Finite Group Databasebeta

← #217095

Subgroups — #217095

Order 512.

767 conjugacy classes of subgroups.

The subgroup lattice has 609 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35283C211yesmaximal, characteristic
2#21978C211yesmaximal, characteristic
3#20812C211yesmaximal, characteristic
4#9905C211yesmaximal, characteristic, complemented
5#9898C211yesmaximal, characteristic, complemented
6#9701C211yesmaximal, characteristic, complemented
7#9693C211yesmaximal, characteristic, complemented
8#9140C211yesmaximal, characteristic
9#9139C211yesmaximal, characteristic
10#9136C211yesmaximal, characteristic
11#9135C211yesmaximal, characteristic
12#8591C211yesmaximal, characteristic, complemented
13#8590C211yesmaximal, characteristic, complemented
14#8587C211yesmaximal, characteristic, complemented
15#8586C211yesmaximal, characteristic, complemented
16C2×#433—21no—
17C2×#432C2211yescharacteristic
18C2×#423C2211yescharacteristic
19#2868C2211yescharacteristic
20#2862C2211yescharacteristic
21#2857C2211yescharacteristic
22#2857C2211yescharacteristic
23#2647—21no—
24#2628—21no—
25#2511—21no—
26#2511C412yes—
27#2492—21no—
28#2492C412yes—
29#2451—21no—
30#2451C412yes—
31#2423—21no—
32#2423C412yes—
33#2331—21no—
34#2314—21no—
35C4×C4⋊Q8C2211yescharacteristic
36C4×C4⋊Q8C2211yescharacteristic
37C4×#117C2211yescharacteristic
38#1940—21no—
39#1940—21no—
40#1851—21no—
41#1851—21no—
42#1842C2211yescharacteristic
43#1842C2211yescharacteristic
44#1842C2211yescharacteristic
45#1842C2211yescharacteristic
46#1832—21no—
47#1831C2211yescharacteristic
48#1831C2211yescharacteristic
49#1798—21no—
50#1797C2211yescharacteristic
51#1797C2211yescharacteristic
52#1768—21no—
53#1767C2211yescharacteristic
54C4×C4⋊C8C2211yescharacteristic
55C2×#333C2211yescharacteristic
56C2×#333C2211yescharacteristic
57C2×#333C2211yescharacteristic
58C2×#333C2211yescharacteristic
59C2×#332C2211yescharacteristic
60C2×#332C2211yescharacteristic
61C2×#332C2211yescharacteristic
62C2×#332C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67#1296C2211yescharacteristic
68#1296C2211yescharacteristic
69#1296C2211yescharacteristic
70#1296C2211yescharacteristic
71C22×M16—21no—
72C2×C4⋊Q8—21no—
73C2×C4⋊Q8—21no—
74C2×C4⋊Q8C2×C412yes—
75C2×C4⋊Q8C2311yescharacteristic
76C2×C4⋊Q8C2311yescharacteristic
77C2×C22⋊Q8—21no—
78C2×C22⋊Q8C2×C412yes—
79C2×C4×Q8—21no—
80C2×C4×Q8C2×C412yes—
81C2×#117D411yescharacteristic
82C2×#117D411yescharacteristic
83C22×C42C2311yescharacteristic
84#433—44no—
85#432—41no—
86#432—41no—
87#423—41no—
88#423—41no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×C22⋊C8—21no—
107C2×C22⋊C8—21no—
108C2×C22⋊C8—21no—
109C2×#97—21no—
110C2×#97—21no—
111C2×C4×C8—21no—
112#400—22no—
113#400—22no—
114#400—22no—
115#399—22no—
116#398—22no—
117#398—22no—
118#395—22no—
119#393—22no—
120#391—21no—
121#391—21no—
122#391—21no—
123#391—21no—
124#391—22no—
125#391—22no—
126#391C2×C412yes—
127#391C2×C412yes—
128#391C2×C412yes—
129#391C2×C412yes—
130#389—21no—
131#389—21no—
132#389—21no—
133#389—21no—
134#389—21no—
135#389—21no—
136#389C2×C412yes—
137#389C2×C412yes—
138#388—22no—
139#388—22no—
140#387—22no—
141#387—22no—
142#387—22no—
143#387—22no—
144#384—21no—
145#384—21no—
146#384C2×C412yes—
147#384C2311yescharacteristic
148#384C2311yescharacteristic
149#382—21no—
150#382—21no—
151#380—21no—
152#380—21no—
153#380—21no—
154#380C2×C412yes—
155C4×C4⋊C4—21no—
156C4×C4⋊C4—21no—
157C4×C4⋊C4—21no—
158C4×C4⋊C4C2×C412yes—
159C4×C4⋊C4C2×C411yescharacteristic
160C4×C4⋊C4C2×C411yescharacteristic
161C4×C4⋊C4D411yescharacteristic
162C4×C4⋊C4D411yescharacteristic
163C4×C22⋊C4C2×C411yescharacteristic
164C4×C22⋊C4C2×C411yescharacteristic
165C4×C22⋊C4D411yescharacteristic
166C4×C22⋊C4D411yescharacteristic
167#376D411yescharacteristic
168#376D411yescharacteristic
169C43C2311yescharacteristic
170C43C2311yescharacteristic
171#336—21no—
172#336—21no—
173#336—21no—
174#336—21no—
175#336—21no—
176#336—21no—
177#336—21no—
178#336—21no—
179#333—41no—
180#333—41no—
181#333—41no—
182#333—41no—
183#332—41no—
184#332—41no—
185#332—41no—
186#332—41no—
187C22×Q8—22no—
188C22×Q8—21no—
189C22×Q8C22×C412yes—
190C23×C4C22×C411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C2×M16—42no—
194C2×M16—81no—
195C2×M16—81no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C4⋊Q8—21no—
205C4⋊Q8—21no—
206C4⋊Q8—21no—
207C4⋊Q8—21no—
208C4⋊Q8—21no—
209C4⋊Q8—21no—
210C4⋊Q8—21no—
211C4⋊Q8—21no—
212C4⋊Q8—42no—
213C4⋊Q8—42no—
214C4⋊Q8—81no—
215C4⋊Q8—81no—
216C4⋊Q8—82no—
217C22⋊Q8—81no—
218C22⋊Q8—81no—
219C22⋊Q8—82no—
220C4×Q8—81no—
221C4×Q8—81no—
222C4×Q8—82no—
223#117—22no—
224#117—22no—
225#117—22no—
226#117—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4C22⋊C411yescharacteristic
242C2×C4⋊C4C22⋊C411yescharacteristic
243C2×C4⋊C4C22⋊C411yescharacteristic
244C2×C4⋊C4C22⋊C411yescharacteristic
245C2×C4⋊C4C22×C412yes—
246C2×C4⋊C4C22×C412yes—
247C2×C4⋊C4C22×C412yes—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4C22⋊C411yescharacteristic
251C2×C22⋊C4C22⋊C411yescharacteristic
252C2×C22⋊C4C22⋊C411yescharacteristic
253C2×C22⋊C4C22⋊C411yescharacteristic
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42C22×C411yescharacteristic
261C2×C42C22×C411yescharacteristic
262C2×C42C22×C411yescharacteristic
263C2×C42C22×C411yescharacteristic
264C2×C42C22×C411yescharacteristic
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42C2411yescharacteristic
276C4⋊C8—21no—
277C4⋊C8—21no—
278C4⋊C8—21no—
279C4⋊C8—21no—
280C4⋊C8—21no—
281C4⋊C8—21no—
282C4⋊C8—21no—
283C4⋊C8—21no—
284C4⋊C8—21no—
285C4⋊C8—21no—
286C4⋊C8—21no—
287C4⋊C8—21no—
288C4⋊C8—21no—
289C4⋊C8—21no—
290C4⋊C8—21no—
291C4⋊C8—21no—
292C4⋊C8—42no—
293C4⋊C8—42no—
294C4⋊C8—42no—
295C4⋊C8—42no—
296C4⋊C8—42no—
297C4⋊C8—81no—
298C4⋊C8—81no—
299C4⋊C8—81no—
300C4⋊C8—81no—
301C22⋊C8—42no—
302C22⋊C8—42no—
303C22⋊C8—42no—
304#97—42no—
305#97—42no—
306C4×C8—42no—
307#95—22no—
308#95—22no—
309#95—22no—
310#95—22no—
311#95—22no—
312#95—22no—
313#95—22no—
314#95—22no—
315#95—22no—
316#95—22no—
317#95—22no—
318#95—22no—
319#95C22⋊C411yescharacteristic
320#95C22⋊C411yescharacteristic
321#95C22⋊C411yescharacteristic
322#95C22⋊C411yescharacteristic
323#95C22⋊C411yescharacteristic
324#95C22⋊C411yescharacteristic
325#95C22⋊C411yescharacteristic
326#95C22⋊C411yescharacteristic
327C24C2×C22⋊C411yescharacteristic
328C2×Q8—22no—
329C2×Q8—22no—
330C2×Q8—42no—
331C2×Q8—42no—
332C2×Q8—41no—
333C2×Q8—41no—
334C2×Q8—41no—
335C2×Q8—41no—
336C2×Q8—41no—
337C2×Q8—41no—
338C2×Q8—44no—
339C2×Q8—82no—
340C2×Q8—82no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4C2×C22⋊C411yescharacteristic
362C22×C4C2×C22⋊C411yescharacteristic
363C22×C4C2×C22⋊C411yescharacteristic
364C22×C4C2×C22⋊C411yescharacteristic
365C22×C4C2×C22⋊C411yescharacteristic
366C22×C4C2×C22⋊C411yescharacteristic
367C22×C4C2×C22⋊C411yescharacteristic
368C22×C4C2×C22⋊C411yescharacteristic
369C22×C4C2×C22⋊C411yescharacteristic
370C22×C4C2×C22⋊C411yescharacteristic
371C22×C4C2×C22⋊C411yescharacteristic
372C22×C4C23×C411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C22×D411yescharacteristic
375M16—84no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—41no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—44no—
423C4⋊C4—82no—
424C4⋊C4—82no—
425C4⋊C4—82no—
426C4⋊C4—82no—
427C4⋊C4—82no—
428C4⋊C4—82no—
429C22⋊C4—22no—
430C22⋊C4—22no—
431C22⋊C4—22no—
432C22⋊C4—22no—
433C22⋊C4—82no—
434C22⋊C4—82no—
435C42—21no—
436C42—21no—
437C42—21no—
438C42—21no—
439C42—21no—
440C42—21no—
441C42—21no—
442C42—21no—
443C42—21no—
444C42—21no—
445C42—21no—
446C42—21no—
447C42—21no—
448C42—21no—
449C42—21no—
450C42—21no—
451C42—21no—
452C42—21no—
453C42—41no—
454C42—41no—
455C42—41no—
456C42—41no—
457C42—82no—
458C42—82no—
459C42AGL(1,ℤ/8)12yes—
460C42#13712yes—
461C42#13712yes—
462C42#13712yes—
463C42H(5,𝔽2)11yescharacteristic
464C42H(5,𝔽2)11yescharacteristic
465C422−1+411yescharacteristic
466C422−1+411yescharacteristic
467C23—21no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23C22×C22⋊C411yescharacteristic
475Q8—82no—
476Q8—82no—
477Q8—81no—
478Q8—81no—
479Q8—81no—
480Q8—81no—
481Q8—164no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4#41712yes—
536C2×C4#41712yes—
537C2×C4#41912yes—
538C2×C4#41912yes—
539C2×C4#52011yescharacteristic
540C2×C4#52011yescharacteristic
541C2×C4#53511yescharacteristic
542C2×C4#53611yescharacteristic
543C2×C4#53711yescharacteristic
544C2×C4#53711yescharacteristic
545C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
546C2×C4C2×#13711yescharacteristic
547C2×C4C2×#13711yescharacteristic
548C2×C4C2×#13711yescharacteristic
549C2×C4#57611yescharacteristic
550C2×C4#57711yescharacteristic
551C2×C4#57711yescharacteristic
552C2×C4#57811yescharacteristic
553C8—81no—
554C8—81no—
555C8—81no—
556C8—81no—
557C8—81no—
558C8—81no—
559C8—81no—
560C8—81no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22#231111yescharacteristic
569C22#288811yescharacteristic
570C22#288911yescharacteristic
571C22C2×#41911yescharacteristic
572C22C2×#41711yescharacteristic
573C22#289711yescharacteristic
574C22#289811yescharacteristic
575C4—22no—
576C4—22no—
577C4—22no—
578C4—22no—
579C4—21no—
580C4—21no—
581C4—41no—
582C4—41no—
583C4—41no—
584C4—41no—
585C4—41no—
586C4—41no—
587C4—41no—
588C4—82no—
589C4—82no—
590C4—82no—
591C4—82no—
592C4#322411yescharacteristic
593C4#322511yescharacteristic
594C4#324311yescharacteristic
595C4#324411yescharacteristic
596C4#325411yescharacteristic
597C4#325411yescharacteristic
598C4#325511yescharacteristic
599C4#325511yescharacteristic
600C2—81no—
601C2#1776011yescharacteristic
602C2#1776111yescharacteristic
603C2#1776211yescharacteristic
604C2#1776311yescharacteristic
605C2#2187111yescharacteristic
606C2#2188111yescharacteristic
607C2#2189411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9905C2semidirect
5. #9898C2semidirect
6. #9701C2semidirect
7. #9693C2semidirect
12. #8591C2semidirect
13. #8590C2semidirect
14. #8587C2semidirect
15. #8586C2semidirect

Maximal subgroups

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

subgroupindex
#352832
#219782
#208122
#99052
#98982
#97012
#96932
#91402
#91392
#91362
#91352
#85912
#85902
#85872
#85862