Finite Group Databasebeta

← #6568662

Subgroups — #6568662

Order 512.

1454 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36407C211yesmaximal, characteristic
2#36188C211yesmaximal, characteristic
3#30230C211yesmaximal, characteristic, complemented
4#30187C212yesmaximal, complemented
5#29338C212yesmaximal, complemented
6#29225C211yesmaximal, characteristic, complemented
7#29132C211yesmaximal, characteristic, complemented
8#27968C212yesmaximal, complemented
9#27823C212yesmaximal, complemented
10#27737C212yesmaximal, complemented
11#27716C214yesmaximal, complemented
12#27714C214yesmaximal, complemented
13#23820C211yesmaximal, characteristic, complemented
14#23744C212yesmaximal, complemented
15#22490C212yesmaximal
16#22490C212yesmaximal
17#20558C211yesmaximal, characteristic
18C2×#556C2212yes—
19C2×C8×D4C2212yes—
20#2868C2211yescharacteristic
21#2859C2211yescharacteristic
22#2857C2212yes—
23#2857C2212yes—
24#2574—22no—
25#2568C2212yes—
26#2556—22no—
27#2552C2211yescharacteristic
28#2551C2211yescharacteristic
29#2547—21no—
30#2547—21no—
31#2499C2214yes—
32#2496—22no—
33#2496—22no—
34#2454—21no—
35#2452—21no—
36#2441C2212yes—
37#2439C2212yes—
38#2370—21no—
39#2369—21no—
40#2365—21no—
41#2365—21no—
42#2354C2211yescharacteristic
43#2354C2211yescharacteristic
44C4×C4⋊Q8C2211yescharacteristic
45C4×#125C2211yescharacteristic
46#2296C2212yes—
47#2285C2212yes—
48C42×D4C2211yescharacteristic
49#2101C2214yes—
50#2101C2214yes—
51#2100C2212yes—
52#2100C2214yes—
53#2100C2212yes—
54#2100C2212yes—
55#2100C2212yes—
56#2099C2211yescharacteristic
57#2099C2211yescharacteristic
58#2099C2212yes—
59#2074C2212yes—
60#2072C2214yes—
61#2043C2211yescharacteristic
62#2043C2211yescharacteristic
63#2043C2212yes—
64#2026C2212yes—
65#2024C2214yes—
66#1993C2214yes—
67#1992C2212yes—
68#1990C2212yes—
69#1990C2212yes—
70#1988C2212yes—
71#1988C2212yes—
72#1935C2214yes—
73#1935C2214yes—
74#1935C2214yes—
75#1892C2214yes—
76#1892C2214yes—
77#1873C2214yes—
78#1873C2214yes—
79#1836C2212yes—
80#1836C2214yes—
81#1825C2214yes—
82#1825C2214yes—
83#1803C2212yes—
84#1803C2212yes—
85#1770C2212yes—
86#1767C2211yescharacteristic
87C4×#103C2212yes—
88C4×#103C2212yes—
89#1752C2214yes—
90#1749C2212yes—
91#1744C2211yescharacteristic
92#1735C2214yes—
93#1735C2214yes—
94C2×#336C2212yes—
95C23×C8C2314yes—
96#556—28no—
97#556—28no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2312yes—
104C2×#126—24no—
105C2×#125—21no—
106C2×#125—21no—
107C2×#125C2311yescharacteristic
108C2×#124—22no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8C2314yes—
111C2×C4×Q8—22no—
112C2×C4×Q8C2312yes—
113C2×C4×D4D411yescharacteristic
114C2×C4×D4D411yescharacteristic
115C2×C4×D4C2311yescharacteristic
116C2×#117C2314yes—
117C2×#117C2312yes—
118C2×#117C2312yes—
119C22×C42C2312yes—
120C8×D4—28no—
121C2×C4⋊C8C2312yes—
122C2×#103C2312yes—
123C2×#103C2314yes—
124C2×#103C2314yes—
125C2×#103C2312yes—
126C2×#103C2312yes—
127C2×#103C2312yes—
128C2×C22⋊C8C2314yes—
129C2×#97C2312yes—
130C2×#97C2314yes—
131C2×C4×C8C2312yes—
132#400—24no—
133#399—22no—
134#398—22no—
135#397—24no—
136#397—24no—
137#396—22no—
138#396—22no—
139#395—22no—
140#393—22no—
141#391—22no—
142#391—22no—
143#391—22no—
144#391—22no—
145#391C2312yes—
146#391C2314yes—
147#390—21no—
148#390—21no—
149#389—21no—
150#389—21no—
151#389—21no—
152#389—21no—
153#389—22no—
154#389—22no—
155#389—22no—
156#389—21no—
157#389—21no—
158#389—22no—
159#389C2311yescharacteristic
160#389C2311yescharacteristic
161#389C2312yes—
162#389C2312yes—
163#389C2312yes—
164#389C2312yes—
165#388—22no—
166#388—22no—
167#387—21no—
168#387—21no—
169#387—22no—
170#387—22no—
171#387—24no—
172#387—24no—
173#385—22no—
174#385—22no—
175#384—21no—
176#384—21no—
177#384C2311yescharacteristic
178#384C2311yescharacteristic
179#384C2311yescharacteristic
180#384C2312yes—
181#384C2312yes—
182#384C2312yes—
183#382—21no—
184#382C2311yescharacteristic
185#382C2314yes—
186#380—22no—
187#380—22no—
188#380—22no—
189#380—22no—
190#380—22no—
191#380C2312yes—
192#380C2312yes—
193C4×C4⋊C4—21no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4D411yescharacteristic
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4C2311yescharacteristic
206C4×C4⋊C4C2311yescharacteristic
207C4×C4⋊C4C2311yescharacteristic
208C4×C22⋊C4D412yes—
209C4×C22⋊C4D412yes—
210C4×C22⋊C4C2312yes—
211C43C2311yescharacteristic
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2312yes—
217#340C2312yes—
218#340C2312yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#336C2314yes—
227#336C2312yes—
228#336C2311yescharacteristic
229#336C2312yes—
230#336C2311yescharacteristic
231C22×Q8—22no—
232C22×Q8—22no—
233C22×Q8C2412yes—
234C22×D4C2×D411yescharacteristic
235C23×C4C2×D412yes—
236C23×C4C2×D412yes—
237C23×C4C2412yes—
238C22×C8—28no—
239C22×C8—44no—
240C22×C8—44no—
241C22×C8C2412yes—
242C22×C8C2414yes—
243C22×C8C2412yes—
244C4⋊Q8—24no—
245C4⋊Q8—24no—
246C4⋊Q8—44no—
247C4⋊Q8—42no—
248C4⋊Q8—44no—
249C4⋊Q8—44no—
250C4⋊Q8—44no—
251#126—416no—
252#125—22no—
253#125—41no—
254#125—44no—
255#125—82no—
256#124—48no—
257C22⋊Q8—28no—
258C22⋊Q8—28no—
259C22⋊Q8—84no—
260C4×Q8—24no—
261C4×Q8—24no—
262C4×Q8—48no—
263C4×D4—42no—
264C4×D4—42no—
265C4×D4#4118yes—
266#117—28no—
267#117—28no—
268#117—24no—
269#117—42no—
270#117—44no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4C2×D411yescharacteristic
298C2×C4⋊C4C2×D411yescharacteristic
299C2×C4⋊C4C2×D411yescharacteristic
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2411yescharacteristic
302C2×C4⋊C4C2414yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2411yescharacteristic
306C2×C4⋊C4C2411yescharacteristic
307C2×C4⋊C4C2411yescharacteristic
308C2×C22⋊C4—24no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4C2×D412yes—
315C2×C22⋊C4C2×D412yes—
316C2×C22⋊C4C2×D412yes—
317C2×C42—24no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—21no—
321C2×C42—21no—
322C2×C42—22no—
323C2×C42—44no—
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D412yes—
329C2×C42C2×D412yes—
330C2×C42C2411yescharacteristic
331C2×C42C2411yescharacteristic
332C2×C42C2412yes—
333C2×C42C2411yescharacteristic
334C4⋊C8—24no—
335#103—24no—
336#103—44no—
337#103—42no—
338#103—42no—
339#103—44no—
340#103—42no—
341C22⋊C8—28no—
342#97—42no—
343#97—44no—
344C4×C8—24no—
345#95—21no—
346#95—21no—
347#95—22no—
348#95—24no—
349#95—24no—
350#95—22no—
351#95—22no—
352#95—21no—
353#95—21no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—21no—
359#95—21no—
360#95—24no—
361#95—24no—
362#95—22no—
363#95C2×D412yes—
364#95C2×D412yes—
365#95C2×D412yes—
366C24C22×D412yes—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—44no—
374C2×Q8—44no—
375C2×Q8—44no—
376C2×Q8—44no—
377C2×Q8—82no—
378C2×Q8—82no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4#12414yes—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D412yes—
402C22×C4C22×D412yes—
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D412yes—
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C2×#4118yes—
410C22×C4C2511yescharacteristic
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—24no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—24no—
417C2×C8—24no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—88no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—21no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—22no—
429C4⋊C4—21no—
430C4⋊C4—22no—
431C4⋊C4—21no—
432C4⋊C4—22no—
433C4⋊C4—24no—
434C4⋊C4—22no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—44no—
438C4⋊C4—44no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—44no—
442C4⋊C4—44no—
443C4⋊C4—44no—
444C4⋊C4—44no—
445C4⋊C4—44no—
446C4⋊C4—44no—
447C4⋊C4—48no—
448C4⋊C4—44no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—44no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—82no—
457C4⋊C4—81no—
458C4⋊C4—81no—
459C4⋊C4—82no—
460C4⋊C4—82no—
461C4⋊C4#12414yes—
462C4⋊C42−1+412yes—
463C22⋊C4—42no—
464C22⋊C4—42no—
465C22⋊C4—44no—
466C22⋊C4—48no—
467C22⋊C4—48no—
468C22⋊C4—82no—
469C22⋊C4—82no—
470C22⋊C4—82no—
471C22⋊C4#12418yes—
472C42—22no—
473C42—22no—
474C42—22no—
475C42—22no—
476C42—21no—
477C42—21no—
478C42—24no—
479C42—22no—
480C42—42no—
481C42—42no—
482C42—44no—
483C42—48no—
484C42—44no—
485C42—42no—
486C42—42no—
487C42—82no—
488C42C2×#4114yes—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23C2×#12414yes—
496C23C23×D411yescharacteristic
497Q8—42no—
498Q8—42no—
499Q8—42no—
500Q8—42no—
501Q8—88no—
502Q8—164no—
503D4—28no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—28no—
513C2×C4—28no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—84no—
554C2×C4—84no—
555C2×C4C2×#12414yes—
556C2×C4C2×#12412yes—
557C2×C4C2×#12412yes—
558C2×C4#57512yes—
559C2×C4#57512yes—
560C2×C4#57712yes—
561C2×C4#57812yes—
562C2×C4C22×#4112yes—
563C2×C4C2×2−1+411yescharacteristic
564C2×C4#58512yes—
565C2×C4#58511yescharacteristic
566C8—42no—
567C8—44no—
568C8—42no—
569C22—24no—
570C22—24no—
571C22—24no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22C22×#12411yescharacteristic
577C22#348511yescharacteristic
578C22#348911yescharacteristic
579C22C2×#57511yescharacteristic
580C22C2×#57511yescharacteristic
581C22C2×#57711yescharacteristic
582C22C2×#57811yescharacteristic
583C4—22no—
584C4—22no—
585C4—24no—
586C4—22no—
587C4—42no—
588C4—42no—
589C4—41no—
590C4—42no—
591C4—41no—
592C4—81no—
593C4—82no—
594C4—81no—
595C4—82no—
596C4—81no—
597C4—84no—
598C4—82no—
599C4—82no—
600C4#313514yes—
601C4#351512yes—
602C4#352112yes—
603C2—44no—
604C2#3604211yescharacteristic
605C2C2×#313511yescharacteristic
606C2#6062311yescharacteristic
607C2#6173011yescharacteristic
608C2#6173911yescharacteristic
609C2#6213311yescharacteristic
610C2#6213511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #30230C2semidirect
4. #30187C2semidirect
5. #29338C2semidirect
6. #29225C2semidirect
7. #29132C2semidirect
8. #27968C2semidirect
9. #27823C2semidirect
10. #27737C2semidirect
11. #27716C2semidirect
12. #27714C2semidirect
13. #23820C2semidirect
14. #23744C2semidirect

Maximal subgroups

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

subgroupindex
#364072
#361882
#302302
#301872
#293382
#292252
#291322
#279682
#278232
#277372
#277162
#277142
#238202
#237442
#224902
#205582