Finite Group Databasebeta

← #393478

Subgroups — #393478

Order 512.

1632 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2274C211yesmaximal, characteristic
2C2×#1854C211yesmaximal, characteristic
3C2×#1824C211yesmaximal, characteristic
4C2×#1823C211yesmaximal, characteristic
5#14264C211yesmaximal, characteristic
6#10908C212yesmaximal, complemented
7#10907C212yesmaximal, complemented
8#10885C212yesmaximal, complemented
9#10884C212yesmaximal, complemented
10#10731C211yesmaximal, characteristic
11#10713C211yesmaximal, characteristic
12C22×#117C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×#107C2211yescharacteristic
16C22×#106C2211yescharacteristic
17C22×C22⋊C8C2211yescharacteristic
18C22×C22⋊C8C2211yescharacteristic
19C2×#389C2212yes—
20C2×#380C2211yescharacteristic
21C2×#380C2211yescharacteristic
22#2274—24no—
23#2274—24no—
24C2×C4×C4⋊C4—21no—
25C2×C4×C4⋊C4—21no—
26C2×C4×C4⋊C4C2212yes—
27C2×C4×C4⋊C4C2212yes—
28C2×C4×C22⋊C4—22no—
29#1854—24no—
30#1824—24no—
31#1823C418yescomplemented
32C2×#340C2211yescharacteristic
33C2×#340C2211yescharacteristic
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2212yes—
41C2×#336C2212yes—
42#1444—21no—
43#1442—21no—
44#1439—22no—
45C4⋊#95C2212yes—
46C24×C4—21no—
47C24×C4C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C2×#117—22no—
52C2×#117—22no—
53C2×#117—22no—
54C2×#117—24no—
55C2×#117—24no—
56C2×#117—24no—
57C2×#117—24no—
58C2×#117D414yes—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4C2×C414yes—
65C22×C4⋊C4D411yescharacteristic
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4D414yes—
68C22×C4⋊C4Q814yes—
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C22⋊C4—21no—
76C22×C22⋊C4—21no—
77C22×C22⋊C4—21no—
78C22×C22⋊C4D412yes—
79C22×C42—21no—
80C22×C42—22no—
81C22×C42—22no—
82C22×C42C2312yes—
83C2×#107—22no—
84C2×#107—24no—
85C2×#106—22no—
86C2×#106—24no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8C2×C414yes—
93#389—24no—
94#389—24no—
95#380—22no—
96#380—22no—
97#380—22no—
98#380C2×C414yes—
99C4×C4⋊C4—28no—
100C4×C4⋊C4—28no—
101C4×C4⋊C4—24no—
102C4×C4⋊C4—24no—
103C4×C4⋊C4—24no—
104C4×C4⋊C4—24no—
105C4×C22⋊C4—216no—
106C2×#95—21no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—21no—
111C2×#95—22no—
112C2×#95—22no—
113#340—24no—
114#340—24no—
115#340—22no—
116#340—22no—
117#340—24no—
118#340—24no—
119#340C2×C418yes—
120#340C2×C418yes—
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—21no—
130C23×C4—22no—
131C23×C4—24no—
132C23×C4—24no—
133C23×C4—24no—
134C23×C4—22no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—81no—
138C23×C4—81no—
139C23×C4—81no—
140C23×C4—81no—
141C23×C4C22×C412yes—
142C23×C4C2×D412yes—
143C23×C4C2×D412yes—
144C23×C4C2×D412yes—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D412yes—
147C23×C4C2×Q812yes—
148C23×C4#4112yes—
149C23×C4#4112yes—
150C23×C4C2411yescharacteristic
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8C22×C412yes—
167C22×C8C22×C412yes—
168#117—216no—
169#117—216no—
170#117—28no—
171#117—28no—
172#117—28no—
173#117—28no—
174C2×C4⋊C4—28no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
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—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—82no—
206C2×C4⋊C4—82no—
207C2×C4⋊C4C4⋊C4116yescomplemented
208C2×C4⋊C4C22×C414yes—
209C2×C4⋊C4C22×C414yes—
210C2×C4⋊C4C2×D414yes—
211C2×C4⋊C4C2×D412yes—
212C2×C4⋊C4C2×D412yes—
213C2×C4⋊C4C2×Q814yes—
214C2×C4⋊C4#4114yes—
215C2×C4⋊C4#4114yes—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—81no—
224C2×C22⋊C4—81no—
225C2×C22⋊C4—82no—
226C2×C42—28no—
227C2×C42—24no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—28no—
233C2×C42—28no—
234C2×C42—82no—
235C2×C42—82no—
236C2×C42—82no—
237C2×C42—82no—
238C2×C42#4114yes—
239C2×C42#4114yes—
240#107—44no—
241#106—44no—
242C22⋊C8—28no—
243C22⋊C8—44no—
244#95—28no—
245#95—28no—
246#95—24no—
247#95—42no—
248#95—44no—
249#95—82no—
250#95—82no—
251C24—22no—
252C24—22no—
253C24—21no—
254C24—21no—
255C24—22no—
256C24—21no—
257C24—21no—
258C24—22no—
259C24C4×D412yes—
260C24#12012yes—
261C24C22⋊Q812yes—
262C24C22×D411yescharacteristic
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—216no—
299C22×C4—24no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—42no—
307C22×C4—42no—
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—41no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—84no—
327C22×C4—84no—
328C22×C4—84no—
329C22×C4—84no—
330C22×C4—84no—
331C22×C4—84no—
332C22×C4C2×C4⋊C414yes—
333C22×C4C4×D414yes—
334C22×C4C4×D412yes—
335C22×C4#12012yes—
336C22×C4C22⋊Q814yes—
337C22×C4C22⋊Q812yes—
338C22×C4#12314yes—
339C22×C4#12314yes—
340C22×C4#12314yes—
341C22×C4C23×C411yescharacteristic
342C22×C4C22×D411yescharacteristic
343C22×C4C22×D411yescharacteristic
344C22×C4C22×Q811yescharacteristic
345C22×C4C2×#4111yescharacteristic
346C22×C4C2×#4112yes—
347C22×C4C2×#4112yes—
348C22×C4C2×#4112yes—
349C22×C4C2×#4112yes—
350C22×C4C2×#4111yescharacteristic
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—24no—
354C2×C8—24no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C4⋊C4—28no—
369C4⋊C4—28no—
370C4⋊C4—24no—
371C4⋊C4—24no—
372C4⋊C4—24no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—216no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—88no—
389C4⋊C4C2×C4⋊C418yes—
390C4⋊C4#12314yes—
391C4⋊C4#12314yes—
392C22⋊C4—216no—
393C22⋊C4—216no—
394C22⋊C4—84no—
395C22⋊C4—84no—
396C42—216no—
397C42—24no—
398C42—24no—
399C42—24no—
400C42—24no—
401C42—88no—
402C42—88no—
403C23—22no—
404C23—22no—
405C23—24no—
406C23—22no—
407C23—22no—
408C23—24no—
409C23—24no—
410C23—22no—
411C23—22no—
412C23—22no—
413C23—22no—
414C23—22no—
415C23—22no—
416C23—24no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23#38514yes—
426C23C2×C4×D411yescharacteristic
427C23C2×#12011yescharacteristic
428C23C2×C22⋊Q811yescharacteristic
429C23C2×#12312yes—
430C23#52512yes—
431C23#57611yescharacteristic
432C23#57611yescharacteristic
433C23#57711yescharacteristic
434C23#57711yescharacteristic
435C23#57712yes—
436C23#57812yes—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—28no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—28no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—28no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
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—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—816no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4#38514yes—
497C2×C4C22×C4⋊C411yescharacteristic
498C2×C4C2×C4×D411yescharacteristic
499C2×C4C4×#4112yes—
500C2×C4C4×#4112yes—
501C2×C4C2×C22⋊Q811yescharacteristic
502C2×C4C2×#12311yescharacteristic
503C2×C4C2×#12311yescharacteristic
504C2×C4C2×#12311yescharacteristic
505C2×C4#52512yes—
506C2×C4#52512yes—
507C2×C4#52511yescharacteristic
508C2×C4#52511yescharacteristic
509C2×C4#52912yes—
510C2×C4#52912yes—
511C2×C4#52912yes—
512C2×C4#52912yes—
513C2×C4#53312yes—
514C2×C4#53312yes—
515C8—42no—
516C8—42no—
517C8—82no—
518C22—24no—
519C22—24no—
520C22—24no—
521C22—44no—
522C22—44no—
523C22—44no—
524C22—44no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22C2×#38511yescharacteristic
530C22C4×#12312yes—
531C22#284911yescharacteristic
532C22#284911yescharacteristic
533C22#285512yes—
534C22#294712yes—
535C22#294811yescharacteristic
536C22#294911yescharacteristic
537C22#301011yescharacteristic
538C22#301111yescharacteristic
539C22#301211yescharacteristic
540C22#301211yescharacteristic
541C22#308111yescharacteristic
542C22#308111yescharacteristic
543C22#308212yes—
544C22#309612yes—
545C22#309712yes—
546C22#309812yes—
547C22#309912yes—
548C22#311912yes—
549C22#312012yes—
550C22#312412yes—
551C22#312512yes—
552C4—24no—
553C4—48no—
554C4—44no—
555C4—44no—
556C4—44no—
557C4—44no—
558C4—84no—
559C4—161no—
560C4—161no—
561C4#229612yes—
562C4C4×#12311yescharacteristic
563C4C4×#12311yescharacteristic
564C4#285312yes—
565C4#285511yescharacteristic
566C4#285511yescharacteristic
567C2—44no—
568C2#1470511yescharacteristic
569C2#2087711yescharacteristic
570C2#2087911yescharacteristic
571C2#2158612yes—
572C2#2158712yes—
573C2#3298711yescharacteristic
574C2#3298911yescharacteristic
575C2#3301311yescharacteristic
576C2#3301711yescharacteristic
577C2#3315812yes—
578C2#3315912yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #10908C2semidirect
7. #10907C2semidirect
8. #10885C2semidirect
9. #10884C2semidirect
31. #1823C4semidirect
207. C2×C4⋊C4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22742
C2×#18542
C2×#18242
C2×#18232
#142642
#109082
#109072
#108852
#108842
#107312
#107132