Finite Group Databasebeta

← #440850

Subgroups — #440850

Order 512.

1454 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2440C211yesmaximal, characteristic
2C2×#2100C211yesmaximal, characteristic, complemented
3C2×#1834C211yesmaximal, characteristic, complemented
4#15195C211yesmaximal, characteristic
5#11384C212yesmaximal, complemented
6#11357C211yesmaximal, characteristic, complemented
7#11293C212yesmaximal, complemented
8#11200C212yesmaximal
9#10921C211yesmaximal, characteristic
10#10880C211yesmaximal, characteristic, complemented
11#10730C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×C22⋊Q8—21no—
14C23×C4⋊C4C2211yescharacteristic
15C22×C22⋊C8C2212yes—
16C22×C4×C8C2211yescharacteristic
17#2440—24no—
18#2440—24no—
19C2×#398—21no—
20C2×#395—21no—
21C2×#389C2211yescharacteristic
22C2×#389C2211yescharacteristic
23C2×#385—21no—
24C2×#384C2211yescharacteristic
25C2×#384C2211yescharacteristic
26C2×#380C2212yes—
27C2×#380C2211yescharacteristic
28C2×C4×C22⋊C4C2211yescharacteristic
29#2100—24no—
30#1834—24no—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41#1442—21no—
42#1442—21no—
43#1439—21no—
44#1439—21no—
45C4⋊#95C2212yes—
46C4⋊#95C2211yescharacteristic
47C23×Q8—21no—
48C24×C4—21no—
49C24×C4C2311yescharacteristic
50C23×C8C2312yes—
51C23×C8C2312yes—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—24no—
55C2×C4⋊Q8—42no—
56C2×C22⋊Q8—28no—
57C2×C22⋊Q8—44no—
58C2×C22⋊Q8—44no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—24no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—42no—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C22⋊C4—21no—
78C22×C22⋊C4—22no—
79C22×C22⋊C4D411yescharacteristic
80C22×C22⋊C4D411yescharacteristic
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C2×C22⋊C8—24no—
84C2×C22⋊C8—24no—
85C2×C22⋊C8—42no—
86C2×C4×C8—24no—
87C2×C4×C8—22no—
88#398—28no—
89#395—28no—
90#389—22no—
91#389—22no—
92#389—22no—
93#389D414yes—
94#385—28no—
95#384—22no—
96#384—22no—
97#384—22no—
98#384D414yes—
99#380—22no—
100#380—22no—
101#380—24no—
102#380—24no—
103C4×C22⋊C4—22no—
104C4×C22⋊C4—22no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95D411yescharacteristic
114C2×#95D411yescharacteristic
115#340—24no—
116#340—24no—
117#340—24no—
118#340—24no—
119#340—24no—
120#340—24no—
121#340—42no—
122#340—42no—
123#336—22no—
124#336—42no—
125C25C2×D411yescharacteristic
126C22×Q8—24no—
127C22×Q8—44no—
128C22×Q8—42no—
129C22×Q8—42no—
130C22×Q8—82no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—24no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—44no—
146C23×C4—42no—
147C23×C4—42no—
148C23×C4—82no—
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4#4114yes—
155C23×C4C2411yescharacteristic
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8#4114yes—
170C4⋊Q8—48no—
171C4⋊Q8—44no—
172C4⋊Q8—44no—
173C22⋊Q8—432no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—48no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—82no—
217C2×C4⋊C4—82no—
218C2×C4⋊C4—82no—
219C2×C4⋊C4C2×D412yes—
220C2×C4⋊C4C2×D412yes—
221C2×C4⋊C4C2×D412yes—
222C2×C4⋊C4C2×D412yes—
223C2×C4⋊C4#4114yes—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—82no—
233C2×C22⋊C4—82no—
234C2×C22⋊C4SD1614yes—
235C2×C22⋊C4Q1614yes—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42C2×D412yes—
243C2×C42C2×D412yes—
244C22⋊C8—48no—
245C4×C8—28no—
246#95—24no—
247#95—24no—
248#95—24no—
249#95—22no—
250#95—22no—
251#95—44no—
252#95—42no—
253#95—42no—
254#95—82no—
255#95—82no—
256C24—21no—
257C24—21no—
258C24—22no—
259C24—22no—
260C24—21no—
261C24—21no—
262C24—22no—
263C24—22no—
264C24#12412yes—
265C24C2×SD1612yes—
266C24C2×Q1612yes—
267C24C22×D411yescharacteristic
268C2×Q8—48no—
269C2×Q8—48no—
270C2×Q8—88no—
271C2×Q8—84no—
272C2×Q8—88no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—24no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—48no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—42no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—48no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—84no—
342C22×C4—88no—
343C22×C4—88no—
344C22×C4#12012yes—
345C22×C4#12012yes—
346C22×C4#12412yes—
347C22×C4C2×SD1612yes—
348C22×C4C2×SD1612yes—
349C22×C4C2×Q1612yes—
350C22×C4C2×Q1612yes—
351C22×C4AGL(1,ℤ/8)12yes—
352C22×C4#13712yes—
353C22×C4C22×D411yescharacteristic
354C22×C4C22×D411yescharacteristic
355C22×C4C2×#4112yes—
356C22×C4C2×#4112yes—
357C22×C4C2×#4112yes—
358C22×C4H(5,𝔽2)12yes—
359C22×C42−1+412yes—
360C22×C42−1+412yes—
361C2×C8—24no—
362C2×C8—24no—
363C2×C8—24no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C4⋊C4—22no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—48no—
381C4⋊C4—48no—
382C4⋊C4—48no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—48no—
390C4⋊C4—48no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—84no—
400C4⋊C4—84no—
401C4⋊C4—84no—
402C4⋊C4#12414yes—
403C22⋊C4—28no—
404C22⋊C4—28no—
405C22⋊C4—48no—
406C22⋊C4—88no—
407C42—22no—
408C42—22no—
409C42—24no—
410C42—41no—
411C42—41no—
412C42—42no—
413C23—24no—
414C23—24no—
415C23—24no—
416C23—22no—
417C23—22no—
418C23—22no—
419C23—22no—
420C23—22no—
421C23—22no—
422C23—24no—
423C23—24no—
424C23—24no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—41no—
429C23—41no—
430C23—42no—
431C23—42no—
432C23—41no—
433C23—41no—
434C23—42no—
435C23#48714yes—
436C23C2×#12011yescharacteristic
437C23C2×#12411yescharacteristic
438C23#53611yescharacteristic
439C23#54612yes—
440C23#54712yes—
441C23C22×SD1611yescharacteristic
442C23C22×Q1611yescharacteristic
443C23C2×AGL(1,ℤ/8)11yescharacteristic
444C23C2×#13711yescharacteristic
445C23#57612yes—
446C23#57712yes—
447Q8—816no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—24no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—42no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—816no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4#45014yes—
511C2×C4#45114yes—
512C2×C4#52511yescharacteristic
513C2×C4C2×#12411yescharacteristic
514C2×C4#52912yes—
515C2×C4#53511yescharacteristic
516C2×C4#53611yescharacteristic
517C2×C4#53912yes—
518C2×C4#54112yes—
519C2×C4#54212yes—
520C2×C4#54612yes—
521C2×C4#54712yes—
522C2×C4#54812yes—
523C2×C4#54812yes—
524C2×C4#55112yes—
525C2×C4#55212yes—
526C2×C4#55212yes—
527C2×C4#56312yes—
528C8—44no—
529C8—82no—
530C22—24no—
531C22—28no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—44no—
538C22—42no—
539C22—44no—
540C22—44no—
541C22#240611yescharacteristic
542C22#243511yescharacteristic
543C22#243611yescharacteristic
544C22#248812yes—
545C22#269812yes—
546C22C2×#45011yescharacteristic
547C22C2×#45111yescharacteristic
548C22#300811yescharacteristic
549C22#301011yescharacteristic
550C22C2×#48711yescharacteristic
551C22#313612yes—
552C22#314111yescharacteristic
553C22#322911yescharacteristic
554C22#323011yescharacteristic
555C22#324112yes—
556C22#329512yes—
557C22#329912yes—
558C22#332512yes—
559C22#332912yes—
560C22#333512yes—
561C22#333812yes—
562C22#334512yes—
563C22#334612yes—
564C4—24no—
565C4—24no—
566C4—42no—
567C4—44no—
568C4—42no—
569C4—42no—
570C4—82no—
571C4—82no—
572C4—82no—
573C4—162no—
574C4—162no—
575C4#247712yes—
576C4#248912yes—
577C4#269012yes—
578C4#272312yes—
579C2—44no—
580C2#1567011yescharacteristic
581C2#2408911yescharacteristic
582C2#2409811yescharacteristic
583C2#2588111yescharacteristic
584C2#2588811yescharacteristic
585C2#2590211yescharacteristic
586C2#2591511yescharacteristic
587C2#2769712yes—
588C2#2770412yes—
589C2#3169012yes—
590C2#3169212yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2100C2semidirect
3. C2×#1834C2semidirect
5. #11384C2semidirect
6. #11357C2semidirect
7. #11293C2semidirect
10. #10880C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24402
C2×#21002
C2×#18342
#151952
#113842
#113572
#112932
#112002
#109212
#108802
#107302