Finite Group Databasebeta

← #6651605

Subgroups — #6651605

Order 512.

1486 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#35179C211yesmaximal, characteristic
2C2×#2584C211yesmaximal, characteristic
3#33100C212yesmaximal
4#32689C214yesmaximal, complemented
5#32641C214yesmaximal, complemented
6#32587C214yesmaximal, complemented
7#32583C214yesmaximal, complemented
8#29643C212yesmaximal
9#27614C211yesmaximal, characteristic
10#27613C211yesmaximal, characteristic
11#24609C212yesmaximal
12#24606C212yesmaximal
13#24064C211yesmaximal, characteristic
14#24063C211yesmaximal, characteristic
15#21822C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17C22×#125C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C2×#484C2212yes—
20C2×#483C2212yes—
21C2×#432C2212yes—
22#2855C2212yes—
23#2855C2212yes—
24#2627—21no—
25#2627—21no—
26#2584—22no—
27#2584—22no—
28#2584C2214yes—
29#2584C2214yes—
30#2582C2212yes—
31#2582C2212yes—
32#2581—22no—
33#2568C2212yes—
34#2567C2212yes—
35#2565—22no—
36C2×#398—21no—
37C2×#395—21no—
38#2317—21no—
39#2310—21no—
40C2×#389—21no—
41C2×#389—21no—
42C2×#389C2211yescharacteristic
43C2×#389C2211yescharacteristic
44C2×#389C2211yescharacteristic
45C2×#389C2211yescharacteristic
46#2285C2211yescharacteristic
47C2×C4×C4⋊C4C2211yescharacteristic
48#2100C2212yes—
49#2100C2212yes—
50#2088C2214yes—
51#2086C2214yes—
52#2084C2212yes—
53#2083C2212yes—
54#2068C2214yes—
55#2066C2214yes—
56#2059C2214yes—
57#2059C2214yes—
58#2058C2212yes—
59#2057C2212yes—
60#2043C2212yes—
61#2043C2212yes—
62#2026C2214yes—
63#2026C2214yes—
64#1985C2214yes—
65#1985C2214yes—
66#1944C2214yes—
67#1943C2214yes—
68#1935C2214yes—
69#1935C2214yes—
70#1928C2214yes—
71#1927C2214yes—
72#1921C2214yes—
73#1919C2214yes—
74#1919C2214yes—
75#1918C2214yes—
76#1872C2212yes—
77#1872C2212yes—
78#1865C2214yes—
79#1863C2214yes—
80#1853C2212yes—
81#1852C2212yes—
82#1824C2212yes—
83#1823C2212yes—
84#1817C2211yescharacteristic
85#1817C2212yes—
86#1817C2211yescharacteristic
87C23×Q8D411yescharacteristic
88#533—44no—
89C2×C4⋊Q8C2311yescharacteristic
90C2×C4⋊Q8C2311yescharacteristic
91C2×#125—21no—
92C2×#125—22no—
93C2×#125—44no—
94C2×#125C2314yes—
95C2×C22⋊Q8—21no—
96C2×C22⋊Q8C2312yes—
97C2×C4×Q8—21no—
98C2×C4×Q8—44no—
99C2×C4×Q8D412yes—
100C2×C4×Q8D412yes—
101C2×C4×Q8C2312yes—
102C2×C4×Q8C2312yes—
103C2×C4×Q8C2312yes—
104C2×#117—21no—
105C2×#117—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4D411yescharacteristic
113C22×C4⋊C4D411yescharacteristic
114C22×C4⋊C4D411yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C42—21no—
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123#484—44no—
124#483—44no—
125#432—44no—
126C2×#107C2314yes—
127C2×#107C2314yes—
128C2×#106C2314yes—
129C2×#106C2314yes—
130C2×C4⋊C8C2314yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×C22⋊C8C2312yes—
136C2×C22⋊C8C2312yes—
137C2×#97C2314yes—
138#400—22no—
139#400—22no—
140#398—22no—
141#398—22no—
142#398—22no—
143#398—22no—
144#398—24no—
145#398—24no—
146#395—24no—
147#395—24no—
148#395—22no—
149#395—22no—
150#391—22no—
151#391—22no—
152#391—22no—
153#391—22no—
154#391C2312yes—
155#391C2312yes—
156#391C2312yes—
157#391C2312yes—
158#389—22no—
159#389—22no—
160#389—24no—
161#389—24no—
162#389—24no—
163#389—24no—
164#389—21no—
165#389—21no—
166#389—22no—
167#389—21no—
168#389—21no—
169#389—22no—
170#389C2312yes—
171#389C2312yes—
172#389C2312yes—
173#389C2312yes—
174#389C2312yes—
175#389C2312yes—
176#389C2314yes—
177#389C2314yes—
178#389C2312yes—
179#389C2312yes—
180#387—22no—
181#387—22no—
182#387—22no—
183#387—22no—
184#385—22no—
185#385—22no—
186#384—22no—
187#384C2311yescharacteristic
188#384C2311yescharacteristic
189#382—22no—
190#382—21no—
191#381—22no—
192#381—22no—
193#380—21no—
194#380C2312yes—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4C2312yes—
198C4×C4⋊C4C2314yes—
199C2×#95—21no—
200C2×#95—21no—
201C2×#95—21no—
202C2×#95—21no—
203C2×#95—21no—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#336C2312yes—
213#336C2312yes—
214#336C2314yes—
215#336C2314yes—
216#336C2312yes—
217#336C2312yes—
218C22×Q8—24no—
219C22×Q8—21no—
220C22×Q8—42no—
221C22×Q8—42no—
222C22×Q8C2×D412yes—
223C22×Q8C2×D412yes—
224C22×Q8C2412yes—
225C23×C4—21no—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2411yescharacteristic
236C22×C8C2414yes—
237C22×C8C2414yes—
238C4⋊Q8—42no—
239C4⋊Q8—42no—
240#125—44no—
241#125—44no—
242#125—48no—
243#125—44no—
244C22⋊Q8—24no—
245C22⋊Q8—24no—
246C22⋊Q8—44no—
247C4×Q8—44no—
248C4×Q8—44no—
249C4×Q8—42no—
250C4×Q8—42no—
251C4×Q8—42no—
252C4×Q8—42no—
253C4×Q8—42no—
254C4×Q8—42no—
255C4×Q8—44no—
256#117—44no—
257#117—44no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—42no—
288C2×C4⋊C4—42no—
289C2×C4⋊C4—42no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—82no—
295C2×C4⋊C4—82no—
296C2×C4⋊C4C2×D412yes—
297C2×C4⋊C4C2×D412yes—
298C2×C4⋊C4C2×D412yes—
299C2×C4⋊C4C2×D412yes—
300C2×C4⋊C4C2×D412yes—
301C2×C4⋊C4C2×D412yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—24no—
317C2×C42—82no—
318C2×C42C2×D411yescharacteristic
319C2×C42C2×D412yes—
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D412yes—
322C2×C42C2411yescharacteristic
323C2×C42C2412yes—
324C2×C42C2411yescharacteristic
325C2×C42C2412yes—
326#107—28no—
327#107—44no—
328#106—28no—
329#106—44no—
330C4⋊C8—44no—
331#103—28no—
332#103—28no—
333#103—44no—
334#103—44no—
335C22⋊C8—24no—
336C22⋊C8—24no—
337#97—44no—
338#95—24no—
339#95—24no—
340#95—22no—
341#95—22no—
342#95—24no—
343#95—22no—
344#95—22no—
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—24no—
361C2×Q8—24no—
362C2×Q8—22no—
363C2×Q8—22no—
364C2×Q8—24no—
365C2×Q8—24no—
366C2×Q8—42no—
367C2×Q8—42no—
368C2×Q8—42no—
369C2×Q8—42no—
370C2×Q8—416no—
371C2×Q8—42no—
372C2×Q8—81no—
373C2×Q8—81no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
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—22no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4#13714yes—
414C22×C4C22×D412yes—
415C22×C4C22×D412yes—
416C22×C4C22×D412yes—
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D412yes—
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4H(5,𝔽2)14yes—
424C22×C4C2511yescharacteristic
425C2×C8—24no—
426C2×C8—24no—
427C2×C8—24no—
428C2×C8—24no—
429C2×C8—24no—
430C2×C8—24no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—48no—
448C4⋊C4—48no—
449C4⋊C4—48no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—44no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—44no—
459C4⋊C4—44no—
460C4⋊C4—44no—
461C4⋊C4—82no—
462C4⋊C4—81no—
463C4⋊C4—82no—
464C4⋊C4—81no—
465C4⋊C4—82no—
466C4⋊C4—82no—
467C4⋊C4—84no—
468C4⋊C4—82no—
469C4⋊C4—82no—
470C4⋊C42−1+414yes—
471C4⋊C42−1+414yes—
472C22⋊C4—44no—
473C22⋊C4—44no—
474C42—24no—
475C42—22no—
476C42—22no—
477C42—24no—
478C42—22no—
479C42—22no—
480C42—22no—
481C42—22no—
482C42—22no—
483C42—22no—
484C42—44no—
485C42—44no—
486C42—84no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23#53412yes—
492C23C2×#13712yes—
493C23#57512yes—
494C23C23×D411yescharacteristic
495Q8—48no—
496Q8—48no—
497Q8—44no—
498Q8—44no—
499Q8—84no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—28no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—28no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—44no—
529C2×C4—48no—
530C2×C4—48no—
531C2×C4—42no—
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—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—84no—
558C2×C4—84no—
559C2×C4#53412yes—
560C2×C4C2×#13712yes—
561C2×C4C2×#13712yes—
562C2×C4C2×H(5,𝔽2)12yes—
563C2×C4C2×2−1+412yes—
564C2×C4C2×2−1+412yes—
565C8—44no—
566C8—44no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22#319614yes—
574C22C2×#53411yescharacteristic
575C22#348611yescharacteristic
576C22#348611yescharacteristic
577C22C22×#13711yescharacteristic
578C22C2×#57511yescharacteristic
579C22#358511yescharacteristic
580C22#358611yescharacteristic
581C4—22no—
582C4—22no—
583C4—22no—
584C4—42no—
585C4—42no—
586C4—42no—
587C4—41no—
588C4—42no—
589C4—41no—
590C4—44no—
591C4—44no—
592C4—82no—
593C4—82no—
594C4—82no—
595C4—82no—
596C4—82no—
597C4—82no—
598C4#354414yes—
599C2—24no—
600C2#3725811yescharacteristic
601C2C2×#319611yescharacteristic
602C2#6075511yescharacteristic
603C2#6182611yescharacteristic
604C2#6182911yescharacteristic
605C2#6183111yescharacteristic
606C2#6183211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #32689C2semidirect
5. #32641C2semidirect
6. #32587C2semidirect
7. #32583C2semidirect

Maximal subgroups

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

subgroupindex
#351792
C2×#25842
#331002
#326892
#326412
#325872
#325832
#296432
#276142
#276132
#246092
#246062
#240642
#240632
#218222