Finite Group Databasebeta

← #6489013

Subgroups — #6489013

Order 512.

1814 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36635C211yesmaximal, characteristic
2C2×C4×C4⋊Q8C211yesmaximal, characteristic
3#32954C212yesmaximal
4#32906C212yesmaximal
5#30132C214yesmaximal, complemented
6#29132C212yesmaximal, complemented
7#28385C212yesmaximal, complemented
8#28283C212yesmaximal, complemented
9#28255C212yesmaximal, complemented
10#27680C212yesmaximal
11#24508C214yesmaximal
12#24212C212yesmaximal
13#22154C214yesmaximal, complemented
14#20810C211yesmaximal, characteristic
15C22×C4⋊Q8C2211yescharacteristic
16C22×C4×Q8C2212yes—
17C2×#475C2214yes—
18C2×#462C2214yes—
19C2×#462C2214yes—
20C2×#451C2214yes—
21C2×#432C2212yes—
22#2864—21no—
23#2863—21no—
24#2863C2212yes—
25#2862C2211yescharacteristic
26#2859C2211yescharacteristic
27#2857C2211yescharacteristic
28#2857C2211yescharacteristic
29#2855—21no—
30#2855C2212yes—
31#2851—21no—
32C4×C4⋊Q8—21no—
33C4×C4⋊Q8—21no—
34C4×C4⋊Q8—21no—
35C4×C4⋊Q8—21no—
36C4×C4⋊Q8C2211yescharacteristic
37C4×C4⋊Q8C2211yescharacteristic
38C4×C4⋊Q8C2212yes—
39C4×C4⋊Q8C2212yes—
40C4×C4⋊Q8C2212yes—
41C4×C4⋊Q8C2212yes—
42C4×#125—21no—
43C4×C22⋊Q8—21no—
44C4×C22⋊Q8C2212yes—
45#2295—21no—
46#2295C2212yes—
47C2×#391C2212yes—
48C2×#389—21no—
49C2×#389—21no—
50C2×#384C2211yescharacteristic
51#2283—21no—
52C2×C4×C4⋊C4—21no—
53C2×C4×C4⋊C4—21no—
54C2×C43C2211yescharacteristic
55#2099C2214yes—
56#2096C2214yes—
57#2066C2218yes—
58#2056C2218yes—
59#2043C2214yes—
60#2016C2214yes—
61#1981C2214yes—
62#1971C2218yes—
63#1948C2214yes—
64#1935C2218yes—
65#1878C2218yes—
66#1872C2214yes—
67#1865C2218yes—
68#1853C2214yes—
69#1842C2212yes—
70#1831C2212yes—
71#1803C2214yes—
72#1803C2214yes—
73#1803C2214yes—
74#1797C2212yes—
75#1797C2212yes—
76#1787C2214yes—
77#1767C2212yes—
78#1767C2212yes—
79C23×Q8C2312yes—
80C22×Q16C2318yes—
81C2×C4⋊Q8—28no—
82C2×C4⋊Q8—21no—
83C2×C4⋊Q8—21no—
84C2×C4⋊Q8C2312yes—
85C2×C4⋊Q8C2311yescharacteristic
86C2×C4⋊Q8C2312yes—
87C2×C4⋊Q8C2311yescharacteristic
88C2×#125—21no—
89C2×C22⋊Q8—21no—
90C2×C22⋊Q8C2312yes—
91C2×C4×Q8—22no—
92C2×C4×Q8—22no—
93C2×C4×Q8—21no—
94C2×C4×Q8—48no—
95C2×C4×Q8C2314yes—
96C2×C4×Q8C2312yes—
97C2×C4×Q8C2314yes—
98C22×C4⋊C4—21no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4—22no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4C2312yes—
104C22×C42—22no—
105C22×C42D411yescharacteristic
106C22×C42D411yescharacteristic
107C22×C42D411yescharacteristic
108C22×C42D411yescharacteristic
109C22×C42C2311yescharacteristic
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112#475—48no—
113#462—216no—
114#462—48no—
115#451—48no—
116#432—44no—
117C2×#106C2318yes—
118C2×C4⋊C8C2314yes—
119C2×C4⋊C8C2314yes—
120C2×C4⋊C8C2314yes—
121C2×#103C2318yes—
122C2×#103C2318yes—
123C2×#103C2318yes—
124C2×#103C2318yes—
125C2×C22⋊C8C2314yes—
126C2×#97C2314yes—
127#391—22no—
128#391—22no—
129#391—21no—
130#391—21no—
131#391—21no—
132#391—21no—
133#391C2314yes—
134#391C2314yes—
135#391C2312yes—
136#391C2312yes—
137#391C2312yes—
138#391C2312yes—
139#389—21no—
140#389—21no—
141#389—22no—
142#389—22no—
143#389—22no—
144#389—22no—
145#389—22no—
146#389—22no—
147#389—22no—
148#389—22no—
149#389—21no—
150#389—21no—
151#389—21no—
152#389—21no—
153#389—21no—
154#389—21no—
155#389C2312yes—
156#389C2312yes—
157#389C2312yes—
158#389C2312yes—
159#387—22no—
160#387—22no—
161#387—22no—
162#387—22no—
163#387—22no—
164#387—22no—
165#387—22no—
166#387—22no—
167#385—22no—
168#385—22no—
169#384—21no—
170#384—21no—
171#384C2312yes—
172#384C2312yes—
173#384C2311yescharacteristic
174#384C2311yescharacteristic
175#383—22no—
176#382—21no—
177#382—21no—
178#382—21no—
179#381—22no—
180#380—21no—
181#380—21no—
182#380—21no—
183#380C2312yes—
184C4×C4⋊C4—21no—
185C4×C4⋊C4—21no—
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—22no—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4C2312yes—
198C4×C22⋊C4—22no—
199#376—22no—
200C2×#95—22no—
201C2×#95—22no—
202C43D411yescharacteristic
203C43D411yescharacteristic
204C43D411yescharacteristic
205C43D411yescharacteristic
206C43C2311yescharacteristic
207C43C2311yescharacteristic
208C43C2311yescharacteristic
209C43C2311yescharacteristic
210#340C2318yes—
211#340C2318yes—
212#336C2314yes—
213#336C2314yes—
214C22×Q8—28no—
215C22×Q8—22no—
216C22×Q8—21no—
217C22×Q8—24no—
218C22×Q8—24no—
219C22×Q8—44no—
220C22×Q8C2412yes—
221C22×Q8C2414yes—
222C23×C4—22no—
223C23×C4—22no—
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2411yescharacteristic
231C2×Q16—216no—
232C2×Q16—48no—
233C2×Q16—48no—
234C22×C8C2418yes—
235C4⋊Q8—28no—
236C4⋊Q8—28no—
237C4⋊Q8—28no—
238C4⋊Q8—24no—
239C4⋊Q8—24no—
240C4⋊Q8—24no—
241C4⋊Q8#4118yes—
242#125—44no—
243C22⋊Q8—24no—
244C22⋊Q8—24no—
245C22⋊Q8—44no—
246C4×Q8—28no—
247C4×Q8—28no—
248C4×Q8—28no—
249C4×Q8—44no—
250C4×Q8—44no—
251C4×Q8—48no—
252C4×Q8—48no—
253C4×Q8—44no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—48no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4C2414yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2412yes—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42—42no—
302C2×C42—48no—
303C2×C42C2×D411yescharacteristic
304C2×C42C2×D411yescharacteristic
305C2×C42C2×D411yescharacteristic
306C2×C42C2×D411yescharacteristic
307C2×C42C2×D411yescharacteristic
308C2×C42C2×D411yescharacteristic
309C2×C42C2×D411yescharacteristic
310C2×C42C2×D411yescharacteristic
311C2×C42C2×D411yescharacteristic
312C2×C42C2×D411yescharacteristic
313C2×C42C2×D411yescharacteristic
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×C42C2411yescharacteristic
326C2×C42C2411yescharacteristic
327C2×C42C2411yescharacteristic
328C2×C42C2411yescharacteristic
329C2×C42C2411yescharacteristic
330C2×C42C2411yescharacteristic
331#106—48no—
332C4⋊C8—28no—
333C4⋊C8—28no—
334C4⋊C8—44no—
335#103—216no—
336#103—48no—
337#103—48no—
338#103—48no—
339C22⋊C8—28no—
340#97—44no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—22no—
346#95—22no—
347#95—22no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—22no—
359C24C22×D411yescharacteristic
360C2×Q8—28no—
361C2×Q8—24no—
362C2×Q8—24no—
363C2×Q8—24no—
364C2×Q8—24no—
365C2×Q8—24no—
366C2×Q8—24no—
367C2×Q8—22no—
368C2×Q8—24no—
369C2×Q8—22no—
370C2×Q8—416no—
371C2×Q8—416no—
372C2×Q8—44no—
373C2×Q8—42no—
374C2×Q8—44no—
375C2×Q8—42no—
376C2×Q8—42no—
377C2×Q8C2×#4118yes—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—28no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—44no—
404C22×C4—44no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4#13714yes—
409C22×C4#13714yes—
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4H(5,𝔽2)12yes—
424C22×C42−1+412yes—
425C22×C4C2511yescharacteristic
426Q16—816no—
427C2×C8—28no—
428C2×C8—28no—
429C2×C8—28no—
430C4⋊C4—28no—
431C4⋊C4—28no—
432C4⋊C4—28no—
433C4⋊C4—28no—
434C4⋊C4—24no—
435C4⋊C4—22no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—22no—
440C4⋊C4—44no—
441C4⋊C4—44no—
442C4⋊C4—48no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—44no—
446C4⋊C4—44no—
447C4⋊C4—48no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—44no—
452C4⋊C4—44no—
453C4⋊C4—44no—
454C4⋊C4—42no—
455C4⋊C4—44no—
456C4⋊C4—42no—
457C4⋊C4—44no—
458C4⋊C4—44no—
459C4⋊C4—44no—
460C22⋊C4—44no—
461C22⋊C4—44no—
462C42—22no—
463C42—22no—
464C42—22no—
465C42—22no—
466C42—22no—
467C42—22no—
468C42—22no—
469C42—22no—
470C42—22no—
471C42—22no—
472C42—22no—
473C42—22no—
474C42—22no—
475C42—22no—
476C42—22no—
477C42—22no—
478C42—22no—
479C42—22no—
480C42—22no—
481C42—22no—
482C42—21no—
483C42—21no—
484C42—22no—
485C42—21no—
486C42—21no—
487C42—41no—
488C42—41no—
489C42—48no—
490C42—44no—
491C42—48no—
492C42—44no—
493C42#13714yes—
494C42C2×#4114yes—
495C42H(5,𝔽2)14yes—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23#53712yes—
501C23C2×#13712yes—
502C23C2×#13712yes—
503C23C23×D411yescharacteristic
504Q8—28no—
505Q8—28no—
506Q8—416no—
507Q8—44no—
508Q8—44no—
509Q8—44no—
510Q8—44no—
511Q8—84no—
512C2×C4—28no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—28no—
528C2×C4—28no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—28no—
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—44no—
545C2×C4—44no—
546C2×C4—416no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—88no—
566C2×C4#52512yes—
567C2×C4#52512yes—
568C2×C4#53512yes—
569C2×C4#53512yes—
570C2×C4#53712yes—
571C2×C4C2×#13712yes—
572C2×C4C2×#13712yes—
573C2×C4C2×#13712yes—
574C2×C4C2×#13712yes—
575C2×C4C2×#13712yes—
576C2×C4C2×#13712yes—
577C2×C4C2×#13712yes—
578C2×C4#57512yes—
579C2×C4C22×#4112yes—
580C2×C4C2×H(5,𝔽2)11yescharacteristic
581C2×C4C2×H(5,𝔽2)12yes—
582C2×C4C2×2−1+411yescharacteristic
583C8—48no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22#325514yes—
591C22C2×#52511yescharacteristic
592C22C2×#53511yescharacteristic
593C22C2×#53711yescharacteristic
594C22C22×#13711yescharacteristic
595C22C22×#13711yescharacteristic
596C22C22×#13711yescharacteristic
597C22C2×#57511yescharacteristic
598C4—22no—
599C4—22no—
600C4—22no—
601C4—22no—
602C4—22no—
603C4—22no—
604C4—22no—
605C4—41no—
606C4—48no—
607C4—41no—
608C4—41no—
609C4—41no—
610C4—82no—
611C4—84no—
612C4—82no—
613C4—82no—
614C4—82no—
615C4#311314yes—
616C4#321814yes—
617C4#351912yes—
618C4#352012yes—
619C2—24no—
620C2#3523611yescharacteristic
621C2C2×#311311yescharacteristic
622C2#6056011yescharacteristic
623C2C2×#321811yescharacteristic
624C2#6080411yescharacteristic
625C2C2×#325511yescharacteristic
626C2#6091811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
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
5. #30132C2semidirect
6. #29132C2semidirect
7. #28385C2semidirect
8. #28283C2semidirect
9. #28255C2semidirect
13. #22154C2semidirect

Maximal subgroups

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

subgroupindex
#366352
C2×C4×C4⋊Q82
#329542
#329062
#301322
#291322
#283852
#282832
#282552
#276802
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