Finite Group Databasebeta

← #6475349

Subgroups — #6475349

Order 512.

1708 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#35280C211yesmaximal, characteristic
2#34268C211yesmaximal, characteristic
3#33157C212yesmaximal, complemented
4#32967C212yesmaximal, complemented
5#29971C212yesmaximal, complemented
6#29951C212yesmaximal, complemented
7#29250C214yesmaximal, complemented
8#29165C212yesmaximal, complemented
9#29159C212yesmaximal, complemented
10#27680C212yesmaximal, complemented
11#24854C214yesmaximal, complemented
12#24306C212yesmaximal, complemented
13#21946C212yesmaximal
14#21771C212yesmaximal
15#20630C211yesmaximal, characteristic
16C2×#536C2211yescharacteristic
17C2×#533C2211yescharacteristic
18C2×#519C2211yescharacteristic
19C2×C4×#41C2211yescharacteristic
20C2×#479C2214yes—
21C2×#424C2212yes—
22C2×#408C2212yes—
23#2857C2212yes—
24#2857C2212yes—
25#2629C2212yes—
26#2629C2212yes—
27#2582—22no—
28#2538—22no—
29#2417—22no—
30#2417—22no—
31#2367—22no—
32#2365—22no—
33#2359—22no—
34#2359—22no—
35#2323C2212yes—
36#2323C2212yes—
37#2316C2212yes—
38#2311C2212yes—
39#2311C2211yescharacteristic
40C4×C22⋊Q8C2212yes—
41C4×C22⋊Q8C2212yes—
42#2296C2211yescharacteristic
43C2×#391C2211yescharacteristic
44C2×#384C2211yescharacteristic
45#2274C2211yescharacteristic
46#2099C2212yes—
47#2099C2212yes—
48#2096C2212yes—
49#2096C2212yes—
50#2088C2214yes—
51#2087C2214yes—
52#2066C2214yes—
53#2066C2214yes—
54#2058C2212yes—
55#2057C2212yes—
56#2024C2212yes—
57#2024C2212yes—
58#1989C2214yes—
59#1948C2214yes—
60#1920C2214yes—
61#1919C2214yes—
62#1894C2214yes—
63#1892C2214yes—
64#1873C2214yes—
65#1872C2214yes—
66#1865C2214yes—
67#1863C2214yes—
68#1854C2214yes—
69#1854C2214yes—
70#1853C2214yes—
71#1853C2214yes—
72#1825C2214yes—
73#1809C2214yes—
74#1807C2214yes—
75#1803C2214yes—
76#1798C2212yes—
77#1798C2212yes—
78#1797C2212yes—
79#1797C2212yes—
80#1787C2214yes—
81#1728C2211yescharacteristic
82C2×#336C2211yescharacteristic
83C22×#41C2311yescharacteristic
84C23×Q8C2311yescharacteristic
85C22×M16C2312yes—
86C23×C8C2312yes—
87#536—24no—
88#536Q818yescomplemented
89#533—44no—
90C2×C4⋊Q8—21no—
91C2×C4⋊Q8C2312yes—
92C2×#125C2312yes—
93C2×#124—21no—
94C2×#123—21no—
95C2×#123—21no—
96C2×C22⋊Q8—22no—
97C2×C22⋊Q8—22no—
98C2×C22⋊Q8—22no—
99C2×C22⋊Q8C2312yes—
100C2×C22⋊Q8C2312yes—
101#519—44no—
102C4×#41—44no—
103C2×C4×Q8—22no—
104C2×C4×Q8—22no—
105C2×C4×Q8D412yes—
106C2×C4×Q8C2312yes—
107C2×C4×Q8C2312yes—
108C2×C4×D4D412yes—
109C2×C4×D4D412yes—
110C2×C4×D4D412yes—
111C2×#117—22no—
112C2×#117C2311yescharacteristic
113C2×#117C2312yes—
114C2×#117C2311yescharacteristic
115C2×#117C2311yescharacteristic
116C2×#117C2311yescharacteristic
117C2×#117C2311yescharacteristic
118C22×C4⋊C4—22no—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124#479—48no—
125C2×#107C2314yes—
126C2×#106C2314yes—
127C2×#106C2314yes—
128C2×#106C2314yes—
129#424—44no—
130C2×C4⋊C8C2314yes—
131C2×C4⋊C8C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136#408—28no—
137C2×C22⋊C8C2314yes—
138C2×C22⋊C8C2314yes—
139#398—24no—
140#398—24no—
141#396—24no—
142#396—24no—
143#391—22no—
144#391—21no—
145#391—22no—
146#391—22no—
147#391—22no—
148#391C2312yes—
149#391C2312yes—
150#390—21no—
151#390—22no—
152#389—24no—
153#389—22no—
154#389—22no—
155#389—22no—
156#389—22no—
157#389C2312yes—
158#389C2314yes—
159#389C2314yes—
160#389C2312yes—
161#388—22no—
162#388—22no—
163#388—22no—
164#388—22no—
165#387—22no—
166#387—22no—
167#387—22no—
168#387—22no—
169#386—21no—
170#386—21no—
171#386—22no—
172#386—22no—
173#385—22no—
174#385—22no—
175#385—22no—
176#384—22no—
177#384C2312yes—
178#384C2312yes—
179#380—22no—
180#380—22no—
181#380—24no—
182#380C2312yes—
183#380C2314yes—
184#380C2312yes—
185#380C2314yes—
186C4×C4⋊C4—22no—
187C4×C4⋊C4—22no—
188C4×C4⋊C4—22no—
189C4×C4⋊C4—22no—
190C4×C4⋊C4C2312yes—
191C4×C4⋊C4C2312yes—
192C4×C22⋊C4—22no—
193C4×C22⋊C4—22no—
194C4×C22⋊C4—22no—
195C2×#95—22no—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204#336C2312yes—
205#336C2312yes—
206#336C2312yes—
207#336C2312yes—
208C2×#41—42no—
209C2×#41—42no—
210C2×#41C2×Q814yes—
211C2×#41#4114yes—
212C22×Q8—21no—
213C22×Q8—22no—
214C22×Q8—22no—
215C22×Q8—42no—
216C22×Q8—42no—
217C22×Q8C2×D411yescharacteristic
218C22×Q8C2×Q814yes—
219C22×Q8C2412yes—
220C22×D4C2×D411yescharacteristic
221C22×D4C2×D411yescharacteristic
222C22×D4C2×D411yescharacteristic
223C23×C4—21no—
224C23×C4—22no—
225C23×C4—21no—
226C23×C4C2×D412yes—
227C23×C4C2×D412yes—
228C23×C4C2×D412yes—
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232C2×M16—24no—
233C2×M16—42no—
234C2×M16—42no—
235C22×C8—24no—
236C22×C8—42no—
237C22×C8—42no—
238C22×C8C2414yes—
239C22×C8C2414yes—
240C4⋊Q8—28no—
241C4⋊Q8—44no—
242#125—28no—
243#124—28no—
244#123—28no—
245#123—28no—
246C22⋊Q8—24no—
247C22⋊Q8—24no—
248C22⋊Q8—48no—
249C22⋊Q8—84no—
250C22⋊Q8—84no—
251C22⋊Q8C2×Q818yes—
252C22⋊Q8C2×Q818yes—
253C4×Q8—28no—
254C4×Q8—44no—
255C4×Q8—44no—
256C4×Q8—48no—
257C4×Q8—48no—
258C4×D4—44no—
259C4×D4—44no—
260C4×D4—44no—
261#117—22no—
262#117—24no—
263#117—24no—
264#117—42no—
265#117—48no—
266#117—42no—
267#117—42no—
268#117—42no—
269#117C2×Q814yes—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4—42no—
287C2×C4⋊C4—42no—
288C2×C4⋊C4—84no—
289C2×C4⋊C4C2×D412yes—
290C2×C4⋊C4C2×D412yes—
291C2×C4⋊C4C2×D412yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2414yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2414yes—
296C2×C4⋊C4C2412yes—
297C2×C22⋊C4—21no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
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⋊C4C2×D412yes—
307C2×C22⋊C4C2×D412yes—
308C2×C22⋊C4C2×D412yes—
309C2×C42—21no—
310C2×C42—24no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—21no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—42no—
318C2×C42—42no—
319C2×C42C2×D412yes—
320C2×C42C2×D412yes—
321C2×C42C2×D412yes—
322C2×C42C2411yescharacteristic
323C2×C42C2411yescharacteristic
324C2×C42C2411yescharacteristic
325C2×C42C2411yescharacteristic
326#107—44no—
327#106—44no—
328#106—44no—
329#106—44no—
330C4⋊C8—28no—
331C4⋊C8—44no—
332#103—44no—
333#103—44no—
334#103—44no—
335#103—44no—
336C22⋊C8—28no—
337C22⋊C8—28no—
338#95—22no—
339#95—22no—
340#95—22no—
341#95—22no—
342#95—22no—
343#95—22no—
344#95—22no—
345#95—22no—
346#95—22no—
347#95—22no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—24no—
352C24C22×D411yescharacteristic
353C24C22×D411yescharacteristic
354C24C22×D411yescharacteristic
355#41—48no—
356#41—48no—
357C2×Q8—24no—
358C2×Q8—22no—
359C2×Q8—24no—
360C2×Q8—24no—
361C2×Q8—41no—
362C2×Q8—41no—
363C2×Q8—48no—
364C2×Q8—42no—
365C2×Q8—42no—
366C2×Q8—48no—
367C2×Q8—44no—
368C2×Q8—44no—
369C2×Q8—44no—
370C2×Q8C22⋊Q814yes—
371C2×Q8C22×Q814yes—
372C2×D4—41no—
373C2×D4—41no—
374C2×D4—41no—
375C2×D4—41no—
376C2×D4—41no—
377C2×D4—41no—
378C2×D4C22⋊Q814yes—
379C2×D4C22⋊Q814yes—
380C2×D4C22⋊Q814yes—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—44no—
406C22×C4—42no—
407C22×C4—84no—
408C22×C4—82no—
409C22×C4C22×D412yes—
410C22×C4C22×D412yes—
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D412yes—
415C22×C4C22×D412yes—
416C22×C4C22×Q812yes—
417C22×C4C22×Q812yes—
418C22×C4C22×Q812yes—
419C22×C4C2×#4112yes—
420C22×C4C2×#4112yes—
421C22×C4C2×#4112yes—
422C22×C4C2511yescharacteristic
423M16—84no—
424C2×C8—24no—
425C2×C8—24no—
426C2×C8—24no—
427C2×C8—24no—
428C2×C8—24no—
429C2×C8—24no—
430C2×C8—84no—
431C4⋊C4—28no—
432C4⋊C4—28no—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—22no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—42no—
443C4⋊C4—44no—
444C4⋊C4—42no—
445C4⋊C4—44no—
446C4⋊C4—44no—
447C4⋊C4—44no—
448C4⋊C4—48no—
449C4⋊C4—44no—
450C4⋊C4—44no—
451C4⋊C4—44no—
452C4⋊C4—82no—
453C4⋊C4—84no—
454C4⋊C4—84no—
455C4⋊C4—84no—
456C4⋊C4C22×Q814yes—
457C22⋊C4—24no—
458C22⋊C4—28no—
459C22⋊C4—28no—
460C22⋊C4—24no—
461C22⋊C4—24no—
462C22⋊C4—42no—
463C22⋊C4—42no—
464C22⋊C4—44no—
465C22⋊C4—82no—
466C22⋊C4—82no—
467C22⋊C4—84no—
468C22⋊C4—84no—
469C42—24no—
470C42—24no—
471C42—22no—
472C42—22no—
473C42—22no—
474C42—42no—
475C42—42no—
476C42—44no—
477C42—42no—
478C42—44no—
479C42—44no—
480C42—82no—
481C42—82no—
482C42H(5,𝔽2)14yes—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23C2×C22⋊Q812yes—
496C23C2×C22⋊Q812yes—
497C23C2×C22⋊Q812yes—
498C23C23×D411yescharacteristic
499Q8—28no—
500Q8—44no—
501Q8—48no—
502Q8—44no—
503Q8—88no—
504D4—44no—
505D4—44no—
506D4—44no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—22no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—48no—
532C2×C4—48no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—84no—
550C2×C4—84no—
551C2×C4—164no—
552C2×C4—162no—
553C2×C4C2×C22⋊Q812yes—
554C2×C4C2×C22⋊Q812yes—
555C2×C4C2×C22⋊Q812yes—
556C2×C4#53414yes—
557C2×C4#57714yes—
558C2×C4C23×Q811yescharacteristic
559C2×C4C22×#4111yescharacteristic
560C2×C4C2×H(5,𝔽2)12yes—
561C2×C4#58512yes—
562C8—44no—
563C8—44no—
564C22—24no—
565C22—24no—
566C22—24no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22C22×C22⋊Q811yescharacteristic
580C22C2×#53411yescharacteristic
581C22#344911yescharacteristic
582C22C2×#57712yes—
583C22#358612yes—
584C4—24no—
585C4—24no—
586C4—22no—
587C4—22no—
588C4—22no—
589C4—44no—
590C4—42no—
591C4—42no—
592C4—44no—
593C4—84no—
594C4—82no—
595C4—82no—
596C4—84no—
597C4#350112yes—
598C4#350512yes—
599C2—42no—
600C2—42no—
601C2—42no—
602C2#3501011yescharacteristic
603C2#6049712yes—
604C2#6076312yes—
605C2#6101211yescharacteristic
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. #33157C2semidirect
4. #32967C2semidirect
5. #29971C2semidirect
6. #29951C2semidirect
7. #29250C2semidirect
8. #29165C2semidirect
9. #29159C2semidirect
10. #27680C2semidirect
11. #24854C2semidirect
12. #24306C2semidirect
88. #536Q8semidirect

Maximal subgroups

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

subgroupindex
#352802
#342682
#331572
#329672
#299712
#299512
#292502
#291652
#291592
#276802
#248542
#243062
#219462
#217712
#206302