Finite Group Databasebeta

← #373749

Subgroups — #373749

Order 512.

1278 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2924C211yesmaximal, characteristic
2C2×#1855C211yesmaximal, characteristic
3#20836C211yesmaximal, characteristic
4#20699C211yesmaximal, characteristic
5C2×#1733C211yesmaximal, characteristic
6#11329C212yesmaximal, complemented
7#11291C212yesmaximal, complemented
8#10910C212yesmaximal, complemented
9#10887C212yesmaximal, complemented
10#10720C211yesmaximal, characteristic
11#10713C211yesmaximal, characteristic
12C23×M16C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#117C2211yescharacteristic
15#2924—24no—
16#2924—24no—
17C2×#433—21no—
18C2×#433—21no—
19C2×#432—21no—
20C2×#431—21no—
21C22×#108C2212yes—
22C2×#424C2211yescharacteristic
23C2×#424C2211yescharacteristic
24C2×#424C2212yes—
25C2×#407C2211yescharacteristic
26C22×C22⋊C8C2211yescharacteristic
27C22×C22⋊C8C2211yescharacteristic
28C2×#405C2211yescharacteristic
29C2×#405C2211yescharacteristic
30#2280C2211yescharacteristic
31#1855—24no—
32#1851—21no—
33#1851—21no—
34#1850—21no—
35#1850—21no—
36#1825C411yescharacteristic, complemented
37#1825C411yescharacteristic, complemented
38#1825C411yescharacteristic, complemented
39#1825C411yescharacteristic, complemented
40#1824C412yescomplemented
41#1823C412yescomplemented
42C2×#341C2212yes—
43#1735C2212yes—
44#1735C2212yes—
45#1733—24no—
46C2×#339C2212yes—
47C2×#339C2212yes—
48C2×#339C2212yes—
49C2×#339C2212yes—
50#1729C2212yes—
51C2×#336C2212yes—
52C2×#336C2212yes—
53C24×C4C2311yescharacteristic
54C22×M16—21no—
55C22×M16—22no—
56C22×M16—24no—
57C22×M16—22no—
58C22×M16—42no—
59C22×M16C2311yescharacteristic
60C22×M16C2311yescharacteristic
61C22×M16C2312yes—
62C23×C8—21no—
63C23×C8C2312yes—
64C23×C8C2311yescharacteristic
65C23×C8C2311yescharacteristic
66C2×#117—21no—
67C2×#117—22no—
68C2×#117—21no—
69C2×#117—22no—
70C2×#117—22no—
71C2×#117—22no—
72C2×#117—24no—
73C2×#117—42no—
74C2×#117C2×C411yescharacteristic
75C2×#117C2×C411yescharacteristic
76C2×#117D414yes—
77C2×#117Q814yes—
78C2×#117C2312yes—
79C2×#117C2311yescharacteristic
80C2×#117C2311yescharacteristic
81C22×C4⋊C4—21no—
82C22×C4⋊C4C2×C411yescharacteristic
83C22×C4⋊C4C2×C411yescharacteristic
84C22×C22⋊C4—21no—
85C22×C22⋊C4—21no—
86C22×C42—21no—
87C22×C42C2312yes—
88#433—28no—
89#433—28no—
90#432—28no—
91#431—28no—
92C2×#108—24no—
93C2×#108—24no—
94C2×#108—42no—
95#424—22no—
96#424—22no—
97#424—22no—
98#424—22no—
99#424—24no—
100#424—42no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8—21no—
105C2×C4⋊C8—22no—
106C2×C4⋊C8—22no—
107#407—22no—
108#407D414yes—
109C2×C22⋊C8—22no—
110C2×C22⋊C8—22no—
111C2×C22⋊C8—22no—
112C2×C22⋊C8—21no—
113C2×C22⋊C8—21no—
114C2×C22⋊C8—22no—
115C2×C22⋊C8C2×C412yes—
116C2×C22⋊C8C2×C411yescharacteristic
117C2×C22⋊C8C2×C411yescharacteristic
118C2×C22⋊C8D411yescharacteristic
119C2×C22⋊C8D411yescharacteristic
120C2×C22⋊C8D411yescharacteristic
121C2×C22⋊C8D411yescharacteristic
122#405—22no—
123#405—22no—
124#405—22no—
125#405—22no—
126C2×#97—21no—
127C2×#97—21no—
128C2×C4×C8—21no—
129C2×C4×C8—21no—
130#381—21no—
131#380C2×C412yes—
132#380C2×C411yescharacteristic
133#380C2×C411yescharacteristic
134#379—21no—
135C4×C22⋊C4—21no—
136C4×C22⋊C4—21no—
137#341—24no—
138#340C2×C412yes—
139#340C2×C412yes—
140#340C2×C412yes—
141#340C2×C412yes—
142#340C2×C412yes—
143#340C2×C412yes—
144#340C2×C412yes—
145#340C2×C412yes—
146#339—24no—
147#339—24no—
148#339—24no—
149#339—42no—
150#336—22no—
151#336—22no—
152#336—24no—
153#336—42no—
154C25#4111yescharacteristic
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—22no—
162C23×C4C22×C411yescharacteristic
163C23×C4C22×C411yescharacteristic
164C23×C4C2×D412yes—
165C23×C4C2×D412yes—
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×Q812yes—
169C23×C4#4111yescharacteristic
170C23×C4#4112yes—
171C23×C4C2411yescharacteristic
172C2×M16—21no—
173C2×M16—21no—
174C2×M16—22no—
175C2×M16—28no—
176C2×M16—22no—
177C2×M16—22no—
178C2×M16—22no—
179C2×M16—22no—
180C2×M16—22no—
181C2×M16—22no—
182C2×M16—22no—
183C2×M16—22no—
184C2×M16—44no—
185C2×M16—48no—
186C2×M16—44no—
187C2×M16—42no—
188C2×M16C2×D412yes—
189C2×M16C2×D412yes—
190C2×M16#4114yes—
191C22×C8—22no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—22no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8—44no—
209C22×C8—42no—
210C22×C8C22×C412yes—
211C22×C8C22×C412yes—
212C22×C8C2×D411yescharacteristic
213C22×C8C2×D411yescharacteristic
214C22×C8C2×D411yescharacteristic
215C22×C8C2×D411yescharacteristic
216#117—22no—
217#117—22no—
218#117—24no—
219#117—24no—
220#117—24no—
221#117—24no—
222#117—21no—
223#117—21no—
224#117—21no—
225#117—21no—
226#117—28no—
227#117—22no—
228#117—22no—
229#117—22no—
230#117—22no—
231#117—48no—
232#117—44no—
233#117C4⋊C414yes—
234#117C4⋊C414yes—
235#117C2×D412yes—
236#117C2×D412yes—
237#117C2×Q812yes—
238#117C2×Q812yes—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4C4⋊C414yes—
246C2×C4⋊C4C4⋊C414yes—
247C2×C4⋊C4C22×C412yes—
248C2×C4⋊C4C22×C412yes—
249C2×C4⋊C4C22×C411yescharacteristic
250C2×C4⋊C4C22×C411yescharacteristic
251C2×C4⋊C4C22×C411yescharacteristic
252C2×C4⋊C4C22×C411yescharacteristic
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—42no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—24no—
268C2×C42—24no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42#4114yes—
278#108—48no—
279C4⋊C8—24no—
280C4⋊C8—24no—
281C4⋊C8—24no—
282C4⋊C8—24no—
283C4⋊C8—44no—
284C4⋊C8—44no—
285C22⋊C8—24no—
286C22⋊C8—22no—
287C22⋊C8—22no—
288C22⋊C8—22no—
289C22⋊C8—22no—
290C22⋊C8—22no—
291C22⋊C8—22no—
292C22⋊C8—24no—
293C22⋊C8—24no—
294#97—24no—
295#97—24no—
296C4×C8—24no—
297C4×C8—24no—
298#95—21no—
299#95—21no—
300#95—22no—
301C24—22no—
302C24—22no—
303C24—41no—
304C24—41no—
305C24—42no—
306C24C4×D411yescharacteristic
307C24C4×D411yescharacteristic
308C24C22⋊Q812yes—
309C24#12312yes—
310C24C2×#4111yescharacteristic
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—82no—
349C22×C4C2×C4⋊C412yes—
350C22×C4C2×C4⋊C412yes—
351C22×C4C4×D412yes—
352C22×C4C4×D412yes—
353C22×C4C4×D411yescharacteristic
354C22×C4C4×D411yescharacteristic
355C22×C4#12012yes—
356C22×C4#12012yes—
357C22×C4C22⋊Q812yes—
358C22×C4C22⋊Q812yes—
359C22×C4C22⋊Q812yes—
360C22×C4#12312yes—
361C22×C4C23×C411yescharacteristic
362C22×C4C22×D411yescharacteristic
363C22×C4C22×D411yescharacteristic
364C22×C4C22×D411yescharacteristic
365C22×C4C22×Q811yescharacteristic
366C22×C4C2×#4111yescharacteristic
367C22×C4C2×#4112yes—
368C22×C4C2×#4112yes—
369C22×C4H(5,𝔽2)12yes—
370C22×C42−1+412yes—
371M16—24no—
372M16—24no—
373M16—24no—
374M16—24no—
375M16—48no—
376M16—41no—
377M16—41no—
378M16—41no—
379M16—42no—
380M16—41no—
381M16—42no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—21no—
389C2×C8—21no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—21no—
393C2×C8—21no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—24no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—48no—
400C2×C8—42no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8—42no—
413C2×C8—42no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—48no—
425C4⋊C4C2×C4⋊C412yes—
426C4⋊C4C2×C4⋊C412yes—
427C4⋊C4C2×C4⋊C412yes—
428C4⋊C4C2×C4⋊C412yes—
429C22⋊C4—22no—
430C22⋊C4—22no—
431C22⋊C4—22no—
432C22⋊C4—22no—
433C22⋊C4—24no—
434C22⋊C4—24no—
435C22⋊C4—42no—
436C22⋊C4—42no—
437C22⋊C4—48no—
438C22⋊C4—42no—
439C22⋊C4—42no—
440C22⋊C4—82no—
441C42—22no—
442C42—22no—
443C42—22no—
444C42—22no—
445C42—24no—
446C42—24no—
447C42—24no—
448C42—24no—
449C42—42no—
450C42—42no—
451C42—42no—
452C42—42no—
453C23—21no—
454C23—21no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—22no—
463C23—22no—
464C23—24no—
465C23—22no—
466C23—22no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—82no—
472C23—82no—
473C23—82no—
474C23—82no—
475C23#38512yes—
476C23#38512yes—
477C23C2×C4×D411yescharacteristic
478C23C2×C22⋊Q811yescharacteristic
479C23C2×#12311yescharacteristic
480C23#55112yes—
481C23#55212yes—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—48no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—84no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4#38512yes—
525C2×C4#38512yes—
526C2×C4C22×C4⋊C411yescharacteristic
527C2×C4C2×C4×D411yescharacteristic
528C2×C4C4×#4112yes—
529C2×C4#52012yes—
530C2×C4C2×#12011yescharacteristic
531C2×C4C2×C22⋊Q811yescharacteristic
532C2×C4#52511yescharacteristic
533C2×C4#52511yescharacteristic
534C2×C4#53312yes—
535C2×C4#53511yescharacteristic
536C2×C4#53611yescharacteristic
537C2×C4#54412yes—
538C2×C4#54612yes—
539C2×C4#54612yes—
540C2×C4#54712yes—
541C2×C4#54712yes—
542C2×C4#55112yes—
543C2×C4#55212yes—
544C8—42no—
545C8—42no—
546C8—41no—
547C8—41no—
548C8—44no—
549C8—42no—
550C8—41no—
551C8—41no—
552C8—42no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—82no—
572C22—84no—
573C22—82no—
574C22C2×#38511yescharacteristic
575C22#235912yes—
576C22#243812yes—
577C22#248211yescharacteristic
578C22#248411yescharacteristic
579C22#295511yescharacteristic
580C22#295511yescharacteristic
581C22#295712yes—
582C4—22no—
583C4—22no—
584C4—42no—
585C4—42no—
586C4—42no—
587C4—42no—
588C4—84no—
589C4—82no—
590C4—82no—
591C4#229611yescharacteristic
592C4#231711yescharacteristic
593C4#235912yes—
594C4#240411yescharacteristic
595C4#240611yescharacteristic
596C4#243812yes—
597C2—21no—
598C2—22no—
599C2—21no—
600C2—42no—
601C2—42no—
602C2#1412811yescharacteristic
603C2#2088411yescharacteristic
604C2#2088611yescharacteristic
605C2#2294912yes—
606C2#2295112yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC25#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C22⋊C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #11329C2semidirect
7. #11291C2semidirect
8. #10910C2semidirect
9. #10887C2semidirect
36. #1825C4semidirect
37. #1825C4semidirect
38. #1825C4semidirect
39. #1825C4semidirect
40. #1824C4semidirect
41. #1823C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29242
C2×#18552
#208362
#206992
C2×#17332
#113292
#112912
#109102
#108872
#107202
#107132