Finite Group Databasebeta

← #205739

Subgroups — #205739

Order 512.

1150 conjugacy classes of subgroups.

The subgroup lattice has 569 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3167C211yesmaximal, characteristic
2#21332C211yesmaximal, characteristic, complemented
3#21212C211yesmaximal, characteristic
4#20816C211yesmaximal, characteristic, complemented
5C8×#117C211yesmaximal, characteristic, complemented
6#17575C211yesmaximal, characteristic, complemented
7#17574C211yesmaximal, characteristic
8#9781C218yesmaximal, complemented
9C2×#533C2211yescharacteristic
10C2×#532C2211yescharacteristic
11#3167—21no—
12#3167—21no—
13#3167—21no—
14#3167—21no—
15#3167—22no—
16#3167—22no—
17C2×#498C411yescharacteristic, complemented
18C2×#492C411yescharacteristic, complemented
19C2×#487C411yescharacteristic, complemented
20C2×#486C411yescharacteristic, complemented
21C2×#465C2214yes—
22C2×C4×SD16C414yescomplemented
23C2×#431C2211yescharacteristic
24C2×#410—22no—
25C22×C4×C8C2211yescharacteristic
26#2285C2211yescharacteristic
27#2285C2211yescharacteristic
28C4×#117C2211yescharacteristic
29#1996—22no—
30#1996C2214yes—
31#1964—22no—
32#1964C2214yes—
33#1844—21no—
34#1834—21no—
35#1807C2214yes—
36C4×#106—22no—
37C8×C4⋊C4—22no—
38C8×C22⋊C4C2214yes—
39#1744—21no—
40#1740—22no—
41C42×C8—21no—
42#1520C2214yes—
43#1519C2214yes—
44#1518—22no—
45#1517—22no—
46#1503—22no—
47#1502—22no—
48C22×#41—21no—
49C22×SD16C2×C414yes—
50C23×C8C2312yes—
51#533—21no—
52#533—21no—
53#533—21no—
54#533—21no—
55#533—21no—
56#533—21no—
57#533—21no—
58#533—21no—
59#532—21no—
60#532—21no—
61#532—21no—
62#532—21no—
63#532—21no—
64#532—21no—
65#532—21no—
66#532—21no—
67C2×C4⋊Q8C2×C411yescharacteristic
68C2×C4⋊Q8C2×C411yescharacteristic
69C2×C4⋊Q8C2×C411yescharacteristic
70C2×#127C2×C411yescharacteristic
71C2×#125—21no—
72C2×#124—21no—
73C2×C22⋊Q8—21no—
74C2×C22⋊Q8C2312yes—
75C2×#121—21no—
76C2×#121C2312yes—
77C2×C4×Q8—21no—
78C2×C4×Q8C2×C412yes—
79C2×C4×D4—21no—
80C2×C4×D4C2×C412yes—
81C2×#117—21no—
82C2×#117—21no—
83C2×#117—21no—
84C2×#117D412yes—
85C22×C42C2311yescharacteristic
86#498—21no—
87#498—22no—
88#498—21no—
89#492—21no—
90#492—22no—
91#492—21no—
92#487—21no—
93#487—21no—
94#487—22no—
95#486—21no—
96#486—21no—
97#486—22no—
98#465—24no—
99#465—24no—
100#465—24no—
101#465—24no—
102C4×SD16—24no—
103C4×SD16—24no—
104C4×SD16—24no—
105C4×SD16—24no—
106#431—21no—
107#431—21no—
108#431D412yes—
109#431D412yes—
110C2×#106—24no—
111C2×#106C2×C414yes—
112C2×C4⋊C8—21no—
113C2×#103C2×C414yes—
114C2×#102C2×C414yes—
115#410—48no—
116C2×#99—28no—
117C2×C22⋊C8C2312yes—
118C2×C4×C8—21no—
119C2×C4×C8—24no—
120C2×C4×C8C2×C411yescharacteristic
121C2×C4×C8C2×C411yescharacteristic
122C2×C4×C8C2×C412yes—
123C2×C4×C8D411yescharacteristic
124C2×C4×C8D411yescharacteristic
125C2×C4×C8D411yescharacteristic
126C2×C4×C8D411yescharacteristic
127C2×C4×C8D412yes—
128C2×C4×C8D412yes—
129#384—21no—
130#384—21no—
131#382—21no—
132#382—21no—
133#380—21no—
134#380—21no—
135#380C2312yes—
136#380C2312yes—
137C4×C4⋊C4—21no—
138C4×C4⋊C4—21no—
139C4×C4⋊C4—21no—
140C4×C4⋊C4—21no—
141C4×C4⋊C4—21no—
142C4×C4⋊C4—21no—
143C4×C22⋊C4D412yes—
144C4×C22⋊C4C2312yes—
145#376—21no—
146C43—21no—
147#342—28no—
148#340—24no—
149#336—24no—
150#328—28no—
151#327—28no—
152C2×#41—41no—
153C2×#41—41no—
154C2×#41—41no—
155C2×#41—41no—
156C2×#41—44no—
157C2×#41—44no—
158C22×Q8—21no—
159C22×Q8—21no—
160C22×Q8C22×C412yes—
161C22×D4—22no—
162C22×D4—21no—
163C22×D4—21no—
164C22×D4C22×C412yes—
165C23×C4—22no—
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2411yescharacteristic
169C2×SD16—24no—
170C2×SD16—24no—
171C2×SD16—48no—
172C22×C8—22no—
173C22×C8—44no—
174C22×C8C22×C412yes—
175C22×C8C22×C412yes—
176C22×C8C2×D412yes—
177C22×C8C2×D412yes—
178C22×C8C2×D412yes—
179C22×C8C2×D412yes—
180C4⋊Q8—21no—
181C4⋊Q8—21no—
182C4⋊Q8—21no—
183C4⋊Q8—21no—
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192#127—21no—
193#127—21no—
194#127—21no—
195#127—21no—
196#125—41no—
197#125—41no—
198#125—41no—
199#125—41no—
200#124—41no—
201#124—41no—
202#124—41no—
203#124—41no—
204C22⋊Q8—22no—
205C22⋊Q8—22no—
206C22⋊Q8—22no—
207C22⋊Q8—22no—
208C22⋊Q8—41no—
209C22⋊Q8—41no—
210C22⋊Q8—41no—
211C22⋊Q8—41no—
212#121—22no—
213#121—22no—
214#121—22no—
215#121—22no—
216#121—41no—
217#121—41no—
218#121—41no—
219#121—41no—
220C4×Q8—22no—
221C4×Q8—22no—
222C4×Q8—22no—
223C4×Q8—22no—
224C4×Q8—41no—
225C4×Q8—41no—
226C4×Q8—41no—
227C4×Q8—41no—
228C4×D4—22no—
229C4×D4—22no—
230C4×D4—22no—
231C4×D4—22no—
232C4×D4—41no—
233C4×D4—41no—
234C4×D4—41no—
235C4×D4—41no—
236#117—24no—
237#117—41no—
238#117—41no—
239#117—41no—
240#117—41no—
241#117—44no—
242#117—44no—
243#117SD1618yescomplemented
244C2×C4⋊C4—22no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—41no—
254C2×C4⋊C4—41no—
255C2×C4⋊C4—41no—
256C2×C4⋊C4—41no—
257C2×C4⋊C4C22×C412yes—
258C2×C4⋊C4C22×C412yes—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4C2×D414yes—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—41no—
270C2×C42—41no—
271C2×C42C22×C411yescharacteristic
272C2×C42C22×C411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42#4111yescharacteristic
280C2×C42#4111yescharacteristic
281C2×C42#4111yescharacteristic
282C2×C42#4111yescharacteristic
283#106—24no—
284#106—24no—
285#106—48no—
286C4⋊C8—22no—
287C4⋊C8—22no—
288#103—24no—
289#103—24no—
290#102—24no—
291#102—24no—
292#99—416no—
293C22⋊C8—22no—
294C22⋊C8C2×D414yes—
295C4×C8—22no—
296C4×C8—22no—
297C4×C8—48no—
298C4×C8C22⋊C412yes—
299C4×C8C22⋊C412yes—
300C4×C8C22⋊C412yes—
301C4×C8C22⋊C412yes—
302C4×C8C22⋊C411yescharacteristic
303C4×C8C22⋊C411yescharacteristic
304C4×C8C22⋊C412yes—
305C4×C8C22⋊C411yescharacteristic
306C4×C8C22⋊C412yes—
307C4×C8C22⋊C411yescharacteristic
308#95—22no—
309#95—22no—
310#95—22no—
311#95—22no—
312#95—41no—
313#95—41no—
314C24—22no—
315C24C22×D411yescharacteristic
316#41—84no—
317#41—84no—
318#41—82no—
319#41—82no—
320#41—82no—
321#41—82no—
322C2×Q8—22no—
323C2×Q8—22no—
324C2×Q8—44no—
325C2×Q8—41no—
326C2×Q8—41no—
327C2×Q8—41no—
328C2×Q8—41no—
329C2×Q8—44no—
330C2×Q8—82no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—44no—
334C2×D4—42no—
335C2×D4—42no—
336C2×D4—44no—
337C2×D4—44no—
338C2×D4—41no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—41no—
342C2×D4—44no—
343C2×D4—82no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—48no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4C4×D411yescharacteristic
356C22×C4C4×D411yescharacteristic
357C22×C4C4×D411yescharacteristic
358C22×C4C4×D411yescharacteristic
359C22×C4#12111yescharacteristic
360C22×C4#12111yescharacteristic
361C22×C4#12111yescharacteristic
362C22×C4#12111yescharacteristic
363C22×C4#12711yescharacteristic
364C22×C4#12711yescharacteristic
365C22×C4#12711yescharacteristic
366C22×C4#12711yescharacteristic
367C22×C4C2×SD1614yes—
368C22×C4#13514yes—
369C22×C4C23×C411yescharacteristic
370C22×C4C22×D411yescharacteristic
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C2×#4111yescharacteristic
374C22×C4C2×#4111yescharacteristic
375SD16—48no—
376SD16—48no—
377C2×C8—24no—
378C2×C8—24no—
379C2×C8—24no—
380C2×C8—24no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—44no—
385C2×C8C2×C22⋊C412yes—
386C2×C8C2×C22⋊C412yes—
387C2×C8C2×C22⋊C412yes—
388C2×C8C2×C22⋊C412yes—
389C4⋊C4—28no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—44no—
395C4⋊C4—44no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—82no—
409C4⋊C4—82no—
410C4⋊C4—82no—
411C4⋊C4—82no—
412C22⋊C4—24no—
413C22⋊C4—42no—
414C22⋊C4—42no—
415C22⋊C4—42no—
416C22⋊C4—42no—
417C22⋊C4—42no—
418C22⋊C4—42no—
419C22⋊C4—44no—
420C22⋊C4—44no—
421C22⋊C4C2×SD1618yes—
422C42—28no—
423C42—44no—
424C42—44no—
425C42—42no—
426C42—42no—
427C42—82no—
428C42—82no—
429C42C2×C22⋊C411yescharacteristic
430C42C2×C22⋊C411yescharacteristic
431C42C2×C22⋊C411yescharacteristic
432C42C2×C22⋊C411yescharacteristic
433C42C4×D411yescharacteristic
434C42C4×D411yescharacteristic
435C42C4×D411yescharacteristic
436C42C4×D411yescharacteristic
437C42#12111yescharacteristic
438C42#12111yescharacteristic
439C42#12111yescharacteristic
440C42#12111yescharacteristic
441C42#12411yescharacteristic
442C42#12411yescharacteristic
443C42#12411yescharacteristic
444C42#12411yescharacteristic
445C23—41no—
446C23—41no—
447C23—42no—
448C23—44no—
449C23—82no—
450C23—82no—
451C23C2×C4×D411yescharacteristic
452C23C2×#12111yescharacteristic
453C23C2×#12711yescharacteristic
454C23C22×SD1612yes—
455C23C2×#13512yes—
456Q8—44no—
457Q8—82no—
458Q8—82no—
459Q8—82no—
460Q8—82no—
461Q8—82no—
462D4—44no—
463D4—82no—
464D4—82no—
465D4—82no—
466D4—82no—
467D4—84no—
468D4—82no—
469D4—82no—
470D4—82no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—28no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—84no—
491C2×C4—84no—
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#39011yescharacteristic
501C2×C4#39011yescharacteristic
502C2×C4#39011yescharacteristic
503C2×C4#39011yescharacteristic
504C2×C4#39011yescharacteristic
505C2×C4#39011yescharacteristic
506C2×C4#39011yescharacteristic
507C2×C4#39011yescharacteristic
508C2×C4C4×SD1614yes—
509C2×C4C4×SD1614yes—
510C2×C4#46514yes—
511C2×C4#46514yes—
512C2×C4#49212yes—
513C2×C4#49212yes—
514C2×C4#49512yes—
515C2×C4#49512yes—
516C2×C4C22×C22⋊C411yescharacteristic
517C2×C4C2×C4×D411yescharacteristic
518C2×C4C2×#12111yescharacteristic
519C2×C4C2×#12411yescharacteristic
520C8—22no—
521C8—22no—
522C8—22no—
523C8—44no—
524C8#41014yescomplemented
525C22—21no—
526C22—21no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—42no—
532C22—42no—
533C22—82no—
534C22—82no—
535C22—82no—
536C22—164no—
537C22C2×#39011yescharacteristic
538C22C2×C4×SD1612yes—
539C22C2×#46512yes—
540C22C2×#49211yescharacteristic
541C22C2×#49511yescharacteristic
542C4—21no—
543C4—21no—
544C4—21no—
545C4—21no—
546C4—21no—
547C4—21no—
548C4—48no—
549C4—82no—
550C4—82no—
551C4—82no—
552C4—162no—
553C4—162no—
554C4#196812yes—
555C4#196812yes—
556C4#196812yes—
557C4#196812yes—
558C4#197612yes—
559C4C2×#41012yes—
560C2—22no—
561C2—21no—
562C2—21no—
563C2—22no—
564C2—162no—
565C2#2121211yescharacteristic
566C2C2×#196811yescharacteristic
567C2C2×#197611yescharacteristic
subgroupG/H
centerC2×C4#390
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22C2×#390
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21332C2semidirect
4. #20816C2semidirect
5. C8×#117C2semidirect
6. #17575C2semidirect
8. #9781C2semidirect
17. C2×#498C4semidirect
18. C2×#492C4semidirect
19. C2×#487C4semidirect
20. C2×#486C4semidirect
22. C2×C4×SD16C4semidirect
243. #117SD16semidirect
524. C8#410semidirect

Maximal subgroups

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

subgroupindex
C2×#31672
#213322
#212122
#208162
C8×#1172
#175752
#175742
#97812