Finite Group Databasebeta

← #6806111

Subgroups — #6806111

Order 512.

1500 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36598C211yesmaximal, characteristic, complemented
2Q8×#124C211yesmaximal, characteristic
3#33236C211yesmaximal, characteristic, complemented
4#31663C214yesmaximal
5#31659C214yesmaximal, complemented
6#30215C212yesmaximal, complemented
7#29224C211yesmaximal, characteristic, complemented
8#29147C212yesmaximal, complemented
9#28620C212yesmaximal
10#28449C211yesmaximal, characteristic
11#25936C212yesmaximal, complemented
12#25929C212yesmaximal
13#25323C212yesmaximal, complemented
14#25319C212yesmaximal
15#22273C212yesmaximal
16C4×#489C211yesmaximal, characteristic
17#21345C211yesmaximal, characteristic, complemented
18C2×Q82C2211yescharacteristic
19C2×D4×Q8C2211yescharacteristic
20C2×#497C2212yes—
21C2×#489C2211yescharacteristic
22C2×#487C2212yes—
23C2×#487C2212yes—
24C2×#462C2212yes—
25C2×#461C2212yes—
26#2868C2211yescharacteristic
27#2862C2212yes—
28#2862C2212yes—
29#2862C2211yescharacteristic
30#2582—21no—
31#2571—21no—
32#2570—22no—
33#2568C2212yes—
34#2568C2212yes—
35#2568C2211yescharacteristic
36#2563C2212yes—
37#2563C2211yescharacteristic
38#2511—22no—
39#2492—22no—
40#2492—21no—
41#2492—21no—
42#2492—22no—
43#2446C2214yes—
44#2365C2212yes—
45#2355—21no—
46#2355—21no—
47Q8×C22⋊C4—21no—
48Q8×C22⋊C4—21no—
49C4×C4⋊Q8C2211yescharacteristic
50C4×C4⋊Q8C2212yes—
51C4×#124C2212yes—
52C4×#124C2211yescharacteristic
53C42×Q8C2211yescharacteristic
54#2101C2214yes—
55#2074C2214yes—
56#2074C2214yes—
57#2056C2214yes—
58#2055C2214yes—
59#2024C2214yes—
60#2024C2214yes—
61#2024C2214yes—
62#2024C2214yes—
63#2016C2212yes—
64#2013C2212yes—
65#1993C2211yescharacteristic
66#1993C2212yes—
67#1988C2212yes—
68#1987C2212yes—
69#1984C2212yes—
70#1960C2212yes—
71#1960C2212yes—
72#1960C2214yes—
73#1960C2212yes—
74#1959C2211yescharacteristic
75#1939C2212yes—
76#1936C2212yes—
77#1935C2212yes—
78#1917C2214yes—
79#1880C2214yes—
80#1878C2214yes—
81#1878C2214yes—
82#1878C2214yes—
83#1877C2214yes—
84#1877C2214yes—
85#1836C2212yes—
86#1836C2211yescharacteristic
87#1805C2214yes—
88#1805C2214yes—
89#1767C2211yescharacteristic
90#1767C2212yes—
91C4×#103C2212yes—
92C4×#102C2212yes—
93#1745C2212yes—
94C4×#97C2211yescharacteristic
95C23×Q8—21no—
96C22×Q16C2314yes—
97C22×SD16C2314yes—
98Q82—44no—
99D4×Q8—44no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2311yescharacteristic
108C2×C4⋊Q8C2312yes—
109C2×C4⋊Q8C2312yes—
110C2×#124C2311yescharacteristic
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8—22no—
113C2×C4×Q8—21no—
114C2×C4×Q8—21no—
115C2×C4×Q8D411yescharacteristic
116C2×C4×Q8D411yescharacteristic
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2314yes—
119C2×C4×Q8C2312yes—
120C2×C4×Q8C2311yescharacteristic
121C2×C4×D4C2311yescharacteristic
122C2×C4×D4C2312yes—
123#497—28no—
124#489—24no—
125#487—44no—
126#487—44no—
127#462—44no—
128#461—44no—
129C2×C4⋊C8C2312yes—
130C2×C4⋊C8C2314yes—
131C2×#103C2312yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2312yes—
139C2×#102C2312yes—
140C2×#102C2312yes—
141C2×#97C2311yescharacteristic
142C2×#97C2312yes—
143C2×#97C2311yescharacteristic
144C2×#97C2312yes—
145C2×C4×C8C2312yes—
146#400—24no—
147#400—22no—
148#400—22no—
149#400—22no—
150#400—24no—
151#398—24no—
152#393—22no—
153#393—22no—
154#393—22no—
155#391—22no—
156#391—21no—
157#391—22no—
158#391—21no—
159#391—22no—
160#391—21no—
161#391—22no—
162#391—21no—
163#391—22no—
164#391—21no—
165#391—22no—
166#391—22no—
167#391—22no—
168#391—22no—
169#391—21no—
170#391—21no—
171#391C2314yes—
172#391C2314yes—
173#391C2312yes—
174#391C2311yescharacteristic
175#391C2314yes—
176#390C2312yes—
177#390C2311yescharacteristic
178#389—22no—
179#389—21no—
180#389—22no—
181#389—22no—
182#389C2312yes—
183#388—22no—
184#388—22no—
185#388—21no—
186#388—21no—
187#387—24no—
188#387—24no—
189#387—24no—
190#384—21no—
191#384—21no—
192#384—21no—
193#384—21no—
194#384C2311yescharacteristic
195#382—22no—
196#382—22no—
197#382—22no—
198#382C2312yes—
199#382C2312yes—
200#382C2312yes—
201#382C2312yes—
202#382C2311yescharacteristic
203#380—21no—
204#380—22no—
205#380—21no—
206#380—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4D412yes—
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4D412yes—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2312yes—
218C4×C22⋊C4—21no—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—21no—
221C4×C22⋊C4—22no—
222C43C2312yes—
223C43C2311yescharacteristic
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#336C2314yes—
233#336C2312yes—
234#336C2314yes—
235#336C2314yes—
236C22×Q8—21no—
237C22×Q8—22no—
238C22×Q8—21no—
239C22×Q8—22no—
240C22×Q8—21no—
241C22×Q8—44no—
242C22×Q8—44no—
243C22×Q8—44no—
244C22×Q8C2×D411yescharacteristic
245C22×Q8C2414yes—
246C22×Q8C2411yescharacteristic
247C22×D4C2411yescharacteristic
248C23×C4—22no—
249C23×C4—21no—
250C2×Q16—28no—
251C2×Q16—44no—
252C2×Q16—44no—
253C2×SD16—28no—
254C2×SD16—44no—
255C2×SD16—44no—
256C22×C8C2414yes—
257C22×C8C2412yes—
258C22×C8C2412yes—
259C4⋊Q8—22no—
260C4⋊Q8—24no—
261C4⋊Q8—24no—
262C4⋊Q8—24no—
263C4⋊Q8—24no—
264C4⋊Q8—24no—
265C4⋊Q8—22no—
266C4⋊Q8—24no—
267C4⋊Q8—44no—
268C4⋊Q8—44no—
269C4⋊Q8—44no—
270C4⋊Q8—48no—
271C4⋊Q8—44no—
272#124—22no—
273#124#4114yes—
274C22⋊Q8—44no—
275C22⋊Q8—48no—
276C4×Q8—24no—
277C4×Q8—24no—
278C4×Q8—24no—
279C4×Q8—42no—
280C4×Q8—44no—
281C4×Q8—44no—
282C4×Q8—44no—
283C4×Q8—44no—
284C4×Q8—44no—
285C4×D4—24no—
286C4×D4—44no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2×D412yes—
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2×D412yes—
309C2×C4⋊C4C2×D412yes—
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2411yescharacteristic
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—21no—
324C2×C42—22no—
325C2×C42—21no—
326C2×C42—22no—
327C2×C42—24no—
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D412yes—
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D412yes—
332C2×C42C2×D412yes—
333C2×C42C2×D412yes—
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2412yes—
337C2×C42C2412yes—
338C2×C42C2411yescharacteristic
339C2×C42C2411yescharacteristic
340C2×C42C2411yescharacteristic
341C4⋊C8—24no—
342C4⋊C8—44no—
343#103—24no—
344#103—42no—
345#103—44no—
346#103—44no—
347#103—44no—
348#103—44no—
349#103—44no—
350#103—44no—
351#102—24no—
352#102—42no—
353#97—22no—
354#97—24no—
355#97—22no—
356#97—42no—
357C4×C8—24no—
358#95—22no—
359#95—21no—
360#95—21no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—21no—
365#95—21no—
366#95—24no—
367#95—24no—
368#95—22no—
369#95—24no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—24no—
378#95—24no—
379#95—24no—
380#95C2×D412yes—
381#95C2×D412yes—
382#95C2×D412yes—
383C24—21no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—21no—
387C2×Q8—24no—
388C2×Q8—22no—
389C2×Q8—24no—
390C2×Q8—24no—
391C2×Q8—24no—
392C2×Q8—22no—
393C2×Q8—24no—
394C2×Q8—24no—
395C2×Q8—42no—
396C2×Q8—44no—
397C2×Q8—42no—
398C2×Q8—44no—
399C2×Q8—44no—
400C2×Q8—42no—
401C2×Q8—42no—
402C2×Q8—42no—
403C2×Q8—44no—
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8—81no—
407C2×Q8—81no—
408C2×Q8—82no—
409C2×Q8—82no—
410C2×Q8—816no—
411C2×Q8#13714yes—
412C2×Q8C2×#4112yes—
413C2×D4—22no—
414C2×D4—21no—
415C2×D4—22no—
416C2×D4C2×#4112yes—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—24no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—42no—
428C22×C4—44no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4C22×D412yes—
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D412yes—
440C22×C4C22×D411yescharacteristic
441C22×C4C2511yescharacteristic
442Q16—88no—
443SD16—88no—
444C2×C8—22no—
445C2×C8—24no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—24no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—24no—
453C4⋊C4—24no—
454C4⋊C4—22no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—24no—
461C4⋊C4—22no—
462C4⋊C4—24no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—42no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—84no—
482C4⋊C4—82no—
483C4⋊C4—82no—
484C22⋊C4—24no—
485C22⋊C4—24no—
486C22⋊C4—44no—
487C22⋊C4—42no—
488C22⋊C4—82no—
489C22⋊C4—81no—
490C22⋊C4—82no—
491C22⋊C4—81no—
492C42—24no—
493C42—22no—
494C42—22no—
495C42—24no—
496C42—22no—
497C42—24no—
498C42—41no—
499C42—42no—
500C42—41no—
501C42—42no—
502C42—42no—
503C42—44no—
504C42—42no—
505C42—44no—
506C42—84no—
507C42C2×#4112yes—
508C42H(5,𝔽2)14yes—
509C422−1+412yes—
510C23—22no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23C23×D411yescharacteristic
518Q8—28no—
519Q8—24no—
520Q8—42no—
521Q8—44no—
522Q8—44no—
523Q8—44no—
524Q8—44no—
525Q8—84no—
526Q8—88no—
527Q8—84no—
528Q8—162no—
529Q8—164no—
530D4—24no—
531D4—42no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—88no—
580C2×C4—84no—
581C2×C4#53512yes—
582C2×C4#53512yes—
583C2×C4#54812yes—
584C2×C4#54812yes—
585C2×C4C2×#13714yes—
586C2×C4C2×#13712yes—
587C2×C4#57512yes—
588C2×C4C22×#4111yescharacteristic
589C2×C4C2×H(5,𝔽2)12yes—
590C2×C4C2×2−1+411yescharacteristic
591C2×C4#58512yes—
592C8—42no—
593C8—44no—
594C8—42no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—81no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22C2×#53511yescharacteristic
603C22C2×#54811yescharacteristic
604C22#350011yescharacteristic
605C22C22×#13711yescharacteristic
606C22C2×#57511yescharacteristic
607C22#358611yescharacteristic
608C22#358611yescharacteristic
609C4—24no—
610C4—22no—
611C4—22no—
612C4—22no—
613C4—42no—
614C4—41no—
615C4—42no—
616C4—42no—
617C4—41no—
618C4—42no—
619C4—84no—
620C4—82no—
621C4—81no—
622C4—82no—
623C4—82no—
624C4—82no—
625C4—81no—
626C4#321514yes—
627C4#351412yes—
628C4#355412yes—
629C2—82no—
630C2#3917111yescharacteristic
631C2C2×#321511yescharacteristic
632C2#6082611yescharacteristic
633C2#6132211yescharacteristic
634C2#6132311yescharacteristic
635C2#6258111yescharacteristic
636C2#6258911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36598C2semidirect
3. #33236C2semidirect
5. #31659C2semidirect
6. #30215C2semidirect
7. #29224C2semidirect
8. #29147C2semidirect
11. #25936C2semidirect
13. #25323C2semidirect
17. #21345C2semidirect

Maximal subgroups

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

subgroupindex
#365982
Q8×#1242
#332362
#316632
#316592
#302152
#292242
#291472
#286202
#284492
#259362
#259292
#253232
#253192
#222732
C4×#4892
#213452