Finite Group Databasebeta

← #6468961

Subgroups — #6468961

Order 512.

1978 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#533C211yesmaximal, characteristic
2C2×C4×C4⋊Q8C211yesmaximal, characteristic
3#29370C214yesmaximal, complemented
4#29259C212yesmaximal, complemented
5#28161C212yesmaximal, complemented
6#27681C212yesmaximal
7#27586C212yesmaximal
8#24212C212yesmaximal
9#23934C212yesmaximal
10#23816C212yesmaximal, complemented
11#23801C212yesmaximal, complemented
12C4×#462C214yesmaximal, complemented
13C4×#451C214yesmaximal
14#20777C211yesmaximal, characteristic
15C2×#533C2211yescharacteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C4×Q8C2212yes—
18C2×#462C2214yes—
19C2×#451C2214yes—
20C2×C4×Q16C2218yes—
21C4×C4⋊Q8—21no—
22C4×C4⋊Q8—21no—
23C4×C4⋊Q8—21no—
24C4×C4⋊Q8—21no—
25C4×C4⋊Q8C2211yescharacteristic
26C4×C4⋊Q8C2211yescharacteristic
27C4×C4⋊Q8C2211yescharacteristic
28C4×C4⋊Q8C2212yes—
29C4×C4⋊Q8C2212yes—
30C4×C4⋊Q8C2212yes—
31C4×C4⋊Q8C2212yes—
32C4×C4⋊Q8C2211yescharacteristic
33C4×#125—21no—
34C4×#125—21no—
35C4×C22⋊Q8—21no—
36C4×C22⋊Q8—21no—
37C4×C22⋊Q8C2212yes—
38C4×C22⋊Q8C2212yes—
39#2296—21no—
40#2295—21no—
41#2295C2212yes—
42C2×#391C2212yes—
43C2×#389—21no—
44C2×#389—21no—
45#2285C2211yescharacteristic
46C2×#384C2211yescharacteristic
47C42×Q8—21no—
48C42×Q8C2212yes—
49C4×#117—21no—
50C2×C4×C4⋊C4—21no—
51C2×C4×C4⋊C4—21no—
52C2×C43C2211yescharacteristic
53#2100C2214yes—
54#2095C2214yes—
55#2095C2214yes—
56#2072C2214yes—
57#2043C2214yes—
58#2024C2214yes—
59#2016C2214yes—
60#2016C2214yes—
61#1981C2214yes—
62#1947C2214yes—
63#1928C2218yes—
64#1878C2218yes—
65#1878C2218yes—
66#1872C2214yes—
67#1863C2218yes—
68#1852C2214yes—
69#1839C2212yes—
70#1831C2212yes—
71#1817C2212yes—
72#1803C2214yes—
73#1797C2212yes—
74#1787C2214yes—
75C4×C4⋊C8C2212yes—
76C4×#103C2218yes—
77C4×C22⋊C8C2212yes—
78#1744C2212yes—
79C23×Q8C2312yes—
80C22×Q16C2318yes—
81#533—28no—
82C2×C4⋊Q8—28no—
83C2×C4⋊Q8—21no—
84C2×C4⋊Q8—21no—
85C2×C4⋊Q8C2312yes—
86C2×C4⋊Q8C2311yescharacteristic
87C2×C4⋊Q8C2312yes—
88C2×C4⋊Q8C2311yescharacteristic
89C2×#125—21no—
90C2×C22⋊Q8—21no—
91C2×C22⋊Q8C2312yes—
92C2×C4×Q8—21no—
93C2×C4×Q8—22no—
94C2×C4×Q8—22no—
95C2×C4×Q8—21no—
96C2×C4×Q8—21no—
97C2×C4×Q8—48no—
98C2×C4×Q8C2314yes—
99C2×C4×Q8C2312yes—
100C2×C4×Q8C2312yes—
101C2×C4×Q8C2312yes—
102C2×C4×Q8C2314yes—
103C2×#117—21no—
104C2×#117—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—22no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4C2312yes—
111C22×C42—22no—
112C22×C42D411yescharacteristic
113C22×C42D411yescharacteristic
114C22×C42D411yescharacteristic
115C22×C42D411yescharacteristic
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119#462—216no—
120#451—216no—
121C4×Q16—416no—
122C2×#107C2318yes—
123C2×C4⋊C8C2314yes—
124C2×#103C2318yes—
125C2×#103C2318yes—
126C2×#103C2318yes—
127C2×#103C2318yes—
128C2×C22⋊C8C2314yes—
129C2×C4×C8C2314yes—
130#391—22no—
131#391—22no—
132#391—21no—
133#391—21no—
134#391C2314yes—
135#391C2314yes—
136#391C2312yes—
137#391C2312yes—
138#389—21no—
139#389—21no—
140#389—21no—
141#389—21no—
142#389—22no—
143#389—22no—
144#389—22no—
145#389—22no—
146#389—22no—
147#389—22no—
148#389—22no—
149#389—22no—
150#389—21no—
151#389—21no—
152#389C2312yes—
153#389C2312yes—
154#387—22no—
155#387—22no—
156#387—22no—
157#387—22no—
158#385—22no—
159#385—22no—
160#384—21no—
161#384—21no—
162#384C2312yes—
163#384C2311yescharacteristic
164#384C2311yescharacteristic
165#384C2312yes—
166#382—21no—
167#380—21no—
168#380C2312yes—
169C4×C4⋊C4—21no—
170C4×C4⋊C4—21no—
171C4×C4⋊C4—21no—
172C4×C4⋊C4—21no—
173C4×C4⋊C4—22no—
174C4×C4⋊C4—22no—
175C4×C4⋊C4—22no—
176C4×C4⋊C4—22no—
177C4×C4⋊C4—22no—
178C4×C4⋊C4—22no—
179C4×C4⋊C4—22no—
180C4×C4⋊C4—22no—
181C4×C4⋊C4—22no—
182C4×C4⋊C4—22no—
183C4×C4⋊C4—22no—
184C4×C4⋊C4—22no—
185C4×C4⋊C4—21no—
186C4×C4⋊C4—21no—
187C4×C4⋊C4—21no—
188C4×C4⋊C4—22no—
189C4×C4⋊C4—22no—
190C4×C4⋊C4C2312yes—
191C4×C4⋊C4C2312yes—
192C4×C4⋊C4C2312yes—
193C4×C22⋊C4—22no—
194C4×C22⋊C4—22no—
195#376—22no—
196C2×#95—22no—
197C2×#95—22no—
198C43—22no—
199C43D411yescharacteristic
200C43D411yescharacteristic
201C43D411yescharacteristic
202C43D411yescharacteristic
203C43C2311yescharacteristic
204C43C2311yescharacteristic
205C43C2311yescharacteristic
206C43C2311yescharacteristic
207#340C2318yes—
208#340C2318yes—
209#336C2314yes—
210#336C2314yes—
211#336C2314yes—
212#336C2314yes—
213C22×Q8—28no—
214C22×Q8—21no—
215C22×Q8—22no—
216C22×Q8—28no—
217C22×Q8—44no—
218C22×Q8C2412yes—
219C22×Q8C2414yes—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C23×C4C2411yescharacteristic
229C2×Q16—216no—
230C2×Q16—216no—
231C2×Q16—48no—
232C22×C8C2418yes—
233C4⋊Q8—28no—
234C4⋊Q8—28no—
235C4⋊Q8—28no—
236C4⋊Q8—24no—
237C4⋊Q8—24no—
238C4⋊Q8—24no—
239C4⋊Q8#4118yes—
240#125—28no—
241C22⋊Q8—28no—
242C22⋊Q8—24no—
243C22⋊Q8#4118yes—
244C4×Q8—28no—
245C4×Q8—24no—
246C4×Q8—24no—
247C4×Q8—28no—
248C4×Q8—216no—
249C4×Q8—28no—
250C4×Q8—48no—
251C4×Q8—44no—
252C4×Q8—48no—
253C4×Q8—48no—
254C4×Q8—44no—
255#117—28no—
256#117—44no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
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—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—48no—
287C2×C4⋊C4C2414yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C42—22no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—22no—
299C2×C42—24no—
300C2×C42—22no—
301C2×C42—22no—
302C2×C42—24no—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42—42no—
306C2×C42—42no—
307C2×C42—48no—
308C2×C42Q1618yescomplemented
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×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2411yescharacteristic
332C2×C42C2411yescharacteristic
333C2×C42C2411yescharacteristic
334C2×C42C2411yescharacteristic
335C2×C42C2411yescharacteristic
336C2×C42C2411yescharacteristic
337#107—48no—
338C4⋊C8—28no—
339#103—216no—
340#103—216no—
341#103—48no—
342#103—48no—
343C22⋊C8—28no—
344C4×C8—28no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—24no—
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—24no—
361C2×Q8—28no—
362C2×Q8—22no—
363C2×Q8—24no—
364C2×Q8—28no—
365C2×Q8—24no—
366C2×Q8—24no—
367C2×Q8—28no—
368C2×Q8—432no—
369C2×Q8—44no—
370C2×Q8—42no—
371C2×Q8—44no—
372C2×Q8—42no—
373C2×Q8C2×#4118yes—
374C2×Q8C2×#4114yes—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
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—28no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—44no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4C2×Q1614yes—
405C22×C4C2×Q1614yes—
406C22×C4C2×Q1614yes—
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D411yescharacteristic
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×C4C2×#4114yes—
421C22×C4C2511yescharacteristic
422Q16—432no—
423C2×C8—28no—
424C2×C8—28no—
425C2×C8—28no—
426C4⋊C4—28no—
427C4⋊C4—28no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—28no—
431C4⋊C4—24no—
432C4⋊C4—28no—
433C4⋊C4—28no—
434C4⋊C4—28no—
435C4⋊C4—22no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—28no—
439C4⋊C4—44no—
440C4⋊C4—44no—
441C4⋊C4—44no—
442C4⋊C4—44no—
443C4⋊C4—48no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—44no—
448C4⋊C4—44no—
449C4⋊C4—48no—
450C4⋊C4—44no—
451C4⋊C4—44no—
452C4⋊C4—42no—
453C4⋊C4—44no—
454C4⋊C4C2×#4114yes—
455C22⋊C4—28no—
456C22⋊C4—44no—
457C42—28no—
458C42—22no—
459C42—22no—
460C42—22no—
461C42—22no—
462C42—22no—
463C42—22no—
464C42—22no—
465C42—22no—
466C42—22no—
467C42—22no—
468C42—22no—
469C42—22no—
470C42—22no—
471C42—22no—
472C42—22no—
473C42—41no—
474C42—41no—
475C42—41no—
476C42—41no—
477C42—41no—
478C42—41no—
479C42—48no—
480C42—44no—
481C42—44no—
482C42—48no—
483C42—44no—
484C42—44no—
485C42—44no—
486C42C2×Q1614yes—
487C42C2×Q1614yes—
488C42C2×Q1614yes—
489C42C2×Q1614yes—
490C42#13714yes—
491C42C2×#4114yes—
492C42H(5,𝔽2)14yes—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23#52512yes—
498C23C22×Q1612yes—
499C23#57512yes—
500C23C23×D411yescharacteristic
501Q8—28no—
502Q8—216no—
503Q8—48no—
504Q8—416no—
505Q8—44no—
506Q8—48no—
507C2×C4—28no—
508C2×C4—28no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
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—28no—
530C2×C4—21no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—28no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—416no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—88no—
558C2×C4#52512yes—
559C2×C4#52512yes—
560C2×C4#52512yes—
561C2×C4#53512yes—
562C2×C4#53512yes—
563C2×C4C22×Q1612yes—
564C2×C4C22×Q1612yes—
565C2×C4C22×Q1612yes—
566C2×C4C22×Q1612yes—
567C2×C4C22×Q1612yes—
568C2×C4C22×Q1612yes—
569C2×C4C2×#13712yes—
570C2×C4C2×#13712yes—
571C2×C4C2×#13712yes—
572C2×C4#57512yes—
573C2×C4C22×#4112yes—
574C2×C4C22×#4112yes—
575C2×C4C2×H(5,𝔽2)12yes—
576C8—48no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22#310414yes—
584C22C2×#52511yescharacteristic
585C22C2×#52511yescharacteristic
586C22C2×#53511yescharacteristic
587C22C23×Q1611yescharacteristic
588C22C22×#13711yescharacteristic
589C22C2×#57511yescharacteristic
590C22C2×#57511yescharacteristic
591C4—22no—
592C4—22no—
593C4—22no—
594C4—22no—
595C4—22no—
596C4—22no—
597C4—22no—
598C4—22no—
599C4—22no—
600C4—41no—
601C4—41no—
602C4—44no—
603C4—48no—
604C4—44no—
605C4—82no—
606C4—84no—
607C4—82no—
608C4#310414yes—
609C4#320914yes—
610C4#348114yes—
611C2—24no—
612C2#3496311yescharacteristic
613C2C2×#310411yescharacteristic
614C2C2×#310411yescharacteristic
615C2#6056011yescharacteristic
616C2#6056211yescharacteristic
617C2C2×#320911yescharacteristic
618C2#6079011yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutator, Frattini, cosocleC22×C4C25
socleC23C23×D4
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #29370C2semidirect
4. #29259C2semidirect
5. #28161C2semidirect
10. #23816C2semidirect
11. #23801C2semidirect
12. C4×#462C2semidirect
308. C2×C42Q16semidirect

Maximal subgroups

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

subgroupindex
C4×#5332
C2×C4×C4⋊Q82
#293702
#292592
#281612
#276812
#275862
#242122
#239342
#238162
#238012
C4×#4622
C4×#4512
#207772