Finite Group Databasebeta

← #399197

Subgroups — #399197

Order 512.

1712 conjugacy classes of subgroups.

The subgroup lattice has 555 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#107C211yesmaximal, characteristic
2C2×#2097C212yesmaximal, complemented
3C2×#1892C211yesmaximal, characteristic
4C2×#1872C211yesmaximal, characteristic
5C2×#1853C211yesmaximal, characteristic
6#14139C211yesmaximal, characteristic
7#13881C211yesmaximal, characteristic
8#11379C212yesmaximal, complemented
9#11325C212yesmaximal, complemented
10#11124C212yesmaximal, complemented
11#10714C211yesmaximal, characteristic
12C24×C8C2211yescharacteristic
13C22×C22⋊Q8C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C22×#107—28no—
17C22×#107—24no—
18C22×#107C2212yes—
19C22×#107C2212yes—
20C22×#106C2212yes—
21C22×#103C2211yescharacteristic
22C22×#103C2211yescharacteristic
23C22×#103C2212yes—
24C22×C22⋊C8C2211yescharacteristic
25C2×#391C2212yes—
26C2×#382C2212yes—
27C2×#380C2211yescharacteristic
28C2×C4×C22⋊C4C2211yescharacteristic
29#2097—28no—
30#1892—24no—
31#1872—24no—
32#1853—24no—
33C2×#340C2211yescharacteristic
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2211yescharacteristic
37C2×#336C2212yes—
38C2×#336C2212yes—
39#1442—22no—
40#1441—22no—
41C4⋊#95C2212yes—
42C4⋊#95C2212yes—
43C22⋊#95—22no—
44C22⋊#95—22no—
45C23×Q8C2311yescharacteristic
46C24×C4—21no—
47C24×C4—21no—
48C24×C4C2311yescharacteristic
49C23×C8—22no—
50C23×C8—22no—
51C23×C8—24no—
52C23×C8—24no—
53C23×C8D414yes—
54C23×C8C2312yes—
55C23×C8C2311yescharacteristic
56C23×C8C2311yescharacteristic
57C2×C22⋊Q8—22no—
58C2×C22⋊Q8—22no—
59C2×C22⋊Q8—22no—
60C2×C22⋊Q8—22no—
61C2×C22⋊Q8—22no—
62C2×C22⋊Q8Q814yes—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—24no—
66C22×C4⋊C4—24no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—42no—
73C22×C4⋊C4—42no—
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2311yescharacteristic
79C22×C22⋊C4—21no—
80C22×C22⋊C4—21no—
81C22×C22⋊C4D412yes—
82C22×C42C2312yes—
83C2×#107—232no—
84C2×#107—24no—
85C2×#107—24no—
86C2×#107—24no—
87C2×#107—24no—
88C2×#107—24no—
89C2×#107D418yes—
90C2×#106—24no—
91C2×#106—24no—
92C2×#106—42no—
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—22no—
96C2×#103—22no—
97C2×#103—24no—
98C2×#103—24no—
99C2×#103—42no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8D414yes—
103#391—24no—
104#391—42no—
105#382—24no—
106#382—42no—
107#380—22no—
108#380—22no—
109C4×C22⋊C4—22no—
110C4×C22⋊C4—22no—
111C2×#95—24no—
112C2×#95—24no—
113C2×#95—21no—
114C2×#95—24no—
115C2×#95—24no—
116C2×#95D412yes—
117#340—24no—
118#340—24no—
119#340—22no—
120#340—22no—
121#336—24no—
122#336—24no—
123C25C2×D411yescharacteristic
124C22×Q8—21no—
125C22×Q8—21no—
126C22×Q8—21no—
127C22×Q8—22no—
128C22×Q8—21no—
129C22×Q8—21no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8C2×Q812yes—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—44no—
145C23×C4—44no—
146C23×C4—42no—
147C23×C4—42no—
148C23×C4—44no—
149C23×C4—44no—
150C23×C4—82no—
151C23×C4—82no—
152C23×C4C2×D414yes—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D412yes—
155C23×C4C2×D412yes—
156C23×C4C2×Q812yes—
157C23×C4#4112yes—
158C23×C4#4112yes—
159C23×C4#4112yes—
160C23×C4C2411yescharacteristic
161C22×C8—216no—
162C22×C8—216no—
163C22×C8—28no—
164C22×C8—22no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—22no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—28no—
171C22×C8—28no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8C2×D414yes—
179C22×C8C2×D414yes—
180C22×C8C2×D412yes—
181C22×C8C2×D412yes—
182C22×C8#4114yes—
183C22⋊Q8—28no—
184C22⋊Q8—28no—
185C22⋊Q8—44no—
186C22⋊Q8—44no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—28no—
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—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—48no—
214C2×C4⋊C4—48no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4C2×D414yes—
224C2×C4⋊C4C2×D414yes—
225C2×C4⋊C4C2×Q812yes—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—44no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240#107—216no—
241#107—216no—
242#106—48no—
243#103—48no—
244#103—44no—
245#103—44no—
246C22⋊C8—28no—
247#95—24no—
248#95—48no—
249#95—48no—
250#95—42no—
251#95—84no—
252#95—84no—
253C24—22no—
254C24—22no—
255C24—22no—
256C24—22no—
257C24C22⋊Q812yes—
258C24#12312yes—
259C24#12712yes—
260C24#13514yes—
261C24#13714yes—
262C24C22×D411yescharacteristic
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—24no—
266C2×Q8—24no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—41no—
272C2×Q8—41no—
273C2×Q8—42no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—42no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—48no—
309C22×C4—48no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—44no—
321C22×C4—44no—
322C22×C4—42no—
323C22×C4—44no—
324C22×C4—44no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—816no—
335C22×C4—816no—
336C22×C4—84no—
337C22×C4—84no—
338C22×C4—84no—
339C22×C4—84no—
340C22×C4C22⋊Q814yes—
341C22×C4C22⋊Q814yes—
342C22×C4C22⋊Q812yes—
343C22×C4#12314yes—
344C22×C4#12314yes—
345C22×C4#12312yes—
346C22×C4#12712yes—
347C22×C4C22×D411yescharacteristic
348C22×C4C22×D412yes—
349C22×C4C22×D411yescharacteristic
350C22×C4C22×Q811yescharacteristic
351C22×C4C2×#4111yescharacteristic
352C22×C4C2×#4111yescharacteristic
353C22×C4C2×#4111yescharacteristic
354C22×C4C2×#4112yes—
355C22×C4H(5,𝔽2)12yes—
356C22×C42−1+412yes—
357C2×C8—232no—
358C2×C8—22no—
359C2×C8—22no—
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—24no—
363C2×C8—24no—
364C2×C8—24no—
365C2×C8—24no—
366C2×C8—24no—
367C2×C8—24no—
368C2×C8—24no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8#12014yes—
374C2×C8#12314yes—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—48no—
382C4⋊C4—48no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—48no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C22⋊C4—48no—
397C22⋊C4—48no—
398C22⋊C4—84no—
399C42—42no—
400C42—42no—
401C42—42no—
402C42—42no—
403C23—28no—
404C23—24no—
405C23—28no—
406C23—24no—
407C23—22no—
408C23—22no—
409C23—22no—
410C23—24no—
411C23—24no—
412C23—22no—
413C23—28no—
414C23—28no—
415C23#39914yes—
416C23#47918yes—
417C23#48518yes—
418C23#49514yes—
419C23#49714yes—
420C23C2×C22⋊Q811yescharacteristic
421C23C2×#12311yescharacteristic
422C23C2×#12711yescharacteristic
423C23D4212yes—
424C23#54712yes—
425C23C2×#13512yes—
426C23C2×#13712yes—
427C23#57612yes—
428C23#57812yes—
429Q8—42no—
430Q8—44no—
431Q8—42no—
432Q8—81no—
433Q8—81no—
434Q8—82no—
435C2×C4—22no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—22no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—28no—
443C2×C4—28no—
444C2×C4—24no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—48no—
459C2×C4—48no—
460C2×C4—816no—
461C2×C4—816no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—81no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—82no—
471C2×C4—81no—
472C2×C4—84no—
473C2×C4—84no—
474C2×C4—84no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—84no—
482C2×C4#39914yes—
483C2×C4C2×#12011yescharacteristic
484C2×C4C2×C22⋊Q811yescharacteristic
485C2×C4C2×C22⋊Q811yescharacteristic
486C2×C4C2×#12311yescharacteristic
487C2×C4C2×#12311yescharacteristic
488C2×C4C2×#12311yescharacteristic
489C2×C4#52512yes—
490C2×C4#52512yes—
491C2×C4#53511yescharacteristic
492C2×C4#53611yescharacteristic
493C2×C4#53912yes—
494C2×C4#53912yes—
495C2×C4D4212yes—
496C2×C4#54712yes—
497C2×C4#54812yes—
498C2×C4D4×Q812yes—
499C2×C4#55312yes—
500C2×C4#55412yes—
501C8—24no—
502C8—24no—
503C8—44no—
504C22—216no—
505C22—24no—
506C22—24no—
507C22—24no—
508C22—24no—
509C22—24no—
510C22—24no—
511C22—216no—
512C22#209718yescomplemented
513C22C2×#39911yescharacteristic
514C22#243311yescharacteristic
515C22#244312yes—
516C22#244912yes—
517C22#250111yescharacteristic
518C22C2×#47912yes—
519C22#308312yes—
520C22C2×#48512yes—
521C22#309812yes—
522C22C2×#49511yescharacteristic
523C22C2×#49711yescharacteristic
524C22#315511yescharacteristic
525C22#315611yescharacteristic
526C22#327014yes—
527C22#328014yes—
528C22#333614yes—
529C22#334614yes—
530C4—24no—
531C4—84no—
532C4—84no—
533C4—84no—
534C4—82no—
535C4—82no—
536C4—161no—
537C4—161no—
538C4#240911yescharacteristic
539C4#244312yes—
540C4#244912yes—
541C4#248711yescharacteristic
542C4#262311yescharacteristic
543C4#284911yescharacteristic
544C2—28no—
545C2#1477711yescharacteristic
546C2C2×#209711yescharacteristic
547C2#2373711yescharacteristic
548C2#2578111yescharacteristic
549C2#2578911yescharacteristic
550C2#2613414yes—
551C2#2634414yes—
552C2#2793111yescharacteristic
553C2#2793811yescharacteristic
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×#2097C2semidirect
8. #11379C2semidirect
9. #11325C2semidirect
10. #11124C2semidirect
512. C22#2097semidirect

Maximal subgroups

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

subgroupindex
C23×#1072
C2×#20972
C2×#18922
C2×#18722
C2×#18532
#141392
#138812
#113792
#113252
#111242
#107142