Finite Group Databasebeta

← #217087

Subgroups — #217087

Order 512.

743 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35283C211yesmaximal, characteristic
2#21978C211yesmaximal, characteristic
3#20786C211yesmaximal, characteristic
4#9903C211yesmaximal, characteristic, complemented
5#9896C211yesmaximal, characteristic, complemented
6#9701C211yesmaximal, characteristic, complemented
7#9693C211yesmaximal, characteristic, complemented
8#9140C211yesmaximal, characteristic
9#9139C211yesmaximal, characteristic
10#9136C211yesmaximal, characteristic
11#9135C211yesmaximal, characteristic
12#8594C211yesmaximal, characteristic, complemented
13#8593C211yesmaximal, characteristic, complemented
14#8590C211yesmaximal, characteristic, complemented
15#8589C211yesmaximal, characteristic, complemented
16C2×#432C2211yescharacteristic
17C2×#423C2211yescharacteristic
18#2868C2211yescharacteristic
19#2862C2211yescharacteristic
20#2857C2211yescharacteristic
21#2857C2211yescharacteristic
22#2647—21no—
23#2628—21no—
24#2511—21no—
25#2511C412yes—
26#2492—21no—
27#2492C412yes—
28#2451—21no—
29#2451C412yes—
30#2423—21no—
31#2423C412yes—
32#2331—21no—
33#2314—21no—
34C4×C4⋊Q8C2211yescharacteristic
35C4×C4⋊Q8C2211yescharacteristic
36C4×#117C2211yescharacteristic
37#1940—21no—
38#1940—21no—
39#1851—21no—
40#1851—21no—
41#1842C2211yescharacteristic
42#1842C2211yescharacteristic
43#1840—21no—
44#1831C2211yescharacteristic
45#1831C2211yescharacteristic
46#1831C2211yescharacteristic
47#1831C2211yescharacteristic
48#1818—21no—
49#1797C2211yescharacteristic
50#1797C2211yescharacteristic
51#1767C2211yescharacteristic
52C4×C4⋊C8C2211yescharacteristic
53#1750—21no—
54#1746—21no—
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#432—41no—
85#432—41no—
86#423—41no—
87#423—41no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C22⋊C8—21no—
102C2×C22⋊C8—21no—
103C2×C22⋊C8—21no—
104C2×#97—21no—
105C2×#97—21no—
106C2×C4×C8—21no—
107#400—22no—
108#400—22no—
109#400—22no—
110#399—22no—
111#398—22no—
112#398—22no—
113#395—22no—
114#393—22no—
115#391—21no—
116#391—21no—
117#391—21no—
118#391—21no—
119#391—22no—
120#391—22no—
121#391C2×C412yes—
122#391C2×C412yes—
123#391C2×C412yes—
124#391C2×C412yes—
125#389—21no—
126#389—21no—
127#389—21no—
128#389—21no—
129#389—21no—
130#389—21no—
131#389C2×C412yes—
132#389C2×C412yes—
133#388—22no—
134#388—22no—
135#387—22no—
136#387—22no—
137#387—22no—
138#387—22no—
139#384—21no—
140#384—21no—
141#384C2×C412yes—
142#384C2311yescharacteristic
143#384C2311yescharacteristic
144#382—21no—
145#382—21no—
146#380—21no—
147#380—21no—
148#380—21no—
149#380C2×C412yes—
150C4×C4⋊C4—21no—
151C4×C4⋊C4—21no—
152C4×C4⋊C4—21no—
153C4×C4⋊C4C2×C412yes—
154C4×C4⋊C4C2×C411yescharacteristic
155C4×C4⋊C4C2×C411yescharacteristic
156C4×C4⋊C4D411yescharacteristic
157C4×C4⋊C4D411yescharacteristic
158C4×C22⋊C4C2×C411yescharacteristic
159C4×C22⋊C4C2×C411yescharacteristic
160C4×C22⋊C4D411yescharacteristic
161C4×C22⋊C4D411yescharacteristic
162#376D411yescharacteristic
163#376D411yescharacteristic
164C43C2311yescharacteristic
165C43C2311yescharacteristic
166#336—21no—
167#336—21no—
168#336—21no—
169#336—21no—
170#336—21no—
171#336—21no—
172#336—21no—
173#336—21no—
174#336—21no—
175#336—21no—
176#336—21no—
177#336—21no—
178#333—41no—
179#333—41no—
180#333—41no—
181#333—41no—
182#332—41no—
183#332—41no—
184#332—41no—
185#332—41no—
186C22×Q8—22no—
187C22×Q8—21no—
188C22×Q8C22×C412yes—
189C23×C4C22×C411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C2×M16—42no—
193C2×M16—81no—
194C2×M16—81no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C4⋊Q8—21no—
204C4⋊Q8—21no—
205C4⋊Q8—21no—
206C4⋊Q8—21no—
207C4⋊Q8—21no—
208C4⋊Q8—21no—
209C4⋊Q8—21no—
210C4⋊Q8—21no—
211C4⋊Q8—42no—
212C4⋊Q8—42no—
213C4⋊Q8—81no—
214C4⋊Q8—81no—
215C4⋊Q8—82no—
216C22⋊Q8—81no—
217C22⋊Q8—81no—
218C22⋊Q8—82no—
219C4×Q8—81no—
220C4×Q8—81no—
221C4×Q8—82no—
222#117—22no—
223#117—22no—
224#117—41no—
225#117—41no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—21no—
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—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4C22⋊C411yescharacteristic
241C2×C4⋊C4C22⋊C411yescharacteristic
242C2×C4⋊C4C22⋊C411yescharacteristic
243C2×C4⋊C4C22⋊C411yescharacteristic
244C2×C4⋊C4C22×C412yes—
245C2×C4⋊C4C22×C412yes—
246C2×C4⋊C4C22×C412yes—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4C22⋊C411yescharacteristic
250C2×C22⋊C4C22⋊C411yescharacteristic
251C2×C22⋊C4C22⋊C411yescharacteristic
252C2×C22⋊C4C22⋊C411yescharacteristic
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42C22×C411yescharacteristic
260C2×C42C22×C411yescharacteristic
261C2×C42C22×C411yescharacteristic
262C2×C42C22×C411yescharacteristic
263C2×C42C22×C411yescharacteristic
264C2×C42C2×D411yescharacteristic
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×C42C2411yescharacteristic
275C4⋊C8—21no—
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—42no—
292C4⋊C8—81no—
293C4⋊C8—81no—
294C4⋊C8—81no—
295C4⋊C8—81no—
296C22⋊C8—42no—
297C22⋊C8—42no—
298C22⋊C8—42no—
299#97—42no—
300#97—42no—
301C4×C8—42no—
302#95—22no—
303#95—22no—
304#95—22no—
305#95—22no—
306#95—22no—
307#95—22no—
308#95—22no—
309#95—22no—
310#95—22no—
311#95—22no—
312#95—22no—
313#95—22no—
314#95C22⋊C411yescharacteristic
315#95C22⋊C411yescharacteristic
316#95C22⋊C411yescharacteristic
317#95C22⋊C411yescharacteristic
318#95C22⋊C411yescharacteristic
319#95C22⋊C411yescharacteristic
320#95C22⋊C411yescharacteristic
321#95C22⋊C411yescharacteristic
322C24C2×C22⋊C411yescharacteristic
323C2×Q8—22no—
324C2×Q8—22no—
325C2×Q8—44no—
326C2×Q8—42no—
327C2×Q8—42no—
328C2×Q8—41no—
329C2×Q8—41no—
330C2×Q8—41no—
331C2×Q8—41no—
332C2×Q8—41no—
333C2×Q8—41no—
334C2×Q8—82no—
335C2×Q8—82no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
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—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4C2×C22⋊C411yescharacteristic
357C22×C4C2×C22⋊C411yescharacteristic
358C22×C4C2×C22⋊C411yescharacteristic
359C22×C4C2×C22⋊C411yescharacteristic
360C22×C4C2×C22⋊C411yescharacteristic
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×C4C23×C411yescharacteristic
368C22×C4C22×D411yescharacteristic
369C22×C4C22×D411yescharacteristic
370M16—84no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—41no—
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—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—44no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—41no—
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—82no—
419C4⋊C4—82no—
420C4⋊C4—82no—
421C4⋊C4—82no—
422C4⋊C4—82no—
423C4⋊C4—82no—
424C22⋊C4—22no—
425C22⋊C4—22no—
426C22⋊C4—41no—
427C22⋊C4—41no—
428C22⋊C4—82no—
429C22⋊C4—82no—
430C42—21no—
431C42—21no—
432C42—21no—
433C42—21no—
434C42—21no—
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—41no—
451C42—41no—
452C42—41no—
453C42—41no—
454C42—82no—
455C42—82no—
456C42#13712yes—
457C42#13712yes—
458C42H(5,𝔽2)11yescharacteristic
459C42H(5,𝔽2)11yescharacteristic
460C422−1+411yescharacteristic
461C422−1+411yescharacteristic
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23C22×C22⋊C411yescharacteristic
470Q8—82no—
471Q8—82no—
472Q8—81no—
473Q8—81no—
474Q8—81no—
475Q8—81no—
476Q8—164no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
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—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
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—81no—
531C2×C4—81no—
532C2×C4#41912yes—
533C2×C4#41912yes—
534C2×C4#52011yescharacteristic
535C2×C4#52011yescharacteristic
536C2×C4#53511yescharacteristic
537C2×C4#53611yescharacteristic
538C2×C4#53711yescharacteristic
539C2×C4#53711yescharacteristic
540C2×C4C2×#13711yescharacteristic
541C2×C4C2×#13711yescharacteristic
542C2×C4#57511yescharacteristic
543C2×C4#57511yescharacteristic
544C2×C4#57611yescharacteristic
545C2×C4#57711yescharacteristic
546C2×C4#57711yescharacteristic
547C2×C4#57811yescharacteristic
548C8—81no—
549C8—81no—
550C8—81no—
551C8—81no—
552C8—81no—
553C8—81no—
554C8—81no—
555C8—81no—
556C22—41no—
557C22—41no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22#231111yescharacteristic
564C22#288811yescharacteristic
565C22#288911yescharacteristic
566C22C2×#41911yescharacteristic
567C22#289611yescharacteristic
568C22#289711yescharacteristic
569C22#289811yescharacteristic
570C4—22no—
571C4—22no—
572C4—21no—
573C4—21no—
574C4—41no—
575C4—41no—
576C4—41no—
577C4—41no—
578C4—41no—
579C4—41no—
580C4—41no—
581C4—41no—
582C4—41no—
583C4—82no—
584C4—82no—
585C4—82no—
586C4—82no—
587C4#322411yescharacteristic
588C4#322511yescharacteristic
589C4#324311yescharacteristic
590C4#324411yescharacteristic
591C4#325211yescharacteristic
592C4#325211yescharacteristic
593C4#325211yescharacteristic
594C4#325211yescharacteristic
595C2—81no—
596C2#1776011yescharacteristic
597C2#1776111yescharacteristic
598C2#1776611yescharacteristic
599C2#1776711yescharacteristic
600C2#2187111yescharacteristic
601C2#2188111yescharacteristic
602C2#2189211yescharacteristic
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. #9903C2semidirect
5. #9896C2semidirect
6. #9701C2semidirect
7. #9693C2semidirect
12. #8594C2semidirect
13. #8593C2semidirect
14. #8590C2semidirect
15. #8589C2semidirect

Maximal subgroups

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

subgroupindex
#352832
#219782
#207862
#99032
#98962
#97012
#96932
#91402
#91392
#91362
#91352
#85942
#85932
#85902
#85892