Finite Group Databasebeta

← #6434458

Subgroups — #6434458

Order 512.

1952 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36631C211yesmaximal, characteristic
2C2×C4×C4⋊Q8C211yesmaximal, characteristic, complemented
3#30130C214yesmaximal, complemented
4#30079C212yesmaximal, complemented
5#30041C212yesmaximal, complemented
6#29953C212yesmaximal, complemented
7#29906C212yesmaximal, complemented
8#29713C212yesmaximal
9#29624C212yesmaximal, complemented
10#23823C212yesmaximal, complemented
11#23803C212yesmaximal, complemented
12#23683C214yesmaximal, complemented
13#21454C214yesmaximal, complemented
14#20778C211yesmaximal, characteristic, complemented
15C22×C4⋊Q8C2211yescharacteristic
16C2×#529—21no—
17C22×C4×Q8C2212yes—
18C2×#448C2214yes—
19C2×#440C2218yes—
20#2861—21no—
21#2860—21no—
22#2860C2212yes—
23#2859C2211yescharacteristic
24#2858C2211yescharacteristic
25#2857C2211yescharacteristic
26#2856C2211yescharacteristic
27#2854—21no—
28#2854C2212yes—
29#2850—21no—
30C4×C4⋊Q8—21no—
31C4×C4⋊Q8—21no—
32C4×C4⋊Q8—21no—
33C4×C4⋊Q8—21no—
34C4×C4⋊Q8C2211yescharacteristic
35C4×C4⋊Q8C2212yes—
36C4×C4⋊Q8C2212yes—
37C4×C4⋊Q8C2212yes—
38C4×C4⋊Q8C2212yes—
39C4×#127C2211yescharacteristic
40C4×#124—21no—
41C4×#121—21no—
42C4×#121C2212yes—
43C2×#391C2212yes—
44#2293—21no—
45#2293C2212yes—
46C2×#389—21no—
47C2×#389—21no—
48C2×#384C2211yescharacteristic
49C2×C4×C4⋊C4—21no—
50C2×C4×C4⋊C4—21no—
51C2×C43C2211yescharacteristic
52#2099C2214yes—
53#2095C2214yes—
54#2093C2214yes—
55#2083C2214yes—
56#2072C2214yes—
57#2071C2214yes—
58#2068C2218yes—
59#2055C2218yes—
60#2043C2214yes—
61#2043C2214yes—
62#2024C2214yes—
63#2014C2214yes—
64#2013C2214yes—
65#2013C2214yes—
66#1880C2218yes—
67#1874C2214yes—
68#1863C2218yes—
69#1852C2214yes—
70#1839C2212yes—
71#1831C2212yes—
72#1817C2212yes—
73#1797C2212yes—
74#1787C2214yes—
75#1767C2212yes—
76#1763C2218yes—
77#1749C2212yes—
78#1745C2212yes—
79C23×Q8C2312yes—
80C22×SD16C2318yes—
81C2×C4⋊Q8—28no—
82C2×C4⋊Q8—21no—
83C2×C4⋊Q8—21no—
84C2×C4⋊Q8C2312yes—
85C2×C4⋊Q8C2312yes—
86C2×C4⋊Q8C2311yescharacteristic
87C2×#127C2311yescharacteristic
88#529—48no—
89C2×#126—22no—
90C2×#125—21no—
91C2×#124—21no—
92C2×#124—21no—
93C2×#123—22no—
94C2×C22⋊Q8—21no—
95C2×#121—21no—
96C2×#121—21no—
97C2×#121C2312yes—
98C2×C4×Q8—22no—
99C2×C4×Q8—22no—
100C2×C4×Q8—21no—
101C2×C4×Q8—48no—
102C2×C4×Q8C2314yes—
103C2×C4×Q8C2314yes—
104C2×C4×D4—22no—
105C2×C4×D4—21no—
106C2×C4×D4—21no—
107C2×C4×D4C2312yes—
108C2×#117—22no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—22no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4C2312yes—
115C22×C42—22no—
116C22×C42D411yescharacteristic
117C22×C42D411yescharacteristic
118C22×C42D411yescharacteristic
119C22×C42D411yescharacteristic
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123#448—216no—
124#440—416no—
125C2×#107C2318yes—
126C2×C4⋊C8C2314yes—
127C2×#103C2318yes—
128C2×#103C2318yes—
129C2×#103C2318yes—
130C2×#102C2318yes—
131C2×C22⋊C8C2314yes—
132C2×#97C2314yes—
133#391—22no—
134#391—22no—
135#391—21no—
136#391—21no—
137#391C2314yes—
138#391C2314yes—
139#390—21no—
140#390—21no—
141#390—21no—
142#390—21no—
143#390—21no—
144#390—21no—
145#390C2312yes—
146#390C2312yes—
147#390C2312yes—
148#390C2312yes—
149#389—22no—
150#389—22no—
151#389—22no—
152#389—22no—
153#389—22no—
154#389—22no—
155#389—22no—
156#389—22no—
157#389—21no—
158#389—21no—
159#389—21no—
160#389—21no—
161#389C2312yes—
162#389C2312yes—
163#389C2312yes—
164#389C2312yes—
165#388—22no—
166#388—22no—
167#388—22no—
168#388—22no—
169#388—22no—
170#388—22no—
171#388—22no—
172#388—22no—
173#386—22no—
174#386—22no—
175#384—21no—
176#384—21no—
177#384C2311yescharacteristic
178#384C2312yes—
179#384C2312yes—
180#384C2311yescharacteristic
181#382—21no—
182#380—21no—
183#380C2312yes—
184C4×C4⋊C4—22no—
185C4×C4⋊C4—22no—
186C4×C4⋊C4—22no—
187C4×C4⋊C4—22no—
188C4×C4⋊C4—22no—
189C4×C4⋊C4—22no—
190C4×C4⋊C4—22no—
191C4×C4⋊C4—22no—
192C4×C4⋊C4—21no—
193C4×C4⋊C4C2312yes—
194C4×C22⋊C4—22no—
195C4×C22⋊C4—22no—
196C2×#95—22no—
197C2×#95—22no—
198C43D411yescharacteristic
199C43D411yescharacteristic
200C43D411yescharacteristic
201C43D411yescharacteristic
202C43C2311yescharacteristic
203C43C2311yescharacteristic
204C43C2311yescharacteristic
205C43C2311yescharacteristic
206#340C2318yes—
207#340C2318yes—
208#336C2314yes—
209#336C2314yes—
210#336C2314yes—
211#336C2314yes—
212C22×Q8—22no—
213C22×Q8—21no—
214C22×Q8—24no—
215C22×Q8—44no—
216C22×Q8—44no—
217C22×Q8#4118yes—
218C22×Q8C2414yes—
219C22×D4—22no—
220C22×D4—21no—
221C22×D4—21no—
222C22×D4C2412yes—
223C23×C4—22no—
224C23×C4—22no—
225C23×C4—22no—
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2411yescharacteristic
233C2×SD16—216no—
234C2×SD16—48no—
235C2×SD16—48no—
236C22×C8C2418yes—
237C4⋊Q8—28no—
238C4⋊Q8—28no—
239C4⋊Q8—28no—
240C4⋊Q8—28no—
241C4⋊Q8—42no—
242#127—42no—
243#126—48no—
244#125—44no—
245#124—44no—
246#124—44no—
247#123—48no—
248C22⋊Q8—44no—
249#121—28no—
250#121—44no—
251#121—44no—
252C4×Q8—216no—
253C4×Q8—48no—
254C4×Q8—48no—
255C4×Q8—48no—
256C4×Q8—44no—
257C4×D4—28no—
258C4×D4—44no—
259C4×D4—44no—
260C4×D4—48no—
261#117—48no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—48no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4C2414yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2412yes—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—22no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—42no—
312C2×C42—42no—
313C2×C42—48no—
314C2×C42C2×D411yescharacteristic
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D411yescharacteristic
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2411yescharacteristic
337C2×C42C2411yescharacteristic
338C2×C42C2411yescharacteristic
339C2×C42C2411yescharacteristic
340C2×C42C2411yescharacteristic
341C2×C42C2411yescharacteristic
342#107—48no—
343C4⋊C8—44no—
344#103—216no—
345#103—48no—
346#103—48no—
347#102—48no—
348C22⋊C8—28no—
349#97—28no—
350#95—24no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—22no—
360C24—22no—
361C24C22×D411yescharacteristic
362C2×Q8—28no—
363C2×Q8—24no—
364C2×Q8—28no—
365C2×Q8—28no—
366C2×Q8—416no—
367C2×Q8—44no—
368C2×Q8—44no—
369C2×Q8—416no—
370C2×Q8—42no—
371C2×Q8—42no—
372C2×Q8—42no—
373C2×Q8C2×#4118yes—
374C2×D4—24no—
375C2×D4—24no—
376C2×D4—24no—
377C2×D4—42no—
378C2×D4—42no—
379C2×D4—42no—
380C2×D4—42no—
381C2×D4—44no—
382C2×D4—44no—
383C2×D4—44no—
384C2×D4—42no—
385C2×D4—42no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—28no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4AGL(1,ℤ/8)14yes—
419C22×C4#13714yes—
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C2×#4114yes—
434C22×C4C2511yescharacteristic
435SD16—432no—
436C2×C8—28no—
437C2×C8—28no—
438C2×C8—28no—
439C4⋊C4—28no—
440C4⋊C4—28no—
441C4⋊C4—28no—
442C4⋊C4—28no—
443C4⋊C4—24no—
444C4⋊C4—28no—
445C4⋊C4—24no—
446C4⋊C4—24no—
447C4⋊C4—44no—
448C4⋊C4—44no—
449C4⋊C4—48no—
450C4⋊C4—44no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—44no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—48no—
457C4⋊C4—44no—
458C4⋊C4—44no—
459C4⋊C4—44no—
460C4⋊C4—44no—
461C4⋊C4—42no—
462C4⋊C4—44no—
463C22⋊C4—44no—
464C22⋊C4—44no—
465C22⋊C4—44no—
466C22⋊C4—44no—
467C22⋊C4—44no—
468C22⋊C4—44no—
469C22⋊C4—44no—
470C22⋊C4—44no—
471C22⋊C4—44no—
472C22⋊C4—44no—
473C22⋊C4—44no—
474C42—22no—
475C42—22no—
476C42—22no—
477C42—22no—
478C42—22no—
479C42—22no—
480C42—22no—
481C42—22no—
482C42—22no—
483C42—22no—
484C42—22no—
485C42—22no—
486C42—22no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—22no—
496C42—41no—
497C42—41no—
498C42—41no—
499C42—41no—
500C42—48no—
501C42—44no—
502C42—48no—
503C42#13714yes—
504C42H(5,𝔽2)14yes—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23#52512yes—
517C23C2×AGL(1,ℤ/8)12yes—
518C23C2×#13712yes—
519C23C23×D411yescharacteristic
520Q8—216no—
521Q8—416no—
522Q8—48no—
523Q8—84no—
524D4—48no—
525D4—84no—
526D4—84no—
527D4—88no—
528C2×C4—28no—
529C2×C4—28no—
530C2×C4—28no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—28no—
554C2×C4—28no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—416no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—88no—
580C2×C4—88no—
581C2×C4#52512yes—
582C2×C4#53512yes—
583C2×C4#53512yes—
584C2×C4C2×AGL(1,ℤ/8)12yes—
585C2×C4C2×AGL(1,ℤ/8)12yes—
586C2×C4C2×#13712yes—
587C2×C4C2×#13712yes—
588C2×C4C2×#13712yes—
589C2×C4C2×#13712yes—
590C2×C4C2×#13712yes—
591C2×C4#57512yes—
592C2×C4C22×#4112yes—
593C2×C4C2×H(5,𝔽2)12yes—
594C2×C4#58512yes—
595C8—48no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—82no—
603C22—82no—
604C22—82no—
605C22—82no—
606C22—82no—
607C22—82no—
608C22—82no—
609C22#311214yes—
610C22C2×#52511yescharacteristic
611C22C2×#53511yescharacteristic
612C22#344911yescharacteristic
613C22C22×AGL(1,ℤ/8)11yescharacteristic
614C22C22×#13711yescharacteristic
615C22C22×#13711yescharacteristic
616C22C2×#57511yescharacteristic
617C4—22no—
618C4—22no—
619C4—22no—
620C4—22no—
621C4—22no—
622C4—22no—
623C4—22no—
624C4—22no—
625C4—22no—
626C4—22no—
627C4—22no—
628C4—48no—
629C4—41no—
630C4—41no—
631C4—82no—
632C4—82no—
633C4—84no—
634C4—82no—
635C4#345914yes—
636C2—24no—
637C2—162no—
638C2#3459311yescharacteristic
639C2C2×#311211yescharacteristic
640C2#6056011yescharacteristic
641C2#6080011yescharacteristic
642C2#6080311yescharacteristic
643C2#6096011yescharacteristic
644C2#6096911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×C4×C4⋊Q8C2semidirect
3. #30130C2semidirect
4. #30079C2semidirect
5. #30041C2semidirect
6. #29953C2semidirect
7. #29906C2semidirect
9. #29624C2semidirect
10. #23823C2semidirect
11. #23803C2semidirect
12. #23683C2semidirect
13. #21454C2semidirect
14. #20778C2semidirect

Maximal subgroups

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

subgroupindex
#366312
C2×C4×C4⋊Q82
#301302
#300792
#300412
#299532
#299062
#297132
#296242
#238232
#238032
#236832
#214542
#207782