Finite Group Databasebeta

← #395850

Subgroups — #395850

Order 512.

1672 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#106C211yesmaximal, characteristic
2C2×#2084C212yesmaximal, complemented
3C2×#1853C211yesmaximal, characteristic
4#13907C211yesmaximal, characteristic
5#13880C211yesmaximal, characteristic
6#11323C212yesmaximal, complemented
7#11257C212yesmaximal, complemented
8#11220C212yesmaximal, complemented
9#10730C211yesmaximal, characteristic
10#10721C211yesmaximal, characteristic
11#10714C211yesmaximal, characteristic
12C24×C8C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×#106—28no—
16C22×#106—24no—
17C22×#106C2212yes—
18C22×#106C2211yescharacteristic
19C22×#106C2212yes—
20C22×#106C2211yescharacteristic
21C22×C22⋊C8C2211yescharacteristic
22C2×#389C2212yes—
23C2×#385—21no—
24C2×#385—21no—
25C2×#384C2212yes—
26C2×#380—21no—
27C2×#380C2211yescharacteristic
28C2×#380C2211yescharacteristic
29C2×C4×C22⋊C4C2211yescharacteristic
30#2084—28no—
31#1853—24no—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2211yescharacteristic
35C2×#340C2211yescharacteristic
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2212yes—
39C2×#336C2212yes—
40#1444—22no—
41#1441—22no—
42C4⋊#95C2212yes—
43C4⋊#95C2212yes—
44C22⋊#95—21no—
45C24×C4—21no—
46C24×C4—21no—
47C24×C4C2311yescharacteristic
48C23×C8—22no—
49C23×C8—22no—
50C23×C8—24no—
51C23×C8—24no—
52C23×C8Q814yes—
53C23×C8C2311yescharacteristic
54C23×C8C2311yescharacteristic
55C23×C8C2312yes—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—24no—
58C22×C4⋊C4—24no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—42no—
67C22×C4⋊C4—42no—
68C22×C4⋊C4D414yes—
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2311yescharacteristic
75C22×C22⋊C4—21no—
76C22×C22⋊C4—21no—
77C22×C22⋊C4D411yescharacteristic
78C22×C22⋊C4D411yescharacteristic
79C22×C42C2312yes—
80C2×#106—232no—
81C2×#106—24no—
82C2×#106—24no—
83C2×#106—24no—
84C2×#106—24no—
85C2×#106—24no—
86C2×#106—24no—
87C2×#106—22no—
88C2×#106—22no—
89C2×#106—42no—
90C2×#106D418yes—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8Q814yes—
94#389—24no—
95#389—42no—
96#385—44no—
97#385—44no—
98#384—24no—
99#384—42no—
100#380—22no—
101#380—22no—
102#380—22no—
103#380—22no—
104#380—44no—
105C4×C22⋊C4—22no—
106C4×C22⋊C4—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—22no—
114C2×#95D412yes—
115#340—24no—
116#340—24no—
117#340—24no—
118#340—22no—
119#340—22no—
120#340—42no—
121#336—24no—
122#336—24no—
123C25C2×D411yescharacteristic
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—21no—
129C23×C4—22no—
130C23×C4—21no—
131C23×C4—24no—
132C23×C4—21no—
133C23×C4—22no—
134C23×C4—21no—
135C23×C4—44no—
136C23×C4—44no—
137C23×C4—42no—
138C23×C4—42no—
139C23×C4—42no—
140C23×C4—42no—
141C23×C4—42no—
142C23×C4—82no—
143C23×C4—82no—
144C23×C4C2×D412yes—
145C23×C4C2×D412yes—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D412yes—
148C23×C4C2×Q812yes—
149C23×C4C2×Q812yes—
150C23×C4#4112yes—
151C23×C4#4112yes—
152C23×C4#4112yes—
153C23×C4C2411yescharacteristic
154C22×C8—216no—
155C22×C8—28no—
156C22×C8—28no—
157C22×C8—28no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—216no—
163C22×C8—22no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—22no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8C2×D414yes—
172C22×C8C2×Q812yes—
173C22×C8C2×Q812yes—
174C22×C8C2×Q814yes—
175C22×C8#4114yes—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—28no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—48no—
205C2×C4⋊C4—48no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—82no—
217C2×C4⋊C4—82no—
218C2×C4⋊C4—82no—
219C2×C4⋊C4C2×D414yes—
220C2×C4⋊C4C2×D412yes—
221C2×C4⋊C4C2×D414yes—
222C2×C4⋊C4C2×D412yes—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4SD1614yes—
234C2×C22⋊C4SD1614yes—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241#106—216no—
242#106—216no—
243#106—44no—
244#106—44no—
245C22⋊C8—28no—
246#95—24no—
247#95—44no—
248#95—44no—
249#95—42no—
250#95—42no—
251#95—44no—
252#95—44no—
253#95—82no—
254C24—22no—
255C24—22no—
256C24—22no—
257C24—22no—
258C24#12112yes—
259C24#12312yes—
260C24C4⋊Q812yes—
261C24C2×SD1612yes—
262C24C2×SD1612yes—
263C24AGL(1,ℤ/8)12yes—
264C24#13712yes—
265C24C22×D411yescharacteristic
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—24no—
270C22×C4—22no—
271C22×C4—24no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—48no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—42no—
302C22×C4—48no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—41no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—48no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—84no—
332C22×C4—88no—
333C22×C4—88no—
334C22×C4—88no—
335C22×C4—88no—
336C22×C4—84no—
337C22×C4—84no—
338C22×C4#12112yes—
339C22×C4C22⋊Q814yes—
340C22×C4C22⋊Q814yes—
341C22×C4#12312yes—
342C22×C4#12514yes—
343C22×C4#12514yes—
344C22×C4C4⋊Q812yes—
345C22×C4C2×SD1614yes—
346C22×C4C2×SD1614yes—
347C22×C4C22×D411yescharacteristic
348C22×C4C22×D412yes—
349C22×C4C22×Q811yescharacteristic
350C22×C4C22×Q811yescharacteristic
351C22×C4C2×#4111yescharacteristic
352C22×C4C2×#4112yes—
353C22×C4C2×#4111yescharacteristic
354C22×C4C2×#4111yescharacteristic
355C22×C4H(5,𝔽2)12yes—
356C22×C42−1+412yes—
357C2×C8—232no—
358C2×C8—24no—
359C2×C8—24no—
360C2×C8—24no—
361C2×C8—24no—
362C2×C8—24no—
363C2×C8—24no—
364C2×C8—22no—
365C2×C8—22no—
366C2×C8—24no—
367C2×C8—24no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8C22⋊Q818yes—
375C4⋊C4—22no—
376C4⋊C4—24no—
377C4⋊C4—22no—
378C4⋊C4—24no—
379C4⋊C4—22no—
380C4⋊C4—24no—
381C4⋊C4—48no—
382C4⋊C4—48no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—42no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—88no—
397C4⋊C4—88no—
398C4⋊C4—88no—
399C4⋊C4#12014yes—
400C22⋊C4—28no—
401C22⋊C4—28no—
402C22⋊C4—84no—
403C22⋊C4—84no—
404C42—42no—
405C42—42no—
406C42—42no—
407C42—42no—
408C23—24no—
409C23—24no—
410C23—24no—
411C23—24no—
412C23—24no—
413C23—24no—
414C23—22no—
415C23—22no—
416C23—24no—
417C23—24no—
418C23—24no—
419C23—24no—
420C23—24no—
421C23—24no—
422C23—22no—
423C23—22no—
424C23#39814yes—
425C23#46014yes—
426C23#46114yes—
427C23#48114yes—
428C23#48314yes—
429C23#49814yes—
430C23#50114yes—
431C23C2×#12111yescharacteristic
432C23C2×#12311yescharacteristic
433C23C2×C4⋊Q811yescharacteristic
434C23D4212yes—
435C23#54712yes—
436C23C22×SD1611yescharacteristic
437C23C22×SD1611yescharacteristic
438C23C2×AGL(1,ℤ/8)11yescharacteristic
439C23C2×#13711yescharacteristic
440C23#57712yes—
441C23#57712yes—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—28no—
451C2×C4—28no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—816no—
478C2×C4—816no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4#39814yes—
503C2×C4C2×#12011yescharacteristic
504C2×C4C2×C22⋊Q811yescharacteristic
505C2×C4C2×C22⋊Q811yescharacteristic
506C2×C4C2×C22⋊Q812yes—
507C2×C4#52511yescharacteristic
508C2×C4#52512yes—
509C2×C4C2×#12511yescharacteristic
510C2×C4C2×#12511yescharacteristic
511C2×C4#53411yescharacteristic
512C2×C4#53611yescharacteristic
513C2×C4#53912yes—
514C2×C4#54412yes—
515C2×C4#54612yes—
516C2×C4#54812yes—
517C2×C4D4×Q812yes—
518C2×C4#55212yes—
519C2×C4#55312yes—
520C2×C4#55412yes—
521C8—28no—
522C8—42no—
523C8—42no—
524C22—24no—
525C22—24no—
526C22—28no—
527C22—28no—
528C22—24no—
529C22—24no—
530C22—24no—
531C22—24no—
532C22—28no—
533C22—28no—
534C22#208418yescomplemented
535C22C2×#39811yescharacteristic
536C22#241011yescharacteristic
537C22#241311yescharacteristic
538C22#244012yes—
539C22#244312yes—
540C22C2×#46111yescharacteristic
541C22C2×#46011yescharacteristic
542C22#304211yescharacteristic
543C22#304311yescharacteristic
544C22C2×#48311yescharacteristic
545C22C2×#48111yescharacteristic
546C22#309912yes—
547C22C2×#49811yescharacteristic
548C22C2×#50111yescharacteristic
549C22#316312yes—
550C22#326512yes—
551C22#326612yes—
552C22D4×SD1612yes—
553C22D4×SD1612yes—
554C22#333512yes—
555C22#333512yes—
556C22#334412yes—
557C22#334512yes—
558C4—24no—
559C4—44no—
560C4—44no—
561C4—44no—
562C4—84no—
563C4—82no—
564C4—161no—
565C4—162no—
566C4—161no—
567C4#240911yescharacteristic
568C4#244212yes—
569C4#247312yes—
570C4#257911yescharacteristic
571C4#262711yescharacteristic
572C4#284811yescharacteristic
573C2—28no—
574C2#1472811yescharacteristic
575C2C2×#208411yescharacteristic
576C2#2366111yescharacteristic
577C2#2435611yescharacteristic
578C2#2435811yescharacteristic
579C2#2457811yescharacteristic
580C2#2458011yescharacteristic
581C2#2606012yes—
582C2#2606812yes—
583C2#2612812yes—
584C2#2613012yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2084C2semidirect
6. #11323C2semidirect
7. #11257C2semidirect
8. #11220C2semidirect
534. C22#2084semidirect

Maximal subgroups

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

subgroupindex
C23×#1062
C2×#20842
C2×#18532
#139072
#138802
#113232
#112572
#112202
#107302
#107212
#107142