4#39512yes—
511C2×C4C2×C22⋊Q811yescharacteristic
512C2×C4C2×C22⋊Q811yescharacteristic
513C2×C4C2×C22⋊Q811yescharacteristic
514C2×C4C2×C4⋊Q811yescharacteristic
515C2×C4C2×C4⋊Q811yescharacteristic
516C2×C4C2×C4⋊Q811yescharacteristic
517C2×C4C2×#13511yescharacteristic
518C2×C4C2×#13511yescharacteristic
519C2×C4#57511yescharacteristic
520C2×C4#57511yescharacteristic
521C2×C4#57611yescharacteristic
522C2×C4#57711yescharacteristic
523C2×C4#57711yescharacteristic
524C2×C4#57811yescharacteristic
525C8—42no—
526C8—42no—
527C8—42no—
528C8—42no—
529C8—42no—
530C8—42no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—41no—
535C22—41no—
536C22—41no—
537C22—41no—
538C22C2×#39511yescharacteristic
539C22#300211yescharacteristic
540C22#300211yescharacteristic
541C22#301211yescharacteristic
542C22#301311yescharacteristic
543C22#302111yescharacteristic
544C22#302211yescharacteristic
545C4—22no—
546C4—21no—
547C4—21no—
548C4—22no—
549C4—22no—
550C4—42no—
551C4—42no—
552C4—82no—
553C4—81no—
554C4—82no—
555C4—81no—
556C4—81no—
557C4—81no—
558C4—81no—
559C4—81no—
560C4—161no—
561C4—161no—
562C4#307611yescharacteristic
563C4#307611yescharacteristic
564C4#308111yescharacteristic
565C4#308211yescharacteristic
566C4#316011yescharacteristic
567C4#316211yescharacteristic
568C4#316311yescharacteristic
569C4#316411yescharacteristic
570C2—42no—
571C2#1889011yescharacteristic
572C2#1890011yescharacteristic
573C2#1891111yescharacteristic
574C2#1892111yescharacteristic
575C2#2355111yescharacteristic
576C2#2355711yescharacteristic
577C2#2356211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10578C2semidirect
6. #10578C2semidirect
7. #10577C2semidirect
8. #10576C2semidirect
9. #10575C2semidirect
10. #10575C2semidirect
11. #10574C2semidirect
12. #10573C2semidirect
83. #431Q8semidirect
84. #431Q8semidirect
98. C2×C4⋊C8Q8semidirect

Maximal subgroups

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

subgroupindex
#351922
#219842
#219622
#213622
#105782
#105772
#105762
#105752
#105742
#105732
#80392
#78942
#78912