Finite Group Databasebeta

← #6475426

Subgroups — #6475426

Order 512.

1580 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35334C211yesmaximal, characteristic
2#35280C211yesmaximal, characteristic
3#33218C212yesmaximal, complemented
4#33217C212yesmaximal, complemented
5#30299C214yesmaximal, complemented
6#29308C214yesmaximal, complemented
7#29307C212yesmaximal, complemented
8#29306C212yesmaximal, complemented
9#27956C212yesmaximal, complemented
10#27953C212yesmaximal, complemented
11#25818C212yesmaximal, complemented
12#25817C212yesmaximal, complemented
13#21947C212yesmaximal
14#21946C212yesmaximal
15#20375C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17C2×#533C2211yescharacteristic
18C2×C4×#41C2211yescharacteristic
19C2×#501C2212yes—
20C2×#501C2212yes—
21C2×#408C2212yes—
22C2×C4×M16C2211yescharacteristic
23C22×#97C2211yescharacteristic
24#2857C2212yes—
25#2857C2212yes—
26#2851C2212yes—
27#2851C2212yes—
28#2629C2212yes—
29#2629C2212yes—
30#2628C2212yes—
31#2628C2212yes—
32#2582—22no—
33#2581—22no—
34#2580—22no—
35#2564—22no—
36#2564—22no—
37#2538—22no—
38#2417—22no—
39#2417—22no—
40#2316C2211yescharacteristic
41#2316C2211yescharacteristic
42#2316C2212yes—
43#2311C2211yescharacteristic
44#2311C2211yescharacteristic
45#2311C2212yes—
46C2×#384C2211yescharacteristic
47C2×#382C2211yescharacteristic
48#2085C2214yes—
49#2085C2214yes—
50#2084C2214yes—
51#2083C2214yes—
52#2059C2214yes—
53#2059C2214yes—
54#2058C2214yes—
55#2057C2214yes—
56#1993C2212yes—
57#1993C2212yes—
58#1992C2212yes—
59#1992C2212yes—
60#1944C2214yes—
61#1943C2214yes—
62#1942C2214yes—
63#1941C2214yes—
64#1854C2214yes—
65#1854C2214yes—
66#1854C2214yes—
67#1854C2214yes—
68#1853C2214yes—
69#1853C2214yes—
70#1852C2214yes—
71#1852C2214yes—
72#1847C2212yes—
73#1847C2212yes—
74#1809C2214yes—
75#1809C2214yes—
76#1808C2214yes—
77#1807C2214yes—
78#1750C2212yes—
79#1750C2212yes—
80#1749C2212yes—
81#1749C2212yes—
82#1729C2212yes—
83C22×#41C2311yescharacteristic
84C22×M16C2312yes—
85C23×C8C2312yes—
86#533—42no—
87#533—42no—
88#533—42no—
89#533—42no—
90C2×C4⋊Q8C2312yes—
91C2×C4⋊Q8C2312yes—
92C2×#125C2312yes—
93C2×#125C2312yes—
94C2×C22⋊Q8—22no—
95C2×C22⋊Q8—22no—
96C2×C22⋊Q8—22no—
97C2×C22⋊Q8—22no—
98C4×#41—42no—
99C4×#41—42no—
100C2×C4×Q8—22no—
101C2×C4×Q8—22no—
102C2×C4×Q8D412yes—
103C2×C4×D4D412yes—
104C2×C4×D4D412yes—
105C2×C4×D4D412yes—
106C2×#117C2311yescharacteristic
107C2×#117C2311yescharacteristic
108C2×#117C2312yes—
109C2×#117C2311yescharacteristic
110C2×#117C2311yescharacteristic
111C2×#117C2311yescharacteristic
112C22×C4⋊C4—22no—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119#501—44no—
120#501—44no—
121C2×#107C2314yes—
122C2×#107C2314yes—
123C2×#107C2314yes—
124C2×#107C2314yes—
125C2×#106C2314yes—
126C2×#106C2314yes—
127C2×#106C2314yes—
128C2×#106C2314yes—
129#408Q8116yescomplemented
130C2×C22⋊C8C2314yes—
131C2×C22⋊C8C2314yes—
132C4×M16—42no—
133C2×#97—42no—
134C2×#97C2312yes—
135C2×#97C2312yes—
136C2×#97C2312yes—
137C2×C4×C8C2312yes—
138#400—22no—
139#400—22no—
140#399—22no—
141#399—22no—
142#398—24no—
143#398—24no—
144#396—24no—
145#396—24no—
146#395—22no—
147#395—22no—
148#393—22no—
149#393—22no—
150#391—21no—
151#391—21no—
152#391—22no—
153#391—22no—
154#391—22no—
155#390—21no—
156#390—21no—
157#390—22no—
158#389—22no—
159#389—22no—
160#389—24no—
161#389—22no—
162#389—22no—
163#389C2314yes—
164#389C2312yes—
165#389C2312yes—
166#389C2312yes—
167#389C2312yes—
168#389C2314yes—
169#387—22no—
170#387—22no—
171#387—22no—
172#387—22no—
173#386—21no—
174#386—21no—
175#386—21no—
176#386—21no—
177#386—22no—
178#386—22no—
179#385—22no—
180#385—22no—
181#385—22no—
182#385—22no—
183#384—22no—
184#384C2312yes—
185#384C2312yes—
186#382—22no—
187#382C2312yes—
188#382C2312yes—
189#380—22no—
190#380—22no—
191#380—22no—
192#380—22no—
193#380—24no—
194#380C2312yes—
195#380C2314yes—
196#380C2312yes—
197#380C2312yes—
198#380C2312yes—
199#380C2314yes—
200C2×#95—22no—
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#336C2314yes—
210#336C2314yes—
211C2×#41—41no—
212C2×#41—41no—
213C2×#41—42no—
214C2×#41C2×Q818yes—
215C22×Q8—22no—
216C22×Q8—22no—
217C22×Q8C2×D411yescharacteristic
218C22×D4C2×D411yescharacteristic
219C22×D4C2×D411yescharacteristic
220C22×D4C2×D411yescharacteristic
221C23×C4—22no—
222C23×C4—21no—
223C23×C4—21no—
224C23×C4C2×D412yes—
225C23×C4C2×D412yes—
226C23×C4C2×D412yes—
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C2×M16—42no—
231C2×M16—42no—
232C2×M16C2×Q818yes—
233C22×C8—42no—
234C22×C8—42no—
235C22×C8C2×Q818yes—
236C22×C8C2414yes—
237C22×C8C2414yes—
238C4⋊Q8—24no—
239C4⋊Q8—24no—
240C4⋊Q8—42no—
241C4⋊Q8—42no—
242#125—24no—
243#125—24no—
244#125—42no—
245#125—42no—
246C22⋊Q8—48no—
247C22⋊Q8—48no—
248C22⋊Q8—84no—
249C22⋊Q8—84no—
250C4×Q8—44no—
251C4×Q8—48no—
252C4×Q8—48no—
253C4×D4—44no—
254C4×D4—44no—
255C4×D4—44no—
256#117—41no—
257#117—41no—
258#117—41no—
259#117—41no—
260#117—41no—
261#117—41no—
262#117—41no—
263#117—41no—
264#117—42no—
265#117—42no—
266#117—41no—
267#117—41no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4—42no—
287C2×C4⋊C4—42no—
288C2×C4⋊C4—42no—
289C2×C4⋊C4—84no—
290C2×C4⋊C4C2×D412yes—
291C2×C4⋊C4C2×D412yes—
292C2×C4⋊C4C2×D412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2414yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2414yes—
299C2×C22⋊C4—21no—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4C2×D412yes—
309C2×C22⋊C4C2×D412yes—
310C2×C22⋊C4C2×D412yes—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—21no—
318C2×C42—21no—
319C2×C42—22no—
320C2×C42C2×D412yes—
321C2×C42C2×D412yes—
322C2×C42C2×D412yes—
323C2×C42C2411yescharacteristic
324C2×C42C2411yescharacteristic
325C2×C42C2411yescharacteristic
326C2×C42C2411yescharacteristic
327#107—44no—
328#107—44no—
329#107—44no—
330#107—44no—
331#106—44no—
332#106—44no—
333#106—44no—
334#106—44no—
335C22⋊C8C2×Q8116yes—
336C22⋊C8C2×Q8116yes—
337#97—24no—
338#97—24no—
339#97—42no—
340C4×C8—42no—
341#95—22no—
342#95—22no—
343#95—22no—
344#95—22no—
345#95—24no—
346#95—24no—
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—24no—
356#95—24no—
357C24C22×D411yescharacteristic
358C24C22×D411yescharacteristic
359C24C22×D411yescharacteristic
360#41—48no—
361#41—48no—
362C2×Q8—41no—
363C2×Q8—41no—
364C2×Q8—44no—
365C2×Q8—44no—
366C2×Q8—44no—
367C2×Q8—44no—
368C2×Q8—82no—
369C2×Q8—82no—
370C2×Q8C4⋊Q814yes—
371C2×D4—41no—
372C2×D4—41no—
373C2×D4—41no—
374C2×D4—41no—
375C2×D4—41no—
376C2×D4—41no—
377C2×D4C4⋊Q814yes—
378C2×D4C4⋊Q814yes—
379C2×D4C4⋊Q814yes—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—44no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—84no—
408C22×C4—82no—
409C22×C4—82no—
410C22×C4C22×D412yes—
411C22×C4C22×D412yes—
412C22×C4C22×D412yes—
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D412yes—
417C22×C4C22×Q814yes—
418C22×C4C22×Q814yes—
419C22×C4C22×Q814yes—
420C22×C4C2511yescharacteristic
421M16—48no—
422C2×C8—24no—
423C2×C8—24no—
424C2×C8—24no—
425C2×C8—24no—
426C2×C8—48no—
427C2×C8C22×Q818yes—
428C2×C8C22×Q818yes—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—44no—
445C4⋊C4—44no—
446C4⋊C4—44no—
447C4⋊C4—48no—
448C4⋊C4—44no—
449C4⋊C4—44no—
450C4⋊C4—44no—
451C4⋊C4—44no—
452C4⋊C4—82no—
453C4⋊C4—82no—
454C4⋊C4—82no—
455C4⋊C4—82no—
456C4⋊C4—84no—
457C4⋊C4—84no—
458C4⋊C4—84no—
459C22⋊C4—42no—
460C22⋊C4—42no—
461C22⋊C4—42no—
462C22⋊C4—42no—
463C22⋊C4—42no—
464C22⋊C4—44no—
465C22⋊C4—82no—
466C22⋊C4—82no—
467C22⋊C4—82no—
468C22⋊C4—82no—
469C22⋊C4—84no—
470C22⋊C4—84no—
471C42—24no—
472C42—24no—
473C42—24no—
474C42—21no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—42no—
481C42—42no—
482C42—44no—
483C42H(5,𝔽2)12yes—
484C42H(5,𝔽2)12yes—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23C2×C4⋊Q812yes—
498C23C2×C4⋊Q812yes—
499C23C2×C4⋊Q812yes—
500C23C23×D411yescharacteristic
501Q8—44no—
502Q8—88no—
503Q8—88no—
504D4—44no—
505D4—44no—
506D4—44no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—48no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—84no—
550C2×C4—84no—
551C2×C4—164no—
552C2×C4—162no—
553C2×C4—162no—
554C2×C4C2×C4⋊Q812yes—
555C2×C4C2×C4⋊Q812yes—
556C2×C4C2×C4⋊Q812yes—
557C2×C4#53414yes—
558C2×C4C23×Q812yes—
559C2×C4C2×H(5,𝔽2)11yescharacteristic
560C2×C4C2×H(5,𝔽2)11yescharacteristic
561C2×C4#58512yes—
562C8—28no—
563C8—28no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—24no—
568C22—24no—
569C22—24no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22C22×C4⋊Q811yescharacteristic
580C22C2×#53411yescharacteristic
581C22#344911yescharacteristic
582C22#358512yes—
583C22#358612yes—
584C4—24no—
585C4—22no—
586C4—22no—
587C4—22no—
588C4—42no—
589C4—42no—
590C4—42no—
591C4—84no—
592C4—82no—
593C4—82no—
594C4—84no—
595C4—82no—
596C4—82no—
597C4#350112yes—
598C4#350112yes—
599C2—42no—
600C2—42no—
601C2—42no—
602C2#3501111yescharacteristic
603C2#6066812yes—
604C2#6075512yes—
605C2#6101711yescharacteristic
606C2#6101811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33218C2semidirect
4. #33217C2semidirect
5. #30299C2semidirect
6. #29308C2semidirect
7. #29307C2semidirect
8. #29306C2semidirect
9. #27956C2semidirect
10. #27953C2semidirect
11. #25818C2semidirect
12. #25817C2semidirect
129. #408Q8semidirect

Maximal subgroups

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

subgroupindex
#353342
#352802
#332182
#332172
#302992
#293082
#293072
#293062
#279562
#279532
#258182
#258172
#219472
#219462
#203752