Finite Group Databasebeta

← #399614

Subgroups — #399614

Order 512.

1794 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×C4⋊C4C211yesmaximal, characteristic
2C2×#2083C211yesmaximal, characteristic, complemented
3C2×#1862C212yesmaximal
4C2×#1862C211yesmaximal, characteristic
5#15149C211yesmaximal, characteristic, complemented
6#11373C211yesmaximal, characteristic, complemented
7#11346C211yesmaximal, characteristic, complemented
8#11328C212yesmaximal, complemented
9#11219C212yesmaximal, complemented
10#10788C212yesmaximal
11#10788C211yesmaximal, characteristic
12C22×C4×D4C2211yescharacteristic
13C22×C4×D4C2212yes—
14C23×C4⋊C4—21no—
15C22×#107C2211yescharacteristic
16C22×#107C2212yes—
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#2083—24no—
31#1862—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—44no—
69C22×C4⋊C4—44no—
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2311yescharacteristic
77C22×C22⋊C4—21no—
78C22×C22⋊C4—22no—
79C22×C42C2312yes—
80C22×C42C2311yescharacteristic
81C2×#107—24no—
82C2×#107—24no—
83C2×#107—22no—
84C2×#107—24no—
85C2×#107—42no—
86C2×#102—22no—
87C2×#102—24no—
88C2×#102—22no—
89C2×#102—24no—
90C2×#102—22no—
91C2×#102—24no—
92C2×#102D414yes—
93C2×#102Q818yes—
94#398—44no—
95#389—24no—
96#389—24no—
97#389—24no—
98#389—22no—
99#389—22no—
100#389—42no—
101#385—216no—
102#384—22no—
103#384D414yes—
104#380—28no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C2×#95—22no—
108C2×#95—21no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95D412yes—
115#340—24no—
116#340—24no—
117#340—22no—
118#340—24no—
119#340—42no—
120#336—24no—
121#336—22no—
122#336—22no—
123#336—42no—
124C25—21no—
125C22×D4—21no—
126C22×D4—22no—
127C22×D4—21no—
128C22×D4—21no—
129C22×D4—22no—
130C22×D4C2×D412yes—
131C22×D4C2×D412yes—
132C22×D4C2×Q814yes—
133C22×D4#4112yes—
134C22×D4#4114yes—
135C23×C4—28no—
136C23×C4—21no—
137C23×C4—24no—
138C23×C4—22no—
139C23×C4—21no—
140C23×C4—22no—
141C23×C4—28no—
142C23×C4—21no—
143C23×C4—44no—
144C23×C4—44no—
145C23×C4—44no—
146C23×C4—44no—
147C23×C4—42no—
148C23×C4—42no—
149C23×C4—44no—
150C23×C4—82no—
151C23×C4—82no—
152C23×C4C2×D412yes—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D412yes—
155C23×C4C2×D411yescharacteristic
156C23×C4C2411yescharacteristic
157C22×C8—22no—
158C22×C8—21no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—21no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—21no—
168C22×C8—22no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—22no—
172C22×C8C2×D412yes—
173C22×C8C2×Q814yes—
174C4×D4—216no—
175C4×D4—28no—
176C4×D4—28no—
177C4×D4—48no—
178C4×D4—48no—
179C4×D4—48no—
180C4×D4—48no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—432no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—81no—
220C2×C4⋊C4—81no—
221C2×C4⋊C4—82no—
222C2×C4⋊C4D814yes—
223C2×C4⋊C4C2×D414yes—
224C2×C4⋊C4C2×D412yes—
225C2×C4⋊C4C2×D412yes—
226C2×C4⋊C4C2×Q814yes—
227C2×C4⋊C4#4114yes—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42C2×D412yes—
242C2×C42C2×D412yes—
243#107—48no—
244#107—44no—
245#102—28no—
246#102—216no—
247#102—44no—
248#95—28no—
249#95—24no—
250#95—44no—
251#95—44no—
252#95—42no—
253#95—42no—
254#95—44no—
255#95—81no—
256C24—22no—
257C24—24no—
258C24—22no—
259C24—22no—
260C24—24no—
261C24—41no—
262C24—41no—
263C24—42no—
264C24—41no—
265C24—41no—
266C24—42no—
267C24C22×D411yescharacteristic
268C2×D4—22no—
269C2×D4—24no—
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—24no—
273C2×D4—24no—
274C2×D4—22no—
275C2×D4—22no—
276C2×D4—24no—
277C2×D4—24no—
278C2×D4—24no—
279C2×D4—24no—
280C2×D4—24no—
281C2×D4#12114yes—
282C2×D4C22⋊Q818yes—
283C2×D4#12314yes—
284C2×D4#12518yes—
285C2×D4C4⋊Q814yes—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—216no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—416no—
312C22×C4—416no—
313C22×C4—48no—
314C22×C4—416no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—42no—
318C22×C4—48no—
319C22×C4—416no—
320C22×C4—41no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—41no—
326C22×C4—42no—
327C22×C4—41no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—41no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—41no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—41no—
337C22×C4—42no—
338C22×C4—41no—
339C22×C4—42no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—42no—
343C22×C4—88no—
344C22×C4—88no—
345C22×C4—88no—
346C22×C4—88no—
347C22×C4—88no—
348C22×C4#12012yes—
349C22×C4#12012yes—
350C22×C4C2×D814yes—
351C22×C4C2×D812yes—
352C22×C4AGL(1,ℤ/8)12yes—
353C22×C4C22×D411yescharacteristic
354C22×C4C22×D411yescharacteristic
355C22×C4C22×D411yescharacteristic
356C22×C4C22×Q812yes—
357C22×C4C2×#4111yescharacteristic
358C22×C4C2×#4112yes—
359C22×C4C2×#4112yes—
360C22×C4H(5,𝔽2)11yescharacteristic
361C22×C4H(5,𝔽2)11yescharacteristic
362C22×C42−1+412yes—
363C2×C8—22no—
364C2×C8—24no—
365C2×C8—24no—
366C2×C8—24no—
367C2×C8—24no—
368C2×C8—41no—
369C2×C8—42no—
370C2×C8—41no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—42no—
379C2×C8—42no—
380C4⋊C4—22no—
381C4⋊C4—24no—
382C4⋊C4—28no—
383C4⋊C4—22no—
384C4⋊C4—24no—
385C4⋊C4—22no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—24no—
389C4⋊C4—44no—
390C4⋊C4—416no—
391C4⋊C4—42no—
392C4⋊C4—41no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—41no—
396C4⋊C4—42no—
397C4⋊C4—48no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—42no—
401C4⋊C4—84no—
402C4⋊C4—88no—
403C4⋊C4—84no—
404C4⋊C4#12314yes—
405C22⋊C4—216no—
406C22⋊C4—416no—
407C42—22no—
408C42—22no—
409C42—24no—
410C42—42no—
411C42—42no—
412C42—42no—
413C42—42no—
414C23—24no—
415C23—24no—
416C23—24no—
417C23—28no—
418C23—28no—
419C23—44no—
420C23—44no—
421C23—44no—
422C23—44no—
423C23—44no—
424C23—44no—
425C23—44no—
426C23—44no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—44no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—44no—
436C23—42no—
437C23C2×#12011yescharacteristic
438C23#52512yes—
439C23D4211yescharacteristic
440C23#54611yescharacteristic
441C23D4×Q812yes—
442C23C22×D811yescharacteristic
443C23C2×AGL(1,ℤ/8)11yescharacteristic
444C23#57512yes—
445C23#57611yescharacteristic
446C23#57711yescharacteristic
447C23#57812yes—
448D4—24no—
449D4—28no—
450D4—28no—
451D4—24no—
452D4—24no—
453D4#39818yescomplemented
454C2×C4—24no—
455C2×C4—22no—
456C2×C4—24no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—432no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—44no—
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—44no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—832no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—81no—
502C2×C4—82no—
503C2×C4—81no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—81no—
507C2×C4—82no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4#44714yes—
511C2×C4C2×#12111yescharacteristic
512C2×C4C2×C22⋊Q812yes—
513C2×C4C2×#12311yescharacteristic
514C2×C4C2×#12311yescharacteristic
515C2×C4#52512yes—
516C2×C4C2×#12512yes—
517C2×C4C2×C4⋊Q811yescharacteristic
518C2×C4#53511yescharacteristic
519C2×C4#53611yescharacteristic
520C2×C4#53912yes—
521C2×C4#53912yes—
522C2×C4D4211yescharacteristic
523C2×C4#54611yescharacteristic
524C2×C4#54712yes—
525C2×C4D4×Q812yes—
526C2×C4#55212yes—
527C2×C4#55412yes—
528C2×C4#55412yes—
529C8—44no—
530C8—42no—
531C8—81no—
532C22—28no—
533C22—48no—
534C22—48no—
535C22—48no—
536C22—44no—
537C22—48no—
538C22—44no—
539C22—44no—
540C22—44no—
541C22—44no—
542C22—44no—
543C22#240811yescharacteristic
544C22#243411yescharacteristic
545C22#244912yes—
546C22#247312yes—
547C22#284911yescharacteristic
548C22C2×#44711yescharacteristic
549C22#300511yescharacteristic
550C22#301111yescharacteristic
551C22#301211yescharacteristic
552C22#310012yes—
553C22#311412yes—
554C22#312012yes—
555C22#312412yes—
556C22#326912yes—
557C22#327211yescharacteristic
558C22D4×D811yescharacteristic
559C22#329411yescharacteristic
560C22#331512yes—
561C22#332511yescharacteristic
562C22Q8×D812yes—
563C22#338612yes—
564C22#338812yes—
565C22#338912yes—
566C4—24no—
567C4—44no—
568C4—44no—
569C4—42no—
570C4—84no—
571C4—82no—
572C4—84no—
573C4—82no—
574C4—161no—
575C4C2×#39811yescharacteristic
576C4#244912yes—
577C4#247312yes—
578C4#248611yescharacteristic
579C4#250211yescharacteristic
580C4#262311yescharacteristic
581C2—48no—
582C2#1478611yescharacteristic
583C2#2423611yescharacteristic
584C2#2425311yescharacteristic
585C2#2583811yescharacteristic
586C2#2584511yescharacteristic
587C2#2636812yes—
588C2#2639812yes—
589C2#2729412yes—
590C2#2731012yes—
591C2#3300211yescharacteristic
592C2#3301711yescharacteristic
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×#2083C2semidirect
5. #15149C2semidirect
6. #11373C2semidirect
7. #11346C2semidirect
8. #11328C2semidirect
9. #11219C2semidirect
453. 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×#20832
C2×#18622
#151492
#113732
#113462
#113282
#112192
#107882