Finite Group Databasebeta

← #6458702

Subgroups — #6458702

Order 512.

1692 conjugacy classes of subgroups.

The subgroup lattice has 571 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35279C211yesmaximal, characteristic
2#34188C211yesmaximal, characteristic
3#27886C218yesmaximal, complemented
4#27878C212yesmaximal, complemented
5#27877C212yesmaximal, complemented
6#27876C212yesmaximal, complemented
7#27875C212yesmaximal, complemented
8#23641C214yesmaximal
9#21324C211yesmaximal, characteristic, complemented
10C2×#1990C211yesmaximal, characteristic, complemented
11#21039C214yesmaximal, complemented
12#20797C211yesmaximal, characteristic, complemented
13#20764C211yesmaximal, characteristic, complemented
14C4×#408C211yesmaximal, characteristic
15C22×C4⋊Q8C2211yescharacteristic
16C2×C4×#41C2211yescharacteristic
17C2×#427C2214yes—
18C2×#408C2212yes—
19C2×C4×M16C2211yescharacteristic
20C22×C4×C8C2211yescharacteristic
21#2857C2212yes—
22#2857C2212yes—
23#2628C2214yes—
24#2568—21no—
25#2567—21no—
26#2563—21no—
27#2542—21no—
28#2487—22no—
29#2409—22no—
30#2355—21no—
31#2355—21no—
32#2354—21no—
33#2354—21no—
34#2352—22no—
35#2352—22no—
36#2317C2212yes—
37#2314C2214yes—
38#2311C2212yes—
39#2311C2212yes—
40#2285C2212yes—
41#2285C2212yes—
42#2285C2211yescharacteristic
43#2285C2211yescharacteristic
44C2×#384C2211yescharacteristic
45#2274C2212yes—
46#2077C2218yes—
47#2077C2218yes—
48#2076C2218yes—
49#2075C2218yes—
50#1993C2214yes—
51#1990C2212yes—
52#1990C2212yes—
53#1990C2214yes—
54#1990C2212yes—
55#1990C2212yes—
56#1921C2218yes—
57#1920C2218yes—
58#1919C2218yes—
59#1918C2218yes—
60#1846C2214yes—
61#1845C2211yescharacteristic
62#1845C2211yescharacteristic
63#1844C2211yescharacteristic
64#1844C2211yescharacteristic
65#1836C2214yes—
66#1834C2212yes—
67#1834C2212yes—
68#1750C2214yes—
69C4×C22⋊C8C2212yes—
70C4×C22⋊C8C2212yes—
71#1736C2214yes—
72#1736C2214yes—
73C2×#340C2214yes—
74#1729C2212yes—
75C22×#41C2311yescharacteristic
76C23×Q8—21no—
77C22×M16C2312yes—
78C23×C8C2312yes—
79C2×C4⋊Q8—21no—
80C2×C4⋊Q8—21no—
81C2×C4⋊Q8—24no—
82C2×C4⋊Q8—24no—
83C2×C4⋊Q8C2312yes—
84C2×C4⋊Q8C2312yes—
85C2×C22⋊Q8—22no—
86C2×C22⋊Q8—22no—
87C4×#41—24no—
88C4×#41—24no—
89C2×C4×Q8D411yescharacteristic
90C2×C4×Q8D411yescharacteristic
91C2×C4×D4D411yescharacteristic
92C2×C4×D4D412yes—
93C2×C4×D4D412yes—
94C2×C4×D4D411yescharacteristic
95C2×#117—22no—
96C2×#117—22no—
97C2×#117C2312yes—
98C2×#117C2312yes—
99C2×#117C2312yes—
100C2×#117C2311yescharacteristic
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C42C2312yes—
107C22×C42C2311yescharacteristic
108#427—216no—
109C2×#107C2314yes—
110C2×#107C2314yes—
111C2×#106C2314yes—
112C2×#106C2314yes—
113#408—28no—
114C2×C22⋊C8C2314yes—
115C2×C22⋊C8C2314yes—
116C4×M16—24no—
117C2×#97C2312yes—
118C2×C4×C8—24no—
119C2×C4×C8C2312yes—
120C2×C4×C8C2311yescharacteristic
121C2×C4×C8C2311yescharacteristic
122C2×C4×C8C2312yes—
123#400—24no—
124#399—24no—
125#395—24no—
126#393—24no—
127#391—22no—
128#391—22no—
129#391—22no—
130#391—22no—
131#390—22no—
132#390—22no—
133#389—22no—
134#389—22no—
135#389C2314yes—
136#389C2314yes—
137#388—24no—
138#388—24no—
139#387—24no—
140#387—24no—
141#386—24no—
142#386—24no—
143#385—24no—
144#384—22no—
145#384—21no—
146#384—21no—
147#384C2312yes—
148#384C2311yescharacteristic
149#384C2311yescharacteristic
150#384C2312yes—
151#384C2311yescharacteristic
152#384C2311yescharacteristic
153#382—22no—
154#382—22no—
155#382—22no—
156#382C2312yes—
157#382C2312yes—
158#380—22no—
159#380—22no—
160#380—21no—
161#380—21no—
162#380—22no—
163#380—22no—
164#380—21no—
165#380—21no—
166#380C2314yes—
167#380C2314yes—
168#380C2314yes—
169#380C2314yes—
170C4×C4⋊C4—22no—
171C4×C4⋊C4—22no—
172C4×C4⋊C4—21no—
173C4×C4⋊C4—21no—
174C4×C4⋊C4—22no—
175C4×C4⋊C4—22no—
176C4×C4⋊C4—21no—
177C4×C4⋊C4—21no—
178C4×C4⋊C4C2314yes—
179C4×C4⋊C4C2314yes—
180C4×C22⋊C4—24no—
181#340C2318yes—
182#340C2318yes—
183#340C2318yes—
184#340C2318yes—
185#340C2318yes—
186#340C2318yes—
187#336C2314yes—
188#336C2314yes—
189C2×#41—22no—
190C2×#41—22no—
191C2×#41—24no—
192C2×#41—24no—
193C22×Q8—22no—
194C22×Q8—22no—
195C22×Q8—44no—
196C22×Q8—82no—
197C22×Q8—82no—
198C22×Q8C2×D411yescharacteristic
199C22×D4C2×D412yes—
200C22×D4C2×D411yescharacteristic
201C23×C4—22no—
202C23×C4—22no—
203C23×C4C2×D412yes—
204C23×C4C2×D412yes—
205C23×C4C2×D411yescharacteristic
206C23×C4C2×D411yescharacteristic
207C23×C4C2412yes—
208C23×C4C2411yescharacteristic
209C2×M16—24no—
210C2×M16—24no—
211C2×M16—24no—
212C22×C8—24no—
213C22×C8—24no—
214C22×C8#4118yes—
215C22×C8C2414yes—
216C22×C8C2412yes—
217C22×C8C2412yes—
218C4⋊Q8—24no—
219C4⋊Q8—24no—
220C4⋊Q8—24no—
221C4⋊Q8—24no—
222C4⋊Q8—44no—
223C4⋊Q8—44no—
224C22⋊Q8—84no—
225C22⋊Q8—84no—
226C4×Q8—24no—
227C4×Q8—24no—
228C4×D4—28no—
229C4×D4—28no—
230C4×D4—24no—
231C4×D4—24no—
232#117—22no—
233#117—22no—
234#117—24no—
235#117—24no—
236#117—28no—
237#117—28no—
238#117—84no—
239#117—84no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—82no—
263C2×C4⋊C4—82no—
264C2×C4⋊C4C2×D411yescharacteristic
265C2×C4⋊C4C2×D411yescharacteristic
266C2×C4⋊C4C2×D412yes—
267C2×C4⋊C4C2×D412yes—
268C2×C4⋊C4C2414yes—
269C2×C4⋊C4C2414yes—
270C2×C4⋊C4C2412yes—
271C2×C4⋊C4C2412yes—
272C2×C4⋊C4C2412yes—
273C2×C4⋊C4C2412yes—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4C2×D412yes—
281C2×C22⋊C4C2×D412yes—
282C2×C22⋊C4C2×D412yes—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42C2×D412yes—
290C2×C42C2×D412yes—
291C2×C42C2×D411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C2×C42#4114yes—
294C2×C42#4114yes—
295C2×C42C2412yes—
296C2×C42C2411yescharacteristic
297C2×C42C2411yescharacteristic
298#107—28no—
299#107—28no—
300#106—28no—
301#106—28no—
302C22⋊C8—28no—
303C22⋊C8—28no—
304#97—24no—
305C4×C8—22no—
306C4×C8—22no—
307C4×C8—24no—
308C4×C8—24no—
309#95—24no—
310#95—24no—
311#95—22no—
312#95—22no—
313#95—22no—
314#95—22no—
315#95—24no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95—24no—
321C24C22×D412yes—
322C24C22×D411yescharacteristic
323#41—48no—
324#41—48no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—44no—
329C2×Q8—44no—
330C2×Q8—44no—
331C2×Q8—44no—
332C2×Q8—82no—
333C2×Q8—82no—
334C2×Q8—816no—
335C2×D4—24no—
336C2×D4—24no—
337C2×D4—24no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—44no—
362C22×C4—44no—
363C22×C4—82no—
364C22×C4—82no—
365C22×C4—82no—
366C22×C4—82no—
367C22×C4#12314yes—
368C22×C4#12314yes—
369C22×C4C22×D412yes—
370C22×C4C22×D412yes—
371C22×C4C22×D412yes—
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C22×D411yescharacteristic
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D411yescharacteristic
377C22×C4C2×#4112yes—
378C22×C4C2×#4112yes—
379C22×C4C2×#4114yes—
380C22×C42−1+414yes—
381C22×C4C2511yescharacteristic
382M16—48no—
383C2×C8—216no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—24no—
389C2×C8—24no—
390C2×C8—24no—
391C2×C8C2×#4114yes—
392C2×C8C2×#4114yes—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—28no—
404C4⋊C4—44no—
405C4⋊C4—44no—
406C4⋊C4—44no—
407C4⋊C4—44no—
408C4⋊C4—44no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—44no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—82no—
419C4⋊C4—82no—
420C4⋊C4—84no—
421C4⋊C4—84no—
422C22⋊C4—24no—
423C22⋊C4—24no—
424C22⋊C4—24no—
425C22⋊C4—44no—
426C22⋊C4—44no—
427C22⋊C4—84no—
428C22⋊C4—84no—
429C22⋊C4—84no—
430C22⋊C4—84no—
431C42—24no—
432C42—24no—
433C42—22no—
434C42—22no—
435C42—44no—
436C42—44no—
437C42—84no—
438C42—84no—
439C42#12314yes—
440C42#12314yes—
441C42C2×#4112yes—
442C42C2×#4112yes—
443C42C2×#4112yes—
444C42C2×#4112yes—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23C2×#12312yes—
457C23#53212yes—
458C23#53612yes—
459C23C23×D411yescharacteristic
460Q8—44no—
461Q8—88no—
462Q8—88no—
463D4—48no—
464D4—44no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—28no—
470C2×C4—28no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—48no—
481C2×C4—48no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—88no—
503C2×C4—164no—
504C2×C4C2×#12312yes—
505C2×C4C2×#12312yes—
506C2×C4C2×#12312yes—
507C2×C4C2×#12312yes—
508C2×C4C2×#12312yes—
509C2×C4#53212yes—
510C2×C4#53612yes—
511C2×C4#55218yes—
512C2×C4#55618yes—
513C2×C4#57612yes—
514C2×C4#57712yes—
515C2×C4#57712yes—
516C2×C4#57812yes—
517C2×C4C22×#4111yescharacteristic
518C2×C4C22×#4112yes—
519C2×C4C22×#4111yescharacteristic
520C2×C4C2×2−1+412yes—
521C8—24no—
522C8—24no—
523C8—44no—
524C22—22no—
525C22—22no—
526C22—22no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22#249914yes—
538C22C22×#12311yescharacteristic
539C22C2×#53211yescharacteristic
540C22C2×#53611yescharacteristic
541C22C2×#57611yescharacteristic
542C22C2×#57711yescharacteristic
543C22C2×#57711yescharacteristic
544C22C2×#57811yescharacteristic
545C4—22no—
546C4—22no—
547C4—42no—
548C4—42no—
549C4—42no—
550C4—82no—
551C4—82no—
552C4—82no—
553C4—82no—
554C4—84no—
555C4—84no—
556C4#249914yes—
557C4#309614yes—
558C4#309714yes—
559C4C2×#55212yes—
560C4C2×#55612yes—
561C2—24no—
562C2—82no—
563C2C2×#249911yescharacteristic
564C2C2×#309611yescharacteristic
565C2C2×#309711yescharacteristic
566C2#6071611yescharacteristic
567C2#6071811yescharacteristic
568C2#6087711yescharacteristic
569C2#6087811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #27886C2semidirect
4. #27878C2semidirect
5. #27877C2semidirect
6. #27876C2semidirect
7. #27875C2semidirect
9. #21324C2semidirect
10. C2×#1990C2semidirect
11. #21039C2semidirect
12. #20797C2semidirect
13. #20764C2semidirect

Maximal subgroups

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

subgroupindex
#352792
#341882
#278862
#278782
#278772
#278762
#278752
#236412
#213242
C2×#19902
#210392
#207972
#207642
C4×#4082