Finite Group Databasebeta

← #241344

Subgroups — #241344

Order 512.

758 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#37624C211yesmaximal, characteristic
2#21984C211yesmaximal, characteristic
3#21964C211yesmaximal, characteristic
4#21962C211yesmaximal, characteristic
5#10582C211yesmaximal, characteristic, complemented
6#10581C211yesmaximal, characteristic, complemented
7#10580C211yesmaximal, characteristic, complemented
8#10579C211yesmaximal, characteristic, complemented
9#10577C211yesmaximal, characteristic, complemented
10#10576C211yesmaximal, characteristic, complemented
11#10575C211yesmaximal, characteristic, complemented
12#10575C211yesmaximal, characteristic, complemented
13#8045C211yesmaximal, characteristic
14#7899C211yesmaximal, characteristic
15#7893C211yesmaximal, characteristic
16C2×#431C2211yescharacteristic
17C2×#423C2211yescharacteristic
18C22×C4⋊C8C2211yescharacteristic
19#2841—22no—
20#2666—22no—
21#2660—21no—
22#2647—21no—
23#2629—21no—
24#2628—21no—
25#2571C2211yescharacteristic
26#2570—21no—
27#2552C2211yescharacteristic
28#2552C2211yescharacteristic
29#2551C2211yescharacteristic
30#2550—21no—
31#2441—21no—
32#2439—21no—
33#2285C2211yescharacteristic
34#2285C2211yescharacteristic
35#2285C2211yescharacteristic
36#2283C2211yescharacteristic
37#1949—21no—
38#1949—21no—
39#1949—22no—
40#1948—21no—
41#1948C2211yescharacteristic
42#1948C2211yescharacteristic
43#1948C2211yescharacteristic
44#1948C2211yescharacteristic
45#1947—21no—
46#1947C2211yescharacteristic
47#1947C2211yescharacteristic
48#1947C2211yescharacteristic
49#1947C2211yescharacteristic
50#1854—22no—
51#1845C2211yescharacteristic
52#1844C2211yescharacteristic
53#1834C2211yescharacteristic
54#1834C2211yescharacteristic
55#1809—21no—
56#1809—21no—
57#1808—21no—
58#1807—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
71C22×M16—21no—
72C23×C8—21no—
73C2×C4⋊Q8—21no—
74C2×C4⋊Q8—21no—
75C2×#125—21no—
76C2×#125—21no—
77C2×#117—21no—
78C2×#117—21no—
79C2×#117—21no—
80C22×C42C2311yescharacteristic
81#431—22no—
82#431Q814yescomplemented
83C2×#107—21no—
84C2×#107—21no—
85C2×#107—21no—
86C2×#107—21no—
87C2×#107—22no—
88C2×#106—21no—
89C2×#106—21no—
90C2×#106—21no—
91C2×#106—21no—
92C2×#106—22no—
93#423—22no—
94#423Q814yescomplemented
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—22no—
99C2×C4⋊C8Q814yescomplemented
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C22⋊C8—21no—
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107#400—22no—
108#400—22no—
109#400—22no—
110#399—22no—
111#398—22no—
112#398—22no—
113#398—22no—
114#398—22no—
115#398—22no—
116#398—22no—
117#396—22no—
118#396—22no—
119#395—22no—
120#395—22no—
121#395—22no—
122#393—22no—
123#391—22no—
124#389—22no—
125#389—21no—
126#389—21no—
127#389—21no—
128#389—21no—
129#389—21no—
130#389—21no—
131#389—21no—
132#389—21no—
133#387—22no—
134#387—22no—
135#384—21no—
136#384—21no—
137#384D411yescharacteristic
138#384D411yescharacteristic
139#384C2311yescharacteristic
140#384C2311yescharacteristic
141#384C2311yescharacteristic
142#384C2311yescharacteristic
143#383—21no—
144#382—21no—
145#382—21no—
146#382—21no—
147#382—21no—
148#382—21no—
149#382D411yescharacteristic
150#382D411yescharacteristic
151#381—21no—
152#380—21no—
153#380—21no—
154#380—21no—
155#380—21no—
156#380D411yescharacteristic
157#380D411yescharacteristic
158#380D411yescharacteristic
159#380D411yescharacteristic
160#380D412yes—
161C4×C4⋊C4—21no—
162C4×C4⋊C4—21no—
163C4×C4⋊C4—21no—
164C4×C4⋊C4—21no—
165C4×C4⋊C4D412yes—
166C4×C4⋊C4C2311yescharacteristic
167C4×C4⋊C4C2311yescharacteristic
168C4×C4⋊C4C2311yescharacteristic
169C4×C4⋊C4C2311yescharacteristic
170C4×C22⋊C4—21no—
171#376—21no—
172#340—21no—
173#340—21no—
174#340—21no—
175#340—21no—
176#340—21no—
177#340—21no—
178#340—21no—
179#340—21no—
180#340—22no—
181#340—22no—
182C22×Q8—22no—
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C2×M16—24no—
187C2×M16—81no—
188C2×M16—81no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—24no—
196C22×C8—81no—
197C22×C8—81no—
198C4⋊Q8—81no—
199C4⋊Q8—81no—
200C4⋊Q8—81no—
201C4⋊Q8—81no—
202#125—81no—
203#125—81no—
204#125—81no—
205#125—81no—
206#117—81no—
207#117—81no—
208#117—81no—
209#117—81no—
210#117—82no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
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—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4C2×D411yescharacteristic
235C2×C4⋊C4C2×D411yescharacteristic
236C2×C4⋊C4C2×D411yescharacteristic
237C2×C4⋊C4C2×D411yescharacteristic
238C2×C4⋊C4C2×D411yescharacteristic
239C2×C4⋊C4C2×D411yescharacteristic
240C2×C4⋊C4C2×D411yescharacteristic
241C2×C4⋊C4C2×D411yescharacteristic
242C2×C4⋊C4C2×D412yes—
243C2×C4⋊C4C2×D412yes—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—41no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—22no—
255C2×C42—41no—
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42C2×Q812yes—
260C2×C42C2×Q812yes—
261C2×C42C2×Q812yes—
262C2×C42#4112yes—
263C2×C42C2411yescharacteristic
264#107—81no—
265#107—81no—
266#107—81no—
267#107—81no—
268#107—82no—
269#106—81no—
270#106—81no—
271#106—81no—
272#106—81no—
273#106—82no—
274C4⋊C8—24no—
275C4⋊C8—24no—
276C4⋊C8—24no—
277C4⋊C8—21no—
278C4⋊C8—21no—
279C4⋊C8—21no—
280C4⋊C8—21no—
281C4⋊C8C2×Q812yes—
282C4⋊C8C2×Q812yes—
283C4⋊C8C2×Q812yes—
284C4⋊C8C2×Q812yes—
285C22⋊C8—24no—
286C4×C8—21no—
287C4×C8—21no—
288C4×C8C2×Q812yes—
289C4×C8C2×Q812yes—
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—82no—
314C2×Q8—82no—
315C2×Q8—82no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
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—22no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—21no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4#12011yescharacteristic
339C22×C4#12011yescharacteristic
340C22×C4#12011yescharacteristic
341C22×C4C22⋊Q812yes—
342C22×C4C22⋊Q812yes—
343C22×C4C22⋊Q812yes—
344C22×C4C4⋊Q812yes—
345C22×C4C4⋊Q812yes—
346C22×C4C4⋊Q812yes—
347C22×C4C22×D411yescharacteristic
348C22×C4C22×D411yescharacteristic
349C22×C4C22×D411yescharacteristic
350M16—84no—
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—22no—
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—22no—
357C2×C8—41no—
358C2×C8—41no—
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—84no—
370C4⋊C4—21no—
371C4⋊C4—21no—
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—42no—
389C4⋊C4—42no—
390C4⋊C4—41no—
391C4⋊C4—41no—
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—81no—
407C4⋊C4—81no—
408C4⋊C4—81no—
409C4⋊C4—81no—
410C4⋊C4—82no—
411C4⋊C4—82no—
412C4⋊C4—82no—
413C4⋊C4—82no—
414C4⋊C4—82no—
415C4⋊C4—82no—
416C4⋊C4—82no—
417C4⋊C4—82no—
418C4⋊C4—82no—
419C22⋊C4—81no—
420C22⋊C4—81no—
421C22⋊C4—81no—
422C22⋊C4—81no—
423C22⋊C4—82no—
424C22⋊C4—82no—
425C42—41no—
426C42—41no—
427C42—41no—
428C42—41no—
429C42—41no—
430C42—41no—
431C42—41no—
432C42—41no—
433C42—82no—
434C42—82no—
435C42C22⋊Q812yes—
436C42C22⋊Q812yes—
437C42C22⋊Q812yes—
438C42C4⋊Q812yes—
439C42C4⋊Q812yes—
440C42C4⋊Q812yes—
441C42C22×Q811yescharacteristic
442C42C22×Q811yescharacteristic
443C42C22×Q811yescharacteristic
444C42C2×#4111yescharacteristic
445C23—21no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23#39512yes—
452C23C2×#12011yescharacteristic
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—42no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—41no—
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×C4#57611yescharacteristic
518C2×C4#57611yescharacteristic
519C2×C4#57711yescharacteristic
520C2×C4#57711yescharacteristic
521C2×C4#57711yescharacteristic
522C2×C4#57711yescharacteristic
523C2×C4#57811yescharacteristic
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#301211yescharacteristic
540C22#301211yescharacteristic
541C22#301311yescharacteristic
542C22#301311yescharacteristic
543C22#302111yescharacteristic
544C22#302211yescharacteristic
545C4—22no—
546C4—22no—
547C4—21no—
548C4—21no—
549C4—22no—
550C4—81no—
551C4—82no—
552C4—81no—
553C4—81no—
554C4—81no—
555C4—81no—
556C4—82no—
557C4—81no—
558C4—81no—
559C4—81no—
560C4—161no—
561C4—161no—
562C4#308111yescharacteristic
563C4#308111yescharacteristic
564C4#308211yescharacteristic
565C4#308211yescharacteristic
566C4#316311yescharacteristic
567C4#316311yescharacteristic
568C4#316411yescharacteristic
569C4#316411yescharacteristic
570C2—42no—
571C2#1896411yescharacteristic
572C2#1896511yescharacteristic
573C2#1897411yescharacteristic
574C2#1897511yescharacteristic
575C2#2355711yescharacteristic
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. #10582C2semidirect
6. #10581C2semidirect
7. #10580C2semidirect
8. #10579C2semidirect
9. #10577C2semidirect
10. #10576C2semidirect
11. #10575C2semidirect
12. #10575C2semidirect
82. #431Q8semidirect
94. #423Q8semidirect
99. C2×C4⋊C8Q8semidirect

Maximal subgroups

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

subgroupindex
#376242
#219842
#219642
#219622
#105822
#105812
#105802
#105792
#105772
#105762
#105752
#80452
#78992
#78932