Finite Group Databasebeta

← #6493783

Subgroups — #6493783

Order 512.

1641 conjugacy classes of subgroups.

The subgroup lattice has 595 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35259C211yesmaximal, characteristic
2#34188C211yesmaximal, characteristic, complemented
3#33157C212yesmaximal, complemented
4#33116C212yesmaximal, complemented
5#29944C212yesmaximal, complemented
6#29921C212yesmaximal, complemented
7#27863C212yesmaximal, complemented
8#27857C212yesmaximal, complemented
9#27678C212yesmaximal, complemented
10#27650C212yesmaximal, complemented
11#27637C214yesmaximal, complemented
12#24567C214yesmaximal, complemented
13#21967C212yesmaximal, complemented
14#21897C212yesmaximal, complemented
15#20630C211yesmaximal, characteristic, complemented
16C2×#540C2211yescharacteristic
17C2×C4×#41C2211yescharacteristic
18C2×#482C2214yes—
19C2×#424C2212yes—
20C2×#408C2212yes—
21#2856C2211yescharacteristic
22#2854C2212yes—
23#2851C2211yescharacteristic
24#2629C2211yescharacteristic
25#2619C2211yescharacteristic
26#2615C2212yes—
27#2464—22no—
28#2458—22no—
29#2456—22no—
30#2456—22no—
31#2355—22no—
32#2354—22no—
33#2352—22no—
34#2352—22no—
35#2324C2212yes—
36#2314C2212yes—
37#2314C2212yes—
38#2314C2211yescharacteristic
39#2311C2212yes—
40C4×#121C2212yes—
41#2285C2212yes—
42#2285C2212yes—
43#2285C2211yescharacteristic
44C2×#384C2211yescharacteristic
45#2274C2212yes—
46#2100C2212yes—
47#2100C2212yes—
48#2094C2212yes—
49#2094C2212yes—
50#2084C2212yes—
51#2083C2212yes—
52#2077C2214yes—
53#2075C2214yes—
54#2071C2212yes—
55#2071C2212yes—
56#2034C2214yes—
57#2034C2214yes—
58#1989C2214yes—
59#1982C2214yes—
60#1949C2214yes—
61#1949C2214yes—
62#1948C2214yes—
63#1948C2214yes—
64#1926C2214yes—
65#1925C2214yes—
66#1920C2214yes—
67#1919C2214yes—
68#1895C2214yes—
69#1893C2214yes—
70#1874C2214yes—
71#1853C2214yes—
72#1825C2214yes—
73#1809C2214yes—
74#1807C2214yes—
75#1804C2214yes—
76#1802C2214yes—
77#1798C2212yes—
78#1798C2212yes—
79#1797C2212yes—
80#1797C2212yes—
81#1728C2211yescharacteristic
82C2×#336C2211yescharacteristic
83C22×#41C2311yescharacteristic
84C22×M16C2312yes—
85C23×C8C2312yes—
86#540—28no—
87C2×#127C2311yescharacteristic
88C2×#125C2311yescharacteristic
89C2×#123—22no—
90C2×#123—22no—
91C2×#121—22no—
92C2×#121—22no—
93C2×#121C2312yes—
94C2×#121C2312yes—
95C2×#121C2312yes—
96C4×#41—28no—
97C2×C4×Q8D412yes—
98C2×C4×D4—22no—
99C2×C4×D4D412yes—
100C2×C4×D4D412yes—
101C2×C4×D4D412yes—
102C2×C4×D4C2312yes—
103C2×#117—22no—
104C2×#117—22no—
105C2×#117C2312yes—
106C2×#117C2311yescharacteristic
107C2×#117C2311yescharacteristic
108C2×#117C2311yescharacteristic
109C22×C4⋊C4—22no—
110C22×C4⋊C4C2312yes—
111C22×C42C2311yescharacteristic
112C22×C42C2311yescharacteristic
113C22×C42C2311yescharacteristic
114#482—48no—
115C2×#107C2314yes—
116C2×#106C2314yes—
117C2×#106C2314yes—
118C2×#106C2314yes—
119#424—28no—
120C2×C4⋊C8C2314yes—
121C2×C4⋊C8C2314yes—
122C2×#102C2314yes—
123C2×#102C2314yes—
124C2×#102C2314yes—
125C2×#102C2314yes—
126#408—44no—
127C2×C22⋊C8C2314yes—
128C2×C22⋊C8C2314yes—
129#398—22no—
130#396—22no—
131#394—22no—
132#393—22no—
133#392—22no—
134#392—22no—
135#392—22no—
136#392—22no—
137#391—22no—
138#390—22no—
139#390—22no—
140#390—22no—
141#390C2312yes—
142#390C2312yes—
143#389C2312yes—
144#389C2312yes—
145#388—22no—
146#388—24no—
147#388—24no—
148#388—22no—
149#388—22no—
150#388—22no—
151#388—22no—
152#387—24no—
153#387—24no—
154#387—22no—
155#387—22no—
156#386—22no—
157#386—22no—
158#386—22no—
159#386—22no—
160#386—22no—
161#385—22no—
162#385—22no—
163#384—22no—
164#384C2312yes—
165#384C2311yescharacteristic
166#384C2312yes—
167#384C2312yes—
168#382—22no—
169#382—22no—
170#382C2312yes—
171#382C2311yescharacteristic
172#380—22no—
173#380—22no—
174#380—22no—
175#380C2314yes—
176#380C2314yes—
177#380C2312yes—
178#380C2312yes—
179#380C2312yes—
180C4×C4⋊C4—24no—
181C4×C4⋊C4—22no—
182C4×C4⋊C4—22no—
183C4×C4⋊C4—22no—
184C4×C4⋊C4C2312yes—
185C4×C4⋊C4C2314yes—
186C4×C4⋊C4C2314yes—
187C4×C22⋊C4—24no—
188C4×C22⋊C4—22no—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#336C2312yes—
198#336C2312yes—
199#336C2312yes—
200#336C2312yes—
201C2×#41—24no—
202C2×#41—22no—
203C2×#41—22no—
204C2×#41—42no—
205C22×Q8C2×D411yescharacteristic
206C22×D4—22no—
207C22×D4—22no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4C2×D411yescharacteristic
211C22×D4C2×D411yescharacteristic
212C22×D4C2×D411yescharacteristic
213C22×D4C2412yes—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4C2×D412yes—
217C23×C4C2×D412yes—
218C23×C4C2×D412yes—
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C2×M16—24no—
223C2×M16—42no—
224C2×M16—42no—
225C22×C8—24no—
226C22×C8—42no—
227C22×C8—42no—
228C22×C8C2414yes—
229C22×C8C2414yes—
230#127—24no—
231#125—24no—
232#123—48no—
233#123—84no—
234#121—28no—
235#121—24no—
236#121—24no—
237#121—44no—
238#121—48no—
239#121—84no—
240C4×Q8—28no—
241C4×D4—28no—
242C4×D4—28no—
243C4×D4—28no—
244C4×D4—28no—
245C4×D4—84no—
246#117—24no—
247#117—24no—
248#117—24no—
249#117—24no—
250#117—84no—
251#117—84no—
252#117#4118yes—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—84no—
265C2×C4⋊C4C2×D412yes—
266C2×C4⋊C4C2×D412yes—
267C2×C4⋊C4C2×D412yes—
268C2×C4⋊C4C2414yes—
269C2×C4⋊C4C2414yes—
270C2×C4⋊C4C2412yes—
271C2×C4⋊C4C2412yes—
272C2×C4⋊C4C2412yes—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4C2×D412yes—
288C2×C22⋊C4C2×D412yes—
289C2×C22⋊C4C2×D412yes—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—22no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—22no—
297C2×C42C2×D412yes—
298C2×C42C2×D412yes—
299C2×C42C2×D412yes—
300C2×C42C2411yescharacteristic
301C2×C42C2411yescharacteristic
302C2×C42C2411yescharacteristic
303C2×C42C2411yescharacteristic
304#107—44no—
305#106—44no—
306#106—44no—
307#106—44no—
308C4⋊C8—28no—
309C4⋊C8—28no—
310#102—44no—
311#102—44no—
312#102—44no—
313#102—44no—
314C22⋊C8—28no—
315C22⋊C8—44no—
316#95—24no—
317#95—24no—
318#95—22no—
319#95—22no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—22no—
324#95—22no—
325#95—22no—
326#95—22no—
327#95—22no—
328C24—22no—
329C24C22×D411yescharacteristic
330C24C22×D411yescharacteristic
331C24C22×D411yescharacteristic
332#41—48no—
333#41—48no—
334C2×Q8—22no—
335C2×Q8—22no—
336C2×Q8—41no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—24no—
344C2×D4—22no—
345C2×D4—22no—
346C2×D4—24no—
347C2×D4—41no—
348C2×D4—41no—
349C2×D4—41no—
350C2×D4—44no—
351C2×D4—44no—
352C2×D4—44no—
353C2×D4—44no—
354C2×D4—82no—
355C2×D4—82no—
356C2×D4—82no—
357C2×D4—82no—
358C2×D4—82no—
359C2×D4—82no—
360C2×D4—82no—
361C2×D4—82no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—82no—
388C22×C4—82no—
389C22×C4—84no—
390C22×C4C22×D411yescharacteristic
391C22×C4C22×D412yes—
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D412yes—
394C22×C4C22×D412yes—
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D412yes—
397C22×C4C2×#4114yes—
398C22×C4H(5,𝔽2)12yes—
399C22×C42−1+412yes—
400C22×C4C2511yescharacteristic
401M16—84no—
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—24no—
405C2×C8—24no—
406C2×C8—24no—
407C2×C8—24no—
408C2×C8—84no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—22no—
414C4⋊C4—24no—
415C4⋊C4—22no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—44no—
420C4⋊C4—44no—
421C4⋊C4—44no—
422C4⋊C4—44no—
423C4⋊C4—44no—
424C4⋊C4—44no—
425C4⋊C4—44no—
426C4⋊C4—84no—
427C4⋊C4—82no—
428C4⋊C4—82no—
429C4⋊C4#12318yes—
430C22⋊C4—24no—
431C22⋊C4—24no—
432C22⋊C4—44no—
433C22⋊C4—44no—
434C22⋊C4—44no—
435C22⋊C4—44no—
436C22⋊C4—44no—
437C22⋊C4—44no—
438C22⋊C4—44no—
439C22⋊C4—82no—
440C22⋊C4—82no—
441C22⋊C4—82no—
442C22⋊C4—84no—
443C22⋊C4—84no—
444C22⋊C4—82no—
445C22⋊C4—82no—
446C22⋊C4#12318yes—
447C42—24no—
448C42—24no—
449C42—24no—
450C42—22no—
451C42—22no—
452C42—44no—
453C42—84no—
454C42—84no—
455C42—82no—
456C42C2×#4114yes—
457C42C2×#4114yes—
458C23—22no—
459C23—22no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—22no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23C2×#12312yes—
479C23#53712yes—
480C23C23×D411yescharacteristic
481Q8—44no—
482D4—44no—
483D4—44no—
484D4—44no—
485D4—44no—
486D4—44no—
487D4—88no—
488D4—164no—
489D4—164no—
490D4—164no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—22no—
511C2×C4—48no—
512C2×C4—48no—
513C2×C4—48no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—164no—
532C2×C4—164no—
533C2×C4C2×#12312yes—
534C2×C4C2×#12314yes—
535C2×C4C2×#12314yes—
536C2×C4#53712yes—
537C2×C4#57714yes—
538C2×C4C22×#4112yes—
539C2×C4C2×H(5,𝔽2)11yescharacteristic
540C2×C4C2×2−1+411yescharacteristic
541C2×C4#58512yes—
542C8—44no—
543C8—44no—
544C22—24no—
545C22—24no—
546C22—22no—
547C22—22no—
548C22—42no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22—42no—
554C22—42no—
555C22—41no—
556C22—41no—
557C22—42no—
558C22—41no—
559C22—82no—
560C22—82no—
561C22—82no—
562C22—82no—
563C22—82no—
564C22—82no—
565C22—82no—
566C22C22×#12311yescharacteristic
567C22C2×#53711yescharacteristic
568C22#344911yescharacteristic
569C22C2×#57712yes—
570C22#358512yes—
571C4—24no—
572C4—24no—
573C4—24no—
574C4—22no—
575C4—22no—
576C4—42no—
577C4—41no—
578C4—84no—
579C4—82no—
580C4—82no—
581C4—84no—
582C4—82no—
583C4#349412yes—
584C4#349512yes—
585C2—42no—
586C2—42no—
587C2—81no—
588C2—162no—
589C2#3526311yescharacteristic
590C2#6052512yes—
591C2#6092912yes—
592C2#6101211yescharacteristic
593C2#6101911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35259C2
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
2. #34188C2semidirect
3. #33157C2semidirect
4. #33116C2semidirect
5. #29944C2semidirect
6. #29921C2semidirect
7. #27863C2semidirect
8. #27857C2semidirect
9. #27678C2semidirect
10. #27650C2semidirect
11. #27637C2semidirect
12. #24567C2semidirect
13. #21967C2semidirect
14. #21897C2semidirect
15. #20630C2semidirect

Maximal subgroups

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

subgroupindex
#352592
#341882
#331572
#331162
#299442
#299212
#278632
#278572
#276782
#276502
#276372
#245672
#219672
#218972
#206302