Finite Group Databasebeta

← #399595

Subgroups — #399595

Order 512.

1646 conjugacy classes of subgroups.

The subgroup lattice has 589 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#106C211yesmaximal, characteristic
2C2×#2088C212yesmaximal, complemented
3C2×#1850C211yesmaximal, characteristic
4#14130C211yesmaximal, characteristic
5#14130C211yesmaximal, characteristic
6#11378C212yesmaximal, complemented
7#11327C212yesmaximal, complemented
8#11231C212yesmaximal, complemented
9#10730C211yesmaximal, characteristic
10#10730C211yesmaximal, characteristic
11#10721C211yesmaximal, characteristic
12C24×C8C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×#106—28no—
16C22×#106—24no—
17C22×#106C2212yes—
18C22×#106C2212yes—
19C22×C4⋊C8C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C22×C22⋊C8C2211yescharacteristic
22C2×#389C2212yes—
23C2×#389C2212yes—
24C2×#385—21no—
25C2×#385—21no—
26C2×#380C2211yescharacteristic
27C2×#380C2211yescharacteristic
28C2×C4×C22⋊C4C2211yescharacteristic
29#2088—28no—
30#1850—24no—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2211yescharacteristic
38C2×#336C2211yescharacteristic
39#1444—21no—
40#1444—21no—
41#1442—21no—
42#1442—21no—
43C4⋊#95C2212yes—
44C4⋊#95C2212yes—
45C22⋊#95—21no—
46C22⋊#95—21no—
47C24×C4—21no—
48C24×C4—21no—
49C24×C4C2311yescharacteristic
50C23×C8—22no—
51C23×C8—22no—
52C23×C8—24no—
53C23×C8—24no—
54C23×C8D414yes—
55C23×C8C2311yescharacteristic
56C23×C8C2311yescharacteristic
57C23×C8C2311yescharacteristic
58C23×C8C2311yescharacteristic
59C22×C4⋊C4—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—24no—
62C22×C4⋊C4—24no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—42no—
70C22×C4⋊C4—42no—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—21no—
77C22×C22⋊C4D411yescharacteristic
78C22×C22⋊C4D411yescharacteristic
79C22×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81C2×#106—232no—
82C2×#106—24no—
83C2×#106—24no—
84C2×#106—24no—
85C2×#106—24no—
86C2×#106—24no—
87C2×#106D418yes—
88C2×C4⋊C8—24no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—24no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95#389—24no—
96#389—24no—
97#389—42no—
98#389—42no—
99#385—44no—
100#385—44no—
101#380—22no—
102#380—22no—
103#380Q814yes—
104#380Q814yes—
105C4×C22⋊C4—22no—
106C4×C22⋊C4—22no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95D411yescharacteristic
116C2×#95D411yescharacteristic
117#340—24no—
118#340—24no—
119#340—24no—
120#340—24no—
121#340—42no—
122#340—42no—
123#336—22no—
124#336—22no—
125C25C2×D411yescharacteristic
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—44no—
135C23×C4—44no—
136C23×C4—42no—
137C23×C4—42no—
138C23×C4—42no—
139C23×C4—42no—
140C23×C4—42no—
141C23×C4—42no—
142C23×C4—82no—
143C23×C4—82no—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D412yes—
150C23×C4C2×Q812yes—
151C23×C4C2×Q812yes—
152C23×C4#4112yes—
153C23×C4#4112yes—
154C23×C4#4112yes—
155C23×C4C2411yescharacteristic
156C22×C8—216no—
157C22×C8—216no—
158C22×C8—28no—
159C22×C8—22no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—22no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—28no—
166C22×C8—28no—
167C22×C8—21no—
168C22×C8—22no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8C2×D414yes—
178C22×C8C2×D412yes—
179C22×C8C2×D412yes—
180C22×C8#4114yes—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—28no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—48no—
204C2×C4⋊C4—48no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—82no—
216C2×C4⋊C4—82no—
217C2×C4⋊C4C2×D414yes—
218C2×C4⋊C4C2×D414yes—
219C2×C4⋊C4C2×Q814yes—
220C2×C4⋊C4C2×Q814yes—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4SD1614yes—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—22no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241#106—216no—
242#106—216no—
243C4⋊C8—44no—
244C4⋊C8—44no—
245C22⋊C8—44no—
246#95—22no—
247#95—22no—
248#95—24no—
249#95—24no—
250#95—44no—
251#95—44no—
252#95—44no—
253#95—44no—
254#95—82no—
255#95—82no—
256C24—22no—
257C24—22no—
258C24—22no—
259C24—22no—
260C24#12112yes—
261C24C22⋊Q812yes—
262C24C22⋊Q812yes—
263C24C2×SD1612yes—
264C24#13512yes—
265C24AGL(1,ℤ/8)12yes—
266C24#13712yes—
267C24C22×D411yescharacteristic
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—48no—
307C22×C4—48no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—88no—
333C22×C4—88no—
334C22×C4—88no—
335C22×C4—88no—
336C22×C4—84no—
337C22×C4—84no—
338C22×C4—84no—
339C22×C4—84no—
340C22×C4#12112yes—
341C22×C4C22⋊Q812yes—
342C22×C4C22⋊Q812yes—
343C22×C4#12314yes—
344C22×C4#12514yes—
345C22×C4#12514yes—
346C22×C4C4⋊Q814yes—
347C22×C4C2×SD1612yes—
348C22×C4C2×SD1612yes—
349C22×C4C22×D411yescharacteristic
350C22×C4C22×D412yes—
351C22×C4C22×Q811yescharacteristic
352C22×C4C22×Q811yescharacteristic
353C22×C4C2×#4111yescharacteristic
354C22×C4C2×#4111yescharacteristic
355C22×C4C2×#4112yes—
356C22×C4C2×#4111yescharacteristic
357C22×C4H(5,𝔽2)12yes—
358C22×C42−1+412yes—
359C2×C8—232no—
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—24no—
365C2×C8—24no—
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×C8—41no—
375C2×C8—41no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8#12014yes—
379C2×C8#12314yes—
380C4⋊C4—24no—
381C4⋊C4—24no—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—48no—
389C4⋊C4—48no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—88no—
399C4⋊C4—88no—
400C22⋊C4—28no—
401C22⋊C4—44no—
402C22⋊C4—48no—
403C22⋊C4—48no—
404C42—41no—
405C42—41no—
406C42—41no—
407C42—41no—
408C42—42no—
409C42—42no—
410C23—24no—
411C23—24no—
412C23—24no—
413C23—24no—
414C23—24no—
415C23—24no—
416C23—22no—
417C23—24no—
418C23—24no—
419C23—22no—
420C23—24no—
421C23—24no—
422C23—24no—
423C23—24no—
424C23—22no—
425C23—22no—
426C23#39814yes—
427C23#46514yes—
428C23#46814yes—
429C23#47514yes—
430C23#47614yes—
431C23#47814yes—
432C23#47914yes—
433C23C2×#12111yescharacteristic
434C23C2×C22⋊Q811yescharacteristic
435C23C2×C22⋊Q811yescharacteristic
436C23D4212yes—
437C23#54712yes—
438C23C22×SD1611yescharacteristic
439C23C2×#13511yescharacteristic
440C23C2×AGL(1,ℤ/8)11yescharacteristic
441C23C2×#13711yescharacteristic
442C23#57712yes—
443C23#57712yes—
444C2×C4—22no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—28no—
452C2×C4—28no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—42no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—42no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—44no—
486C2×C4—816no—
487C2×C4—816no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4#39814yes—
505C2×C4C2×#12011yescharacteristic
506C2×C4C2×#12311yescharacteristic
507C2×C4C2×#12311yescharacteristic
508C2×C4#52512yes—
509C2×C4#52512yes—
510C2×C4C2×#12511yescharacteristic
511C2×C4C2×#12511yescharacteristic
512C2×C4C2×C4⋊Q811yescharacteristic
513C2×C4#53511yescharacteristic
514C2×C4#53611yescharacteristic
515C2×C4#53912yes—
516C2×C4#53912yes—
517C2×C4#54612yes—
518C2×C4D4×Q812yes—
519C2×C4D4×Q812yes—
520C2×C4#55212yes—
521C2×C4#55412yes—
522C2×C4#55412yes—
523C8—24no—
524C8—24no—
525C8—81no—
526C8—81no—
527C22—24no—
528C22—24no—
529C22—24no—
530C22—24no—
531C22—24no—
532C22—24no—
533C22—28no—
534C22—28no—
535C22—28no—
536C22—28no—
537C22#208818yescomplemented
538C22C2×#39811yescharacteristic
539C22#240811yescharacteristic
540C22#244912yes—
541C22#244912yes—
542C22#248611yescharacteristic
543C22C2×#46511yescharacteristic
544C22C2×#46811yescharacteristic
545C22#305711yescharacteristic
546C22#305711yescharacteristic
547C22C2×#47611yescharacteristic
548C22C2×#47811yescharacteristic
549C22C2×#47511yescharacteristic
550C22C2×#47911yescharacteristic
551C22#308412yes—
552C22#308412yes—
553C22#326512yes—
554C22#326612yes—
555C22D4×SD1612yes—
556C22#328212yes—
557C22#333512yes—
558C22#333712yes—
559C22#334412yes—
560C22#334512yes—
561C4—24no—
562C4—42no—
563C4—42no—
564C4—81no—
565C4—84no—
566C4—81no—
567C4—84no—
568C4—84no—
569C4—84no—
570C4#243411yescharacteristic
571C4#247312yes—
572C4#247312yes—
573C4#250211yescharacteristic
574C4#262311yescharacteristic
575C4#284911yescharacteristic
576C2—28no—
577C2#1478611yescharacteristic
578C2C2×#208811yescharacteristic
579C2#2366811yescharacteristic
580C2#2426411yescharacteristic
581C2#2426911yescharacteristic
582C2#2633812yes—
583C2#2634012yes—
584C2#2634612yes—
585C2#2634812yes—
586C2#2765811yescharacteristic
587C2#2766311yescharacteristic
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×#2088C2semidirect
6. #11378C2semidirect
7. #11327C2semidirect
8. #11231C2semidirect
537. C22#2088semidirect

Maximal subgroups

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

subgroupindex
C23×#1062
C2×#20882
C2×#18502
#141302
#113782
#113272
#112312
#107302
#107212