Finite Group Databasebeta

← #185085

Subgroups — #185085

Order 512.

2304 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3507C211yesmaximal, characteristic
2#21818C211yesmaximal, characteristic, complemented
3C2×#1518C211yesmaximal, characteristic, complemented
4C2×#1517C211yesmaximal, characteristic
5C2×#1503C211yesmaximal, characteristic, complemented
6C2×#1502C211yesmaximal, characteristic
7#13942C211yesmaximal, characteristic, complemented
8#8330C218yesmaximal, complemented
9#3507—24no—
10#3507—24no—
11#3507—24no—
12#3507—24no—
13C2×#559C412yescomplemented
14C2×#557C412yescomplemented
15C2×#556—22no—
16C2×#556C412yescomplemented
17C2×#554C412yescomplemented
18C2×#552—22no—
19C2×#539—22no—
20C2×#536—21no—
21C2×#533C2211yescharacteristic
22C2×#532C2211yescharacteristic
23C2×#529C2214yes—
24C2×#520—21no—
25C22×C4×Q8C2211yescharacteristic
26C2×#431C2211yescharacteristic
27C2×#431C2211yescharacteristic
28C2×#391—21no—
29C2×#382C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31#1984—22no—
32#1984—22no—
33#1817C2214yes—
34#1518—24no—
35#1517—24no—
36#1503—24no—
37#1502—24no—
38C2×#328C2214yes—
39C2×#328C2214yes—
40C2×#327C2214yes—
41C2×#327C2214yes—
42#1441—22no—
43C4⋊#95—21no—
44#1434C2214yes—
45C22×#41—21no—
46C23×Q8D411yescharacteristic
47#559—28no—
48#559—28no—
49#557—28no—
50#557—28no—
51#556—216no—
52#556—216no—
53#556—28no—
54#556—28no—
55#554—28no—
56#554—28no—
57#552—216no—
58#552—216no—
59#539—216no—
60#539—216no—
61#536—28no—
62#536—28no—
63#533—22no—
64#533—22no—
65#533D414yes—
66#533D414yes—
67#532—22no—
68#532—22no—
69#532D414yes—
70#532D414yes—
71C2×C4⋊Q8—22no—
72C2×C4⋊Q8C2×C411yescharacteristic
73C2×C4⋊Q8C2×C411yescharacteristic
74C2×C4⋊Q8C2×C411yescharacteristic
75C2×#127C2×C411yescharacteristic
76#529—28no—
77#529—28no—
78#529—28no—
79#529—28no—
80C2×#126—24no—
81C2×#126—24no—
82C2×#125—24no—
83C2×#125—21no—
84C2×#125C2×C414yes—
85C2×#124—22no—
86C2×#124—21no—
87C2×#124C2×C414yes—
88C2×#123—24no—
89C2×#123—24no—
90C2×C22⋊Q8—22no—
91C2×C22⋊Q8—22no—
92C2×C22⋊Q8—21no—
93C2×C22⋊Q8—24no—
94C2×C22⋊Q8C2312yes—
95C2×#121—21no—
96C2×#121C2312yes—
97#520—28no—
98#520—28no—
99C2×C4×Q8—22no—
100C2×C4×Q8—22no—
101C2×C4×Q8—22no—
102C2×C4×Q8—22no—
103C2×C4×Q8—22no—
104C2×C4×Q8—21no—
105C2×C4×Q8C2×C414yes—
106C2×C4×Q8C2×C412yes—
107C2×C4×Q8C2×C412yes—
108C2×C4×Q8C2×C412yes—
109C2×C4×Q8D412yes—
110C2×C4×Q8D412yes—
111C2×C4×D4—24no—
112C2×C4×D4—22no—
113C2×C4×D4—21no—
114C2×C4×D4C2×C414yes—
115C2×C4×D4C2×C412yes—
116C2×#117—21no—
117C2×#117—22no—
118C2×#117—24no—
119C22×C4⋊C4—22no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4D411yescharacteristic
123C22×C4⋊C4D412yes—
124C22×C42—21no—
125C22×C42C2312yes—
126C22×C42C2311yescharacteristic
127#431—22no—
128#431—22no—
129#431—22no—
130#431—22no—
131C2×C4⋊C8—21no—
132C2×C4⋊C8—21no—
133C2×C22⋊C8C2312yes—
134C2×C22⋊C8C2312yes—
135C2×C4×C8—21no—
136C2×C4×C8—21no—
137#391—44no—
138#382—22no—
139#382—22no—
140C4×C4⋊C4—22no—
141C4×C4⋊C4—22no—
142C2×#95—24no—
143C2×#95—24no—
144C2×#95—21no—
145C2×#95C2312yes—
146C2×#95C2312yes—
147#336—28no—
148#328—28no—
149#328—28no—
150#327—28no—
151#327—28no—
152C2×#41—24no—
153C2×#41—24no—
154C2×#41—42no—
155C2×#41—42no—
156C2×#41—42no—
157C2×#41—42no—
158C22×Q8—21no—
159C22×Q8—21no—
160C22×Q8—22no—
161C22×Q8—22no—
162C22×Q8—22no—
163C22×Q8—22no—
164C22×Q8—22no—
165C22×Q8—22no—
166C22×Q8C22⋊C412yes—
167C22×Q8C22⋊C412yes—
168C22×Q8C22×C412yes—
169C22×D4—22no—
170C22×D4—21no—
171C22×D4—21no—
172C22×D4C22×C412yes—
173C23×C4—22no—
174C23×C4—24no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4C2×D412yes—
178C23×C4C2×D412yes—
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2411yescharacteristic
182C22×C8—22no—
183C22×C8—22no—
184C4⋊Q8—216no—
185C4⋊Q8—22no—
186C4⋊Q8—22no—
187C4⋊Q8—22no—
188C4⋊Q8C22⋊C414yes—
189C4⋊Q8C22⋊C414yes—
190C4⋊Q8C22⋊C414yes—
191#127—22no—
192#127C22⋊C414yes—
193#126—216no—
194#126—216no—
195#126—232no—
196#125—232no—
197#125—28no—
198#125—28no—
199#125—24no—
200#125—24no—
201#124—216no—
202#124—28no—
203#124—28no—
204#124—24no—
205#124—24no—
206#123—216no—
207#123—216no—
208#123—232no—
209C22⋊Q8—216no—
210C22⋊Q8—216no—
211C22⋊Q8—232no—
212C22⋊Q8—24no—
213C22⋊Q8—24no—
214C22⋊Q8—22no—
215C22⋊Q8—22no—
216C22⋊Q8C2×D414yes—
217C22⋊Q8C2×D414yes—
218#121—24no—
219#121—24no—
220#121—22no—
221#121—22no—
222#121C2×D414yes—
223#121C2×D414yes—
224C4×Q8—216no—
225C4×Q8—28no—
226C4×Q8—28no—
227C4×Q8—24no—
228C4×Q8—24no—
229C4×Q8—24no—
230C4×Q8—24no—
231C4×Q8—24no—
232C4×Q8—24no—
233C4×Q8—24no—
234C4×Q8—24no—
235C4×Q8—24no—
236C4×Q8—24no—
237C4×Q8—24no—
238C4×Q8C22⋊C418yes—
239C4×D4—216no—
240C4×D4—216no—
241C4×D4—216no—
242C4×D4—28no—
243C4×D4—28no—
244C4×D4—24no—
245C4×D4—24no—
246C4×D4—24no—
247C4×D4C22⋊C418yes—
248#117—216no—
249#117—216no—
250#117—216no—
251#117—24no—
252#117—24no—
253C2×C4⋊C4—216no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4C22⋊C412yes—
273C2×C4⋊C4C22⋊C412yes—
274C2×C4⋊C4C22⋊C414yes—
275C2×C4⋊C4C22⋊C414yes—
276C2×C4⋊C4C22×C412yes—
277C2×C4⋊C4C22×C412yes—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—22no—
284C2×C42—24no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—42no—
291C2×C42—42no—
292C2×C42—42no—
293C2×C42C22×C414yes—
294C2×C42C22×C411yescharacteristic
295C2×C42C22×C411yescharacteristic
296C2×C42C2×D414yes—
297C2×C42C2×D412yes—
298C2×C42C2×D412yes—
299C2×C42C2×D412yes—
300C2×C42C2×D412yes—
301C2×C42C2×D411yescharacteristic
302C2×C42C2×D411yescharacteristic
303C4⋊C8—42no—
304C4⋊C8—42no—
305C22⋊C8—22no—
306C22⋊C8—22no—
307C22⋊C8—22no—
308C22⋊C8—22no—
309C4×C8—42no—
310C4×C8—42no—
311#95—22no—
312#95—22no—
313#95—22no—
314#95—22no—
315#95—48no—
316#95—48no—
317#95—42no—
318C24—22no—
319C24C22×D411yescharacteristic
320#41—48no—
321#41—48no—
322#41—48no—
323#41—48no—
324C2×Q8—22no—
325C2×Q8—24no—
326C2×Q8—24no—
327C2×Q8—24no—
328C2×Q8—24no—
329C2×Q8—24no—
330C2×Q8—24no—
331C2×Q8—24no—
332C2×Q8—24no—
333C2×Q8—44no—
334C2×Q8—42no—
335C2×Q8—42no—
336C2×Q8—44no—
337C2×Q8—44no—
338C2×Q8—44no—
339C2×Q8—44no—
340C2×Q8—82no—
341C2×Q8C2×C22⋊C414yes—
342C2×D4—24no—
343C2×D4—24no—
344C2×D4—22no—
345C2×D4—24no—
346C2×D4—24no—
347C2×D4—44no—
348C2×D4—42no—
349C2×D4—42no—
350C2×D4—42no—
351C2×D4—42no—
352C2×D4—42no—
353C2×D4—42no—
354C2×D4—82no—
355C2×D4C2×C22⋊C414yes—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—216no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—28no—
369C22×C4—24no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—44no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—88no—
387C22×C4C2×C22⋊C414yes—
388C22×C4C2×C22⋊C414yes—
389C22×C4C2×C22⋊C411yescharacteristic
390C22×C4C2×C22⋊C411yescharacteristic
391C22×C4C2×C22⋊C411yescharacteristic
392C22×C4C2×C22⋊C411yescharacteristic
393C22×C4#12012yes—
394C22×C4#12012yes—
395C22×C4#12012yes—
396C22×C4#12012yes—
397C22×C4#13512yes—
398C22×C4#13512yes—
399C22×C4AGL(1,ℤ/8)12yes—
400C22×C4#13712yes—
401C22×C4C23×C411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D411yescharacteristic
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C2×C8—44no—
408C2×C8—44no—
409C2×C8—42no—
410C2×C8—42no—
411C4⋊C4—216no—
412C4⋊C4—216no—
413C4⋊C4—216no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—28no—
422C4⋊C4—28no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—28no—
427C4⋊C4—48no—
428C4⋊C4—84no—
429C4⋊C4C2×C22⋊C414yes—
430C4⋊C4C2×C22⋊C414yes—
431C22⋊C4—216no—
432C22⋊C4—216no—
433C22⋊C4—216no—
434C22⋊C4—24no—
435C22⋊C4—24no—
436C22⋊C4—24no—
437C22⋊C4—24no—
438C22⋊C4—28no—
439C42—216no—
440C42—24no—
441C42—24no—
442C42—24no—
443C42—24no—
444C42—28no—
445C42—84no—
446C42C2×C22⋊C412yes—
447C42C2×C22⋊C412yes—
448C42C2×C22⋊C412yes—
449C42C2×C22⋊C412yes—
450C42#12012yes—
451C42#12012yes—
452C42#12012yes—
453C42#12012yes—
454C23—22no—
455C23—22no—
456C23—24no—
457C23—44no—
458C23—42no—
459C23—42no—
460C23—44no—
461C23C22×C22⋊C411yescharacteristic
462C23C2×#12011yescharacteristic
463C23C2×#12011yescharacteristic
464C23C2×#13511yescharacteristic
465C23C2×#13511yescharacteristic
466C23C2×AGL(1,ℤ/8)11yescharacteristic
467C23C2×#13711yescharacteristic
468Q8—44no—
469Q8—44no—
470Q8—48no—
471Q8—48no—
472Q8—48no—
473Q8—84no—
474D4—44no—
475D4—44no—
476D4—44no—
477D4—44no—
478D4—44no—
479D4—44no—
480D4—48no—
481D4—84no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—28no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—416no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—816no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—84no—
522C2×C4—84no—
523C2×C4—82no—
524C2×C4#37912yes—
525C2×C4#37912yes—
526C2×C4#37912yes—
527C2×C4#37912yes—
528C2×C4#41612yes—
529C2×C4#41612yes—
530C2×C4#41712yes—
531C2×C4#41712yes—
532C2×C4#44914yes—
533C2×C4#44914yes—
534C2×C4#45214yes—
535C2×C4#45214yes—
536C2×C4C22×C22⋊C411yescharacteristic
537C2×C4C22×C22⋊C411yescharacteristic
538C2×C4C2×#12011yescharacteristic
539C2×C4C2×#12011yescharacteristic
540C8—82no—
541C8—82no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—44no—
551C22—44no—
552C22—44no—
553C22—84no—
554C22—84no—
555C22C2×#37911yescharacteristic
556C22#288412yes—
557C22C2×#41611yescharacteristic
558C22C2×#41711yescharacteristic
559C22#290112yes—
560C22C2×#44911yescharacteristic
561C22C2×#44911yescharacteristic
562C22C2×#45211yescharacteristic
563C22C2×#45211yescharacteristic
564C4—21no—
565C4—21no—
566C4—21no—
567C4—21no—
568C4—22no—
569C4—22no—
570C4—44no—
571C4—44no—
572C4—44no—
573C4—44no—
574C4—44no—
575C4—84no—
576C4—162no—
577C4#178814yes—
578C4#178814yes—
579C2—21no—
580C2—21no—
581C2—21no—
582C2—21no—
583C2—84no—
584C2#1761612yes—
585C2#1762812yes—
586C2#2058711yescharacteristic
587C2C2×#178811yescharacteristic
588C2#2060611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21818C2semidirect
3. C2×#1518C2semidirect
5. C2×#1503C2semidirect
7. #13942C2semidirect
8. #8330C2semidirect
13. C2×#559C4semidirect
14. C2×#557C4semidirect
16. C2×#556C4semidirect
17. C2×#554C4semidirect

Maximal subgroups

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

subgroupindex
C2×#35072
#218182
C2×#15182
C2×#15172
C2×#15032
C2×#15022
#139422
#83302