Finite Group Databasebeta

← #399610

Subgroups — #399610

Order 512.

1794 conjugacy classes of subgroups.

The subgroup lattice has 591 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×C4⋊C4C211yesmaximal, characteristic
2C2×#2084C211yesmaximal, characteristic, complemented
3C2×#1864C212yesmaximal
4C2×#1862C211yesmaximal, characteristic
5#15149C211yesmaximal, characteristic, complemented
6#11371C211yesmaximal, characteristic, complemented
7#11346C211yesmaximal, characteristic, complemented
8#11327C212yesmaximal, complemented
9#11219C212yesmaximal, complemented
10#10788C212yesmaximal
11#10788C211yesmaximal, characteristic
12C22×C4×D4C2211yescharacteristic
13C22×C4×D4C2212yes—
14C23×C4⋊C4—21no—
15C22×#107C2211yescharacteristic
16C22×#106C2212yes—
17C22×#102C2211yescharacteristic
18C22×#102C2211yescharacteristic
19C22×#102C2212yes—
20C2×#398—21no—
21D4×C4⋊C4—24no—
22D4×C4⋊C4—24no—
23C2×#389C2211yescharacteristic
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#385—22no—
27C2×#384C2211yescharacteristic
28C2×#380—21no—
29C2×C4×C4⋊C4C2211yescharacteristic
30#2084—24no—
31#1864—28no—
32#1862—24no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2211yescharacteristic
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2211yescharacteristic
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41C2×#336C2212yes—
42#1442—22no—
43#1441—21no—
44C4⋊#95C2211yescharacteristic
45C4⋊#95C2212yes—
46C23×D4C2311yescharacteristic
47C24×C4—21no—
48C24×C4—22no—
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C23×C8C2312yes—
52C2×C4×D4—24no—
53C2×C4×D4—24no—
54C2×C4×D4—24no—
55C2×C4×D4—24no—
56C2×C4×D4—24no—
57C2×C4×D4—24no—
58C2×C4×D4—22no—
59C2×C4×D4—42no—
60C2×C4×D4—42no—
61C2×C4×D4D414yes—
62C22×C4⋊C4—28no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—48no—
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C22⋊C4—21no—
77C22×C22⋊C4—22no—
78C22×C42C2312yes—
79C22×C42C2311yescharacteristic
80C2×#107—22no—
81C2×#107—24no—
82C2×#106—24no—
83C2×#106—24no—
84C2×#106—42no—
85C2×#102—22no—
86C2×#102—24no—
87C2×#102—22no—
88C2×#102—24no—
89C2×#102—22no—
90C2×#102—24no—
91C2×#102D414yes—
92C2×#102Q818yes—
93#398—44no—
94#389—24no—
95#389—24no—
96#389—24no—
97#389—22no—
98#389—22no—
99#389—42no—
100#385—216no—
101#384—22no—
102#384D414yes—
103#380—28no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106C2×#95—22no—
107C2×#95—21no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95D412yes—
114#340—24no—
115#340—24no—
116#340—22no—
117#340—24no—
118#340—42no—
119#336—24no—
120#336—22no—
121#336—22no—
122#336—42no—
123C25—21no—
124C22×D4—21no—
125C22×D4—22no—
126C22×D4—21no—
127C22×D4—21no—
128C22×D4—22no—
129C22×D4C2×D412yes—
130C22×D4C2×D412yes—
131C22×D4C2×Q814yes—
132C22×D4#4112yes—
133C22×D4#4114yes—
134C23×C4—28no—
135C23×C4—21no—
136C23×C4—24no—
137C23×C4—22no—
138C23×C4—21no—
139C23×C4—22no—
140C23×C4—28no—
141C23×C4—21no—
142C23×C4—44no—
143C23×C4—44no—
144C23×C4—44no—
145C23×C4—44no—
146C23×C4—42no—
147C23×C4—42no—
148C23×C4—44no—
149C23×C4—82no—
150C23×C4—82no—
151C23×C4C2×D412yes—
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D412yes—
154C23×C4C2×D411yescharacteristic
155C23×C4C2411yescharacteristic
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—21no—
159C22×C8—22no—
160C22×C8—21no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—21no—
167C22×C8—22no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—22no—
171C22×C8C2×D412yes—
172C22×C8C2×Q814yes—
173C4×D4—216no—
174C4×D4—28no—
175C4×D4—28no—
176C4×D4—48no—
177C4×D4—48no—
178C4×D4—48no—
179C4×D4—48no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—432no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—82no—
219C2×C4⋊C4—82no—
220C2×C4⋊C4C2×D414yes—
221C2×C4⋊C4C2×D412yes—
222C2×C4⋊C4C2×D412yes—
223C2×C4⋊C4C2×Q814yes—
224C2×C4⋊C4#4114yes—
225C2×C22⋊C4—28no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42C2×D412yes—
239C2×C42C2×D412yes—
240#107—44no—
241#106—48no—
242#102—28no—
243#102—216no—
244#102—44no—
245#95—28no—
246#95—24no—
247#95—44no—
248#95—44no—
249#95—42no—
250#95—44no—
251#95—42no—
252#95—81no—
253C24—22no—
254C24—24no—
255C24—22no—
256C24—22no—
257C24—24no—
258C24—41no—
259C24—42no—
260C24—41no—
261C24—42no—
262C24—41no—
263C24—41no—
264C24C22×D411yescharacteristic
265C2×D4—22no—
266C2×D4—24no—
267C2×D4—22no—
268C2×D4—22no—
269C2×D4—24no—
270C2×D4—24no—
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—24no—
274C2×D4—24no—
275C2×D4—24no—
276C2×D4—24no—
277C2×D4—24no—
278C2×D4#12114yes—
279C2×D4C22⋊Q818yes—
280C2×D4#12314yes—
281C2×D4#12518yes—
282C2×D4C4⋊Q814yes—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—216no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—24no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—24no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—48no—
309C22×C4—416no—
310C22×C4—416no—
311C22×C4—416no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—48no—
316C22×C4—416no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—41no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—42no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—42no—
340C22×C4—88no—
341C22×C4—88no—
342C22×C4—88no—
343C22×C4—88no—
344C22×C4—88no—
345C22×C4#12012yes—
346C22×C4#12012yes—
347C22×C4#13512yes—
348C22×C4AGL(1,ℤ/8)14yes—
349C22×C4AGL(1,ℤ/8)12yes—
350C22×C4C22×D411yescharacteristic
351C22×C4C22×D411yescharacteristic
352C22×C4C22×D411yescharacteristic
353C22×C4C22×Q812yes—
354C22×C4C2×#4111yescharacteristic
355C22×C4C2×#4112yes—
356C22×C4C2×#4112yes—
357C22×C4H(5,𝔽2)11yescharacteristic
358C22×C4H(5,𝔽2)11yescharacteristic
359C22×C42−1+412yes—
360C2×C8—22no—
361C2×C8—24no—
362C2×C8—24no—
363C2×C8—24no—
364C2×C8—24no—
365C2×C8—41no—
366C2×C8—42no—
367C2×C8—41no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—42no—
376C2×C8—42no—
377C4⋊C4—28no—
378C4⋊C4—22no—
379C4⋊C4—24no—
380C4⋊C4—22no—
381C4⋊C4—24no—
382C4⋊C4—22no—
383C4⋊C4—24no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—48no—
387C4⋊C4—416no—
388C4⋊C4—42no—
389C4⋊C4—41no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—41no—
393C4⋊C4—42no—
394C4⋊C4—44no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—42no—
398C4⋊C4—84no—
399C4⋊C4—88no—
400C4⋊C4—84no—
401C4⋊C4#12314yes—
402C22⋊C4—216no—
403C22⋊C4—416no—
404C42—22no—
405C42—22no—
406C42—24no—
407C42—42no—
408C42—42no—
409C42—42no—
410C42—42no—
411C23—24no—
412C23—24no—
413C23—24no—
414C23—28no—
415C23—28no—
416C23—42no—
417C23—44no—
418C23—44no—
419C23—44no—
420C23—44no—
421C23—44no—
422C23—44no—
423C23—44no—
424C23—44no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—44no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—44no—
434C23C2×#12011yescharacteristic
435C23#52512yes—
436C23D4211yescharacteristic
437C23#54611yescharacteristic
438C23D4×Q812yes—
439C23C2×#13511yescharacteristic
440C23C2×AGL(1,ℤ/8)11yescharacteristic
441C23C2×AGL(1,ℤ/8)12yes—
442C23#57712yes—
443C23#57711yescharacteristic
444C23#57811yescharacteristic
445D4—24no—
446D4—28no—
447D4—28no—
448D4—24no—
449D4—24no—
450D4#39818yescomplemented
451C2×C4—22no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—22no—
461C2×C4—24no—
462C2×C4—432no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—832no—
490C2×C4—81no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—81no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—81no—
498C2×C4—82no—
499C2×C4—81no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—81no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—81no—
506C2×C4—81no—
507C2×C4#44914yes—
508C2×C4C2×#12111yescharacteristic
509C2×C4C2×C22⋊Q812yes—
510C2×C4C2×#12311yescharacteristic
511C2×C4C2×#12311yescharacteristic
512C2×C4#52512yes—
513C2×C4C2×#12512yes—
514C2×C4C2×C4⋊Q811yescharacteristic
515C2×C4#53511yescharacteristic
516C2×C4#53611yescharacteristic
517C2×C4#53912yes—
518C2×C4#53912yes—
519C2×C4D4211yescharacteristic
520C2×C4#54611yescharacteristic
521C2×C4#54712yes—
522C2×C4D4×Q812yes—
523C2×C4#55212yes—
524C2×C4#55412yes—
525C2×C4#55412yes—
526C8—44no—
527C8—42no—
528C8—81no—
529C22—28no—
530C22—48no—
531C22—48no—
532C22—48no—
533C22—44no—
534C22—48no—
535C22—44no—
536C22—44no—
537C22—44no—
538C22—44no—
539C22—44no—
540C22#240811yescharacteristic
541C22#243411yescharacteristic
542C22#244912yes—
543C22#247312yes—
544C22#284911yescharacteristic
545C22C2×#44911yescharacteristic
546C22#300311yescharacteristic
547C22#300911yescharacteristic
548C22#301311yescharacteristic
549C22#310512yes—
550C22#310912yes—
551C22#312012yes—
552C22#312512yes—
553C22#326512yes—
554C22#327211yescharacteristic
555C22#328311yescharacteristic
556C22#330311yescharacteristic
557C22#331012yes—
558C22#333011yescharacteristic
559C22#338112yes—
560C22#338412yes—
561C22#338412yes—
562C22#338612yes—
563C4—24no—
564C4—44no—
565C4—44no—
566C4—42no—
567C4—84no—
568C4—82no—
569C4—82no—
570C4—84no—
571C4—161no—
572C4C2×#39811yescharacteristic
573C4#244912yes—
574C4#247312yes—
575C4#248611yescharacteristic
576C4#250211yescharacteristic
577C4#262311yescharacteristic
578C2—48no—
579C2#1478611yescharacteristic
580C2#2423711yescharacteristic
581C2#2425211yescharacteristic
582C2#2583511yescharacteristic
583C2#2584411yescharacteristic
584C2#2637412yes—
585C2#2639212yes—
586C2#2729912yes—
587C2#2731512yes—
588C2#3300411yescharacteristic
589C2#3301611yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2084C2semidirect
5. #15149C2semidirect
6. #11371C2semidirect
7. #11346C2semidirect
8. #11327C2semidirect
9. #11219C2semidirect
450. D4#398semidirect

Maximal subgroups

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

subgroupindex
C2×D4×C4⋊C42
C2×#20842
C2×#18642
C2×#18622
#151492
#113712
#113462
#113272
#112192
#107882