Finite Group Databasebeta

← #6434687

Subgroups — #6434687

Order 512.

2866 conjugacy classes of subgroups.

The subgroup lattice has 621 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3458C211yesmaximal, characteristic
2C2×#3095C211yesmaximal, characteristic
3C22×#485C211yesmaximal, characteristic
4#35311C211yesmaximal, characteristic
5#29649C214yesmaximal, complemented
6#29648C214yesmaximal, complemented
7#29645C218yesmaximal, complemented
8#23556C212yesmaximal, complemented
9#23555C212yesmaximal, complemented
10#23457C212yesmaximal, complemented
11#23457C212yesmaximal, complemented
12#20701C211yesmaximal, characteristic
13#20699C211yesmaximal, characteristic
14#20685C211yesmaximal, characteristic
15#3458—28no—
16#3458—28no—
17C2×#544—21no—
18C2×#544—21no—
19C2×#544C2212yes—
20C2×#544C2212yes—
21C2×#544C2212yes—
22C2×#544C2212yes—
23C2×#543—22no—
24C2×#533—22no—
25C2×#533—22no—
26C2×#533C2214yes—
27C22×C22⋊Q8C2211yescharacteristic
28C22×#117C2211yescharacteristic
29C22×#117C2211yescharacteristic
30C22×#117C2211yescharacteristic
31#3095—28no—
32C2×#484C2212yes—
33C2×#485—28no—
34C2×#485C2214yes—
35C2×#485C2214yes—
36C2×#485C2214yes—
37C2×#483C2212yes—
38C2×#428C2212yes—
39C22×#107C2211yescharacteristic
40C22×#106C2211yescharacteristic
41C2×#419C2212yes—
42C22×#103C2211yescharacteristic
43C22×#103C2211yescharacteristic
44C2×#407C2211yescharacteristic
45C22×C22⋊C8C2211yescharacteristic
46#2855C2214yes—
47#2625C2214yes—
48#2503—21no—
49#2502—21no—
50#2499—22no—
51#2484—22no—
52#2483—22no—
53#2317C2212yes—
54#2310C2212yes—
55#2280C2211yescharacteristic
56#2059C2218yescomplemented
57#2059C2218yescomplemented
58#2058C2218yescomplemented
59#2057C2218yescomplemented
60#2026C2218yescomplemented
61#2026C2218yescomplemented
62#2024C2218yescomplemented
63#2024C2218yescomplemented
64#1825C2214yes—
65#1825C2214yes—
66#1825C2214yes—
67#1824C2212yes—
68#1823C2212yes—
69#1818C2214yes—
70#1817C2214yes—
71#1736C2212yes—
72#1736C2212yes—
73#1735C2212yes—
74#1735C2212yes—
75#1729C2212yes—
76#1729C2212yes—
77C23×Q8C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C22×M16C2312yes—
80C23×C8C2311yescharacteristic
81C23×C8C2311yescharacteristic
82#544—216no—
83#544—216no—
84#544—28no—
85#544—28no—
86#544—28no—
87#544—28no—
88#544—28no—
89#544—28no—
90#544—28no—
91#544—28no—
92#543—232no—
93#533—232no—
94#533—232no—
95#533—216no—
96#533—216no—
97C2×C4⋊Q8—22no—
98C2×C4⋊Q8—22no—
99C2×C4⋊Q8—22no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8C2314yes—
102C2×C4⋊Q8C2314yes—
103C2×#125—22no—
104C2×#125—22no—
105C2×#125—22no—
106C2×#125—22no—
107C2×#125C2314yes—
108C2×#125C2314yes—
109C2×C22⋊Q8—24no—
110C2×C22⋊Q8—24no—
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8—21no—
113C2×C22⋊Q8—24no—
114C2×C22⋊Q8—24no—
115C2×C22⋊Q8—24no—
116C2×C22⋊Q8—24no—
117C2×C22⋊Q8—24no—
118C2×C22⋊Q8—24no—
119C2×C22⋊Q8C2312yes—
120C2×C22⋊Q8C2312yes—
121C2×C4×Q8—24no—
122C2×C4×Q8—24no—
123C2×#117—21no—
124C2×#117—21no—
125C2×#117—21no—
126C2×#117—21no—
127C2×#117—24no—
128C2×#117—24no—
129C2×#117—24no—
130C2×#117—24no—
131C2×#117D418yes—
132C2×#117D418yes—
133C2×#117D412yes—
134C2×#117D412yes—
135C2×#117C2312yes—
136C2×#117C2312yes—
137C2×#117C2312yes—
138C2×#117C2312yes—
139C2×#117C2312yes—
140C2×#117C2312yes—
141C2×#117C2312yes—
142C22×C4⋊C4—21no—
143C22×C4⋊C4D411yescharacteristic
144C22×C4⋊C4D411yescharacteristic
145C22×C4⋊C4C2311yescharacteristic
146C22×C4⋊C4C2311yescharacteristic
147C22×C4⋊C4C2311yescharacteristic
148C22×C4⋊C4C2311yescharacteristic
149C22×C4⋊C4C2311yescharacteristic
150C22×C22⋊C4—21no—
151C22×C22⋊C4—21no—
152C22×C22⋊C4—21no—
153C22×C22⋊C4D412yes—
154C22×C42—21no—
155C22×C42—21no—
156C22×C42C2312yes—
157#485—216no—
158#485—216no—
159#485—216no—
160#484—28no—
161#483—28no—
162#428—28no—
163C2×#107—24no—
164C2×#107C2314yes—
165C2×#107C2314yes—
166C2×#106—24no—
167C2×#106C2314yes—
168C2×#106C2314yes—
169#419—28no—
170C2×#103—24no—
171C2×#103—24no—
172C2×#103C2314yes—
173C2×#103C2314yes—
174C2×#103C2314yes—
175C2×#103C2314yes—
176#407—24no—
177C2×C22⋊C8—24no—
178C2×C22⋊C8C2312yes—
179C2×C22⋊C8C2312yes—
180C2×C22⋊C8C2312yes—
181C2×C22⋊C8C2312yes—
182#400—24no—
183#398—24no—
184#397—28no—
185#391—22no—
186#391—22no—
187#391C2318yes—
188#389C2318yes—
189#387—28no—
190#385—24no—
191#385—24no—
192#384—22no—
193#382—22no—
194#381—24no—
195#381—21no—
196#381—24no—
197#380C2312yes—
198#380C2312yes—
199#379—21no—
200C4×C22⋊C4—21no—
201C4×C22⋊C4—21no—
202#340C2314yes—
203#340C2318yes—
204#340C2314yes—
205#340C2318yes—
206#340C2314yes—
207#340C2314yes—
208#336C2318yes—
209#336C2318yes—
210C25C2×D411yescharacteristic
211C22×Q8—24no—
212C22×Q8—24no—
213C22×Q8—21no—
214C22×Q8—22no—
215C22×Q8—22no—
216C22×Q8—24no—
217C22×Q8—24no—
218C22×Q8C2412yes—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—21no—
222C23×C4—21no—
223C23×C4—21no—
224C23×C4—21no—
225C23×C4—21no—
226C23×C4—21no—
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D414yes—
231C23×C4C2×D414yes—
232C23×C4C2×D412yes—
233C23×C4C2×D412yes—
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C2×M16—24no—
238C2×M16—24no—
239C2×M16—24no—
240C22×C8—22no—
241C22×C8—22no—
242C22×C8—24no—
243C22×C8—24no—
244C22×C8C2412yes—
245C22×C8C2412yes—
246C22×C8C2414yes—
247C4⋊Q8—216no—
248C4⋊Q8—216no—
249C4⋊Q8—216no—
250C4⋊Q8—216no—
251C4⋊Q8—28no—
252C4⋊Q8—28no—
253C4⋊Q8—28no—
254C4⋊Q8—28no—
255#125—216no—
256#125—216no—
257#125—216no—
258#125—216no—
259#125—28no—
260#125—28no—
261#125—28no—
262#125—28no—
263C22⋊Q8—232no—
264C22⋊Q8—232no—
265C22⋊Q8—232no—
266C22⋊Q8—232no—
267C22⋊Q8—232no—
268C22⋊Q8—232no—
269C22⋊Q8—24no—
270C22⋊Q8—24no—
271C22⋊Q8—24no—
272C22⋊Q8—24no—
273C22⋊Q8—44no—
274C22⋊Q8—44no—
275C4×Q8—232no—
276C4×Q8—232no—
277#117—28no—
278#117—28no—
279#117—28no—
280#117—28no—
281#117—28no—
282#117—28no—
283#117—24no—
284#117—24no—
285#117—24no—
286#117—24no—
287#117—24no—
288#117—24no—
289#117—24no—
290#117—24no—
291#117—24no—
292#117—24no—
293#117C2×D418yes—
294#117C2×D418yes—
295#117C2×D418yes—
296#117C2×D418yes—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—28no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2×D412yes—
323C2×C4⋊C4C2×D412yes—
324C2×C4⋊C4C2×D414yes—
325C2×C4⋊C4C2×D414yes—
326C2×C4⋊C4C2×D414yes—
327C2×C4⋊C4C2×D414yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2414yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C22⋊C4—28no—
335C2×C22⋊C4—28no—
336C2×C22⋊C4—28no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—44no—
347C2×C22⋊C4C2×D414yes—
348C2×C22⋊C4C2×D414yes—
349C2×C42—28no—
350C2×C42—28no—
351C2×C42—24no—
352C2×C42—24no—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42—22no—
356C2×C42—22no—
357C2×C42—22no—
358C2×C42C2×D414yes—
359C2×C42C2414yes—
360#107—28no—
361#107—28no—
362#106—28no—
363#106—28no—
364#103—28no—
365#103—28no—
366#103—28no—
367#103—28no—
368C22⋊C8—24no—
369C22⋊C8—24no—
370C22⋊C8—24no—
371C22⋊C8—24no—
372#95—22no—
373#95—22no—
374#95—28no—
375#95—28no—
376#95—28no—
377C24—24no—
378C24—42no—
379C24—42no—
380C24#12014yes—
381C24C22×D411yescharacteristic
382C24C22×D411yescharacteristic
383C24C22×D411yescharacteristic
384C2×Q8—216no—
385C2×Q8—216no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—24no—
389C2×Q8—24no—
390C2×Q8—42no—
391C2×Q8—44no—
392C2×Q8—44no—
393C2×Q8—44no—
394C2×Q8—48no—
395C2×Q8—88no—
396C2×Q8—88no—
397C22×C4—24no—
398C22×C4—28no—
399C22×C4—28no—
400C22×C4—216no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—44no—
425C22×C4—82no—
426C22×C4—82no—
427C22×C4—82no—
428C22×C4—82no—
429C22×C4—82no—
430C22×C4#12014yes—
431C22×C4#12014yes—
432C22×C4#12014yes—
433C22×C4C22×D414yes—
434C22×C4C22×D414yes—
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C42−1+414yes—
445C22×C42−1+414yes—
446C22×C4C2511yescharacteristic
447M16—48no—
448C2×C8—22no—
449C2×C8—24no—
450C2×C8—22no—
451C2×C8—24no—
452C2×C8—24no—
453C2×C8—24no—
454C2×C8—24no—
455C2×C8—44no—
456C2×C8—44no—
457C4⋊C4—216no—
458C4⋊C4—216no—
459C4⋊C4—216no—
460C4⋊C4—216no—
461C4⋊C4—216no—
462C4⋊C4—216no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—42no—
480C4⋊C4—44no—
481C4⋊C4—44no—
482C4⋊C4—88no—
483C4⋊C4C22×D414yes—
484C4⋊C4C22×D414yes—
485C4⋊C4C22×D414yes—
486C4⋊C4C22×D414yes—
487C22⋊C4—28no—
488C22⋊C4—28no—
489C22⋊C4—28no—
490C22⋊C4—28no—
491C22⋊C4—28no—
492C22⋊C4—28no—
493C22⋊C4—44no—
494C22⋊C4—44no—
495C22⋊C4—44no—
496C22⋊C4—44no—
497C22⋊C4—82no—
498C42—28no—
499C42—28no—
500C42—28no—
501C42—28no—
502C42—28no—
503C42—24no—
504C42—24no—
505C42—44no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—28no—
515C23—44no—
516C23—44no—
517C23—88no—
518C23C2×#12012yes—
519C23C2×#12012yes—
520C23C2×#12012yes—
521C23#53614yes—
522C23C23×D411yescharacteristic
523Q8—42no—
524Q8—42no—
525Q8—44no—
526Q8—816no—
527Q8—816no—
528Q8—84no—
529C2×C4—28no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—28no—
533C2×C4—28no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—28no—
541C2×C4—28no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—84no—
560C2×C4—84no—
561C2×C4—84no—
562C2×C4—84no—
563C2×C4—84no—
564C2×C4—84no—
565C2×C4—84no—
566C2×C4—84no—
567C2×C4C2×#12012yes—
568C2×C4C2×#12012yes—
569C2×C4C2×#12012yes—
570C2×C4#53614yes—
571C2×C4#53614yes—
572C2×C4#53614yes—
573C2×C4#53618yes—
574C2×C4C23×D411yescharacteristic
575C2×C4C23×D411yescharacteristic
576C2×C4C2×2−1+412yes—
577C2×C4C2×2−1+412yes—
578C8—42no—
579C8—42no—
580C8—44no—
581C22—28no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—44no—
591C22—44no—
592C22—88no—
593C22#261414yes—
594C22C22×#12011yescharacteristic
595C22C2×#53612yes—
596C22#358412yes—
597C22#358612yes—
598C4—22no—
599C4—22no—
600C4—44no—
601C4—44no—
602C4—44no—
603C4—44no—
604C4—44no—
605C4—44no—
606C4—82no—
607C4—84no—
608C4—82no—
609C4#261414yes—
610C4C2×#53611yescharacteristic
611C4C2×#53611yescharacteristic
612C4C2×#53612yes—
613C2—24no—
614C2—42no—
615C2—42no—
616C2C2×#261411yescharacteristic
617C2#6036611yescharacteristic
618C2#6037011yescharacteristic
619C2#6087614yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #29649C2semidirect
6. #29648C2semidirect
7. #29645C2semidirect
8. #23556C2semidirect
9. #23555C2semidirect
10. #23457C2semidirect
11. #23457C2semidirect
56. #2059C22semidirect
57. #2059C22semidirect
58. #2058C22semidirect
59. #2057C22semidirect
60. #2026C22semidirect
61. #2026C22semidirect
62. #2024C22semidirect
63. #2024C22semidirect

Maximal subgroups

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

subgroupindex
C2×#34582
C2×#30952
C22×#4852
#353112
#296492
#296482
#296452
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