Finite Group Databasebeta

← #216947

Subgroups — #216947

Order 512.

814 conjugacy classes of subgroups.

The subgroup lattice has 615 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35283C211yesmaximal, characteristic
2#21922C211yesmaximal, characteristic
3#20793C211yesmaximal, characteristic
4#9898C211yesmaximal, characteristic, complemented
5#9896C211yesmaximal, characteristic, complemented
6#9697C211yesmaximal, characteristic, complemented
7#9697C211yesmaximal, characteristic, complemented
8#9137C211yesmaximal, characteristic
9#9135C211yesmaximal, characteristic
10#9129C211yesmaximal, characteristic
11#9127C211yesmaximal, characteristic
12#8591C211yesmaximal, characteristic, complemented
13#8590C211yesmaximal, characteristic, complemented
14#8587C211yesmaximal, characteristic, complemented
15#8586C211yesmaximal, characteristic, complemented
16C2×#424—21no—
17C2×#424—21no—
18C2×#423C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20#2868C2211yescharacteristic
21#2862C2211yescharacteristic
22#2857C2211yescharacteristic
23#2857C2211yescharacteristic
24#2647—21no—
25#2628—21no—
26#2511—21no—
27#2511C412yes—
28#2492—21no—
29#2492C412yes—
30#2451—21no—
31#2451C412yes—
32#2423—21no—
33#2423C412yes—
34#2331—21no—
35#2314—21no—
36C4×C4⋊Q8C2211yescharacteristic
37C4×C4⋊Q8C2211yescharacteristic
38C4×#117C2211yescharacteristic
39#1923—21no—
40#1923—21no—
41#1916—21no—
42#1916—21no—
43#1843—21no—
44#1843—21no—
45#1842C2211yescharacteristic
46#1842C2211yescharacteristic
47#1842C2211yescharacteristic
48#1842C2211yescharacteristic
49#1842C2211yescharacteristic
50#1831C2211yescharacteristic
51#1797C2211yescharacteristic
52#1797C2211yescharacteristic
53C4×C4⋊C8C2211yescharacteristic
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—
72C23×C8—21no—
73C2×C4⋊Q8—21no—
74C2×C4⋊Q8—21no—
75C2×C4⋊Q8C2×C412yes—
76C2×C4⋊Q8C2311yescharacteristic
77C2×C4⋊Q8C2311yescharacteristic
78C2×C22⋊Q8—21no—
79C2×C22⋊Q8C2×C412yes—
80C2×C4×Q8—21no—
81C2×C4×Q8C2×C412yes—
82C2×#117D411yescharacteristic
83C2×#117D411yescharacteristic
84C22×C42C2311yescharacteristic
85#424—44no—
86#424—44no—
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⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—41no—
102C2×C4⋊C8—41no—
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×C4⋊C8C2311yescharacteristic
107C2×C4⋊C8C2311yescharacteristic
108C2×C4⋊C8C2311yescharacteristic
109C2×C4⋊C8C2311yescharacteristic
110C2×C4⋊C8C2311yescharacteristic
111C2×C22⋊C8—21no—
112C2×C22⋊C8—21no—
113C2×C4×C8—21no—
114C2×C4×C8—21no—
115#400—22no—
116#400—22no—
117#400—22no—
118#399—22no—
119#398—22no—
120#398—22no—
121#395—22no—
122#393—22no—
123#391—21no—
124#391—21no—
125#391—21no—
126#391—21no—
127#391—22no—
128#391—22no—
129#391C2×C412yes—
130#391C2×C412yes—
131#391C2×C412yes—
132#391C2×C412yes—
133#389—21no—
134#389—21no—
135#389—21no—
136#389—21no—
137#389—21no—
138#389—21no—
139#389C2×C412yes—
140#389C2×C412yes—
141#388—22no—
142#388—22no—
143#387—22no—
144#387—22no—
145#387—22no—
146#387—22no—
147#384—21no—
148#384—21no—
149#384C2×C412yes—
150#384C2311yescharacteristic
151#384C2311yescharacteristic
152#382—21no—
153#382—21no—
154#380—21no—
155#380—21no—
156#380—21no—
157#380C2×C412yes—
158C4×C4⋊C4—21no—
159C4×C4⋊C4—21no—
160C4×C4⋊C4—21no—
161C4×C4⋊C4C2×C412yes—
162C4×C4⋊C4C2×C411yescharacteristic
163C4×C4⋊C4C2×C411yescharacteristic
164C4×C4⋊C4D411yescharacteristic
165C4×C4⋊C4D411yescharacteristic
166C4×C22⋊C4C2×C411yescharacteristic
167C4×C22⋊C4C2×C411yescharacteristic
168C4×C22⋊C4D411yescharacteristic
169C4×C22⋊C4D411yescharacteristic
170#376D411yescharacteristic
171#376D411yescharacteristic
172C43C2311yescharacteristic
173C43C2311yescharacteristic
174#336—21no—
175#336—21no—
176#336—21no—
177#336—21no—
178#336—21no—
179#336—21no—
180#333—41no—
181#333—41no—
182#333—41no—
183#333—41no—
184#332—41no—
185#332—41no—
186#332—41no—
187#332—41no—
188C22×Q8—22no—
189C22×Q8—21no—
190C22×Q8C22×C412yes—
191C23×C4C22×C411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C2×M16—42no—
195C2×M16—42no—
196C2×M16—42no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—42no—
206C22×C8—42no—
207C22×C8—42no—
208C4⋊Q8—21no—
209C4⋊Q8—21no—
210C4⋊Q8—21no—
211C4⋊Q8—21no—
212C4⋊Q8—21no—
213C4⋊Q8—21no—
214C4⋊Q8—21no—
215C4⋊Q8—21no—
216C4⋊Q8—42no—
217C4⋊Q8—42no—
218C4⋊Q8—81no—
219C4⋊Q8—81no—
220C4⋊Q8—82no—
221C22⋊Q8—81no—
222C22⋊Q8—81no—
223C22⋊Q8—82no—
224C4×Q8—81no—
225C4×Q8—81no—
226C4×Q8—82no—
227#117—22no—
228#117—22no—
229#117SD1614yes—
230#117Q1614yes—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4C22⋊C411yescharacteristic
246C2×C4⋊C4C22⋊C411yescharacteristic
247C2×C4⋊C4C22⋊C411yescharacteristic
248C2×C4⋊C4C22⋊C411yescharacteristic
249C2×C4⋊C4C22×C412yes—
250C2×C4⋊C4C22×C412yes—
251C2×C4⋊C4C22×C412yes—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4C22⋊C411yescharacteristic
255C2×C22⋊C4C22⋊C411yescharacteristic
256C2×C22⋊C4C22⋊C411yescharacteristic
257C2×C22⋊C4C22⋊C411yescharacteristic
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42C22×C411yescharacteristic
265C2×C42C22×C411yescharacteristic
266C2×C42C22×C411yescharacteristic
267C2×C42C22×C411yescharacteristic
268C2×C42C22×C411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42C2411yescharacteristic
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—21no—
293C4⋊C8—21no—
294C4⋊C8—21no—
295C4⋊C8—21no—
296C4⋊C8—42no—
297C4⋊C8—42no—
298C4⋊C8—42no—
299C4⋊C8—42no—
300C4⋊C8—42no—
301C4⋊C8—42no—
302C4⋊C8—42no—
303C4⋊C8—42no—
304C4⋊C8—42no—
305C4⋊C8—42no—
306C4⋊C8—81no—
307C4⋊C8—81no—
308C22⋊C8—81no—
309C22⋊C8—81no—
310C4×C8—81no—
311C4×C8—81no—
312#95—22no—
313#95—22no—
314#95—22no—
315#95—22no—
316#95—22no—
317#95—22no—
318#95—22no—
319#95—22no—
320#95—22no—
321#95—22no—
322#95—22no—
323#95—22no—
324#95C22⋊C411yescharacteristic
325#95C22⋊C411yescharacteristic
326#95C22⋊C411yescharacteristic
327#95C22⋊C411yescharacteristic
328#95C22⋊C411yescharacteristic
329#95C22⋊C411yescharacteristic
330#95C22⋊C411yescharacteristic
331#95C22⋊C411yescharacteristic
332C24C2×C22⋊C411yescharacteristic
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8—42no—
336C2×Q8—42no—
337C2×Q8—41no—
338C2×Q8—41no—
339C2×Q8—41no—
340C2×Q8—41no—
341C2×Q8—41no—
342C2×Q8—41no—
343C2×Q8—44no—
344C2×Q8—82no—
345C2×Q8—82no—
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—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—41no—
363C22×C4—41no—
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×C4C2×C22⋊C411yescharacteristic
373C22×C4C2×C22⋊C411yescharacteristic
374C22×C4C2×C22⋊C411yescharacteristic
375C22×C4C2×SD1612yes—
376C22×C4C2×Q1612yes—
377C22×C4C23×C411yescharacteristic
378C22×C4C22×D411yescharacteristic
379C22×C4C22×D411yescharacteristic
380M16—84no—
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—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—84no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—42no—
411C4⋊C4—42no—
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—41no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—44no—
427C4⋊C4—82no—
428C4⋊C4—82no—
429C4⋊C4—82no—
430C4⋊C4—82no—
431C4⋊C4—82no—
432C4⋊C4—82no—
433C4⋊C4#10314yes—
434C4⋊C4#10314yes—
435C22⋊C4—22no—
436C22⋊C4—22no—
437C22⋊C4—82no—
438C22⋊C4—82no—
439C22⋊C4#10314yes—
440C22⋊C4#10314yes—
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—21no—
454C42—21no—
455C42—21no—
456C42—21no—
457C42—41no—
458C42—41no—
459C42—41no—
460C42—41no—
461C42—82no—
462C42—82no—
463C42C2×SD1612yes—
464C42C2×SD1612yes—
465C42C2×Q1612yes—
466C42C2×Q1612yes—
467C42AGL(1,ℤ/8)12yes—
468C42#13712yes—
469C42H(5,𝔽2)11yescharacteristic
470C42H(5,𝔽2)11yescharacteristic
471C422−1+411yescharacteristic
472C422−1+411yescharacteristic
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23C2×#10312yes—
480C23C22×C22⋊C411yescharacteristic
481Q8—82no—
482Q8—82no—
483Q8—81no—
484Q8—81no—
485Q8—81no—
486Q8—81no—
487Q8—164no—
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—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
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—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—81no—
537C2×C4—81no—
538C2×C4C2×#10312yes—
539C2×C4C2×#10312yes—
540C2×C4C2×#10312yes—
541C2×C4C2×#10312yes—
542C2×C4C2×#10312yes—
543C2×C4#41712yes—
544C2×C4#41712yes—
545C2×C4#52011yescharacteristic
546C2×C4#52011yescharacteristic
547C2×C4#53511yescharacteristic
548C2×C4#53611yescharacteristic
549C2×C4#53711yescharacteristic
550C2×C4#53711yescharacteristic
551C2×C4C22×SD1611yescharacteristic
552C2×C4C22×Q1611yescharacteristic
553C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
554C2×C4C2×#13711yescharacteristic
555C2×C4#57611yescharacteristic
556C2×C4#57711yescharacteristic
557C2×C4#57711yescharacteristic
558C2×C4#57811yescharacteristic
559C8—81no—
560C8—81no—
561C8—81no—
562C8—81no—
563C8—81no—
564C8—81no—
565C8—81no—
566C8—81no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22#231111yescharacteristic
575C22#288811yescharacteristic
576C22#288911yescharacteristic
577C22C22×#10311yescharacteristic
578C22C2×#41711yescharacteristic
579C22#289711yescharacteristic
580C22#289811yescharacteristic
581C4—22no—
582C4—22no—
583C4—22no—
584C4—22no—
585C4—22no—
586C4—22no—
587C4—22no—
588C4—21no—
589C4—21no—
590C4—41no—
591C4—41no—
592C4—41no—
593C4—41no—
594C4—82no—
595C4—82no—
596C4—82no—
597C4—82no—
598C4#320811yescharacteristic
599C4#320911yescharacteristic
600C4#322911yescharacteristic
601C4#323011yescharacteristic
602C4#325711yescharacteristic
603C4#325811yescharacteristic
604C4#326211yescharacteristic
605C4#326311yescharacteristic
606C2—42no—
607C2#1772211yescharacteristic
608C2#1772311yescharacteristic
609C2#1776211yescharacteristic
610C2#1776311yescharacteristic
611C2#2186011yescharacteristic
612C2#2188111yescharacteristic
613C2#2189611yescharacteristic
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. #9898C2semidirect
5. #9896C2semidirect
6. #9697C2semidirect
7. #9697C2semidirect
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
#219222
#207932
#98982
#98962
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