Finite Group Databasebeta

← #464459

Subgroups — #464459

Order 512.

1336 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#498C211yesmaximal, characteristic, complemented
2C2×#2077C212yesmaximal
3C2×#1923C211yesmaximal, characteristic
4#15125C211yesmaximal, characteristic
5#15125C211yesmaximal, characteristic
6#11384C211yesmaximal, characteristic, complemented
7#11384C211yesmaximal, characteristic, complemented
8#11378C212yesmaximal, complemented
9#11231C212yesmaximal, complemented
10#11095C212yesmaximal
11#10820C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C2×#498—28no—
15C2×#498—24no—
16C22×#106C2212yes—
17C22×#106C2212yes—
18C22×C4⋊C8C2211yescharacteristic
19C22×C22⋊C8C2211yescharacteristic
20C22×C22⋊C8C2211yescharacteristic
21C22×C4×C8C2211yescharacteristic
22C2×#393—21no—
23C2×#393—21no—
24C2×#380C2212yes—
25C2×#380C2212yes—
26C2×C4×C22⋊C4C2211yescharacteristic
27#2077—28no—
28#1923—24no—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2211yescharacteristic
36C2×#336C2211yescharacteristic
37#1446—21no—
38#1446—21no—
39#1442—21no—
40#1442—21no—
41#1439—21no—
42#1439—21no—
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2211yescharacteristic
45C4⋊#95C2212yes—
46C4⋊#95C2212yes—
47C23×Q8—21no—
48C23×Q8—21no—
49C24×C4C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C2×C4⋊Q8—24no—
55C2×C4⋊Q8—24no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—42no—
61C2×C4⋊Q8—42no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2312yes—
68C22×C22⋊C4—22no—
69C22×C22⋊C4—22no—
70C22×C22⋊C4D411yescharacteristic
71C22×C22⋊C4D411yescharacteristic
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74#498—232no—
75C2×#106—24no—
76C2×#106—24no—
77C2×#106—24no—
78C2×#106—24no—
79C2×#106—24no—
80C2×#106D418yes—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—24no—
83C2×C22⋊C8—24no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—24no—
87C2×C4×C8—24no—
88C2×C4×C8D414yes—
89#393—44no—
90#393—44no—
91#380—24no—
92#380—24no—
93#380—42no—
94#380—42no—
95C4×C22⋊C4—22no—
96C4×C22⋊C4—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95—21no—
102C2×#95—21no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95D411yescharacteristic
108C2×#95D411yescharacteristic
109#340—24no—
110#340—24no—
111#340—24no—
112#340—24no—
113#340—42no—
114#340—42no—
115#336—22no—
116#336—22no—
117C25C2×D411yescharacteristic
118C22×Q8—44no—
119C22×Q8—44no—
120C22×Q8—42no—
121C22×Q8—42no—
122C22×Q8—42no—
123C22×Q8—42no—
124C22×Q8—42no—
125C22×Q8—42no—
126C22×Q8—82no—
127C22×Q8—82no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4—21no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2411yescharacteristic
148C22×C8—22no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—22no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8C2×D412yes—
165C22×C8C2×D414yes—
166C22×C8C2×D412yes—
167C22×C8#4114yes—
168C4⋊Q8—28no—
169C4⋊Q8—28no—
170C4⋊Q8—48no—
171C4⋊Q8—48no—
172C4⋊Q8—44no—
173C4⋊Q8—44no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
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—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4C2×D414yes—
201C2×C4⋊C4C2×D414yes—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—44no—
208C2×C22⋊C4—44no—
209C2×C22⋊C4—44no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4—82no—
212C2×C22⋊C4—82no—
213C2×C22⋊C4SD1614yes—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42C2×D412yes—
221C2×C42#4112yes—
222#106—216no—
223#106—216no—
224C4⋊C8—44no—
225C22⋊C8—44no—
226C22⋊C8—44no—
227C4×C8—28no—
228#95—22no—
229#95—22no—
230#95—44no—
231#95—44no—
232#95—44no—
233#95—44no—
234#95—42no—
235#95—42no—
236#95—42no—
237#95—42no—
238#95—82no—
239#95—82no—
240C24—21no—
241C24—21no—
242C24—21no—
243C24—22no—
244C24—22no—
245C24—21no—
246C24—21no—
247C24—21no—
248C24—22no—
249C24—22no—
250C24C2×SD1612yes—
251C24C22×D411yescharacteristic
252C2×Q8—48no—
253C2×Q8—48no—
254C2×Q8—88no—
255C2×Q8—88no—
256C2×Q8—88no—
257C2×Q8—88no—
258C2×Q8—84no—
259C2×Q8—84no—
260C2×Q8—84no—
261C2×Q8—84no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—21no—
267C22×C4—22no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—42no—
305C22×C4—42no—
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—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#12112yes—
327C22×C4#12112yes—
328C22×C4C2×SD1612yes—
329C22×C4C2×SD1612yes—
330C22×C4#13512yes—
331C22×C4AGL(1,ℤ/8)12yes—
332C22×C4#13712yes—
333C22×C4C22×D411yescharacteristic
334C22×C4C22×D412yes—
335C22×C4C2×#4112yes—
336C22×C4C2×#4111yescharacteristic
337C22×C4H(5,𝔽2)12yes—
338C22×C42−1+411yescharacteristic
339C22×C42−1+411yescharacteristic
340C22×C42−1+412yes—
341C22×C42−1+411yescharacteristic
342C22×C42−1+411yescharacteristic
343C2×C8—22no—
344C2×C8—22no—
345C2×C8—22no—
346C2×C8—24no—
347C2×C8—24no—
348C2×C8—22no—
349C2×C8—24no—
350C2×C8—24no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8#12014yes—
362C2×C8#12414yes—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—24no—
366C4⋊C4—24no—
367C4⋊C4—48no—
368C4⋊C4—48no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C22⋊C4—28no—
382C22⋊C4—44no—
383C22⋊C4—88no—
384C22⋊C4—88no—
385C42—22no—
386C42—22no—
387C42—24no—
388C42—41no—
389C42—41no—
390C42—42no—
391C23—24no—
392C23—22no—
393C23—22no—
394C23—24no—
395C23—22no—
396C23—42no—
397C23—42no—
398C23—42no—
399C23—41no—
400C23—41no—
401C23—42no—
402C23—42no—
403C23—41no—
404C23—41no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—42no—
410C23—41no—
411C23—41no—
412C23—41no—
413C23—41no—
414C23C2×#12111yescharacteristic
415C23#53611yescharacteristic
416C23#53611yescharacteristic
417C23D4212yes—
418C23#54712yes—
419C23C22×SD1611yescharacteristic
420C23C2×#13511yescharacteristic
421C23C2×AGL(1,ℤ/8)11yescharacteristic
422C23C2×#13711yescharacteristic
423C23#57712yes—
424C23#57712yes—
425Q8—816no—
426Q8—816no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—24no—
436C2×C4—22no—
437C2×C4—42no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—41no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4#46514yes—
485C2×C4#46814yes—
486C2×C4C2×#12011yescharacteristic
487C2×C4C2×#12411yescharacteristic
488C2×C4#53511yescharacteristic
489C2×C4#53611yescharacteristic
490C2×C4#53611yescharacteristic
491C2×C4#53611yescharacteristic
492C2×C4#53612yes—
493C2×C4#53612yes—
494C2×C4#54111yescharacteristic
495C2×C4#54111yescharacteristic
496C2×C4#54212yes—
497C2×C4#54212yes—
498C2×C4#54612yes—
499C2×C4#54812yes—
500C2×C4#54812yes—
501C2×C4#55612yes—
502C2×C4#56312yes—
503C2×C4#56312yes—
504C8—24no—
505C8—24no—
506C8—81no—
507C8—81no—
508C22—42no—
509C22—44no—
510C22—44no—
511C22—42no—
512C22—42no—
513C22—81no—
514C22—82no—
515C22—82no—
516C22—81no—
517C22—81no—
518C22—81no—
519C22#240811yescharacteristic
520C22#254711yescharacteristic
521C22#262111yescharacteristic
522C22#267412yes—
523C22#267412yes—
524C22C2×#46511yescharacteristic
525C22C2×#46811yescharacteristic
526C22#305711yescharacteristic
527C22#305711yescharacteristic
528C22#322711yescharacteristic
529C22#322811yescharacteristic
530C22#322911yescharacteristic
531C22#323111yescharacteristic
532C22#324412yes—
533C22#324412yes—
534C22#326512yes—
535C22#326612yes—
536C22D4×SD1612yes—
537C22#328212yes—
538C22#333512yes—
539C22#334012yes—
540C22#335412yes—
541C22#335612yes—
542C4—24no—
543C4—24no—
544C4—42no—
545C4—81no—
546C4—81no—
547C4—84no—
548C4—84no—
549C4—162no—
550C4—162no—
551C4#243611yescharacteristic
552C4#261411yescharacteristic
553C4#264011yescharacteristic
554C4#271612yes—
555C4#271612yes—
556C4#282111yescharacteristic
557C2—82no—
558C2#1650611yescharacteristic
559C2#2426411yescharacteristic
560C2#2426911yescharacteristic
561C2#2870611yescharacteristic
562C2#2871311yescharacteristic
563C2#2987711yescharacteristic
564C2#2988111yescharacteristic
565C2#3077412yes—
566C2#3077512yes—
567C2#3077712yes—
568C2#3078012yes—
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
1. C22×#498C2semidirect
6. #11384C2semidirect
7. #11384C2semidirect
8. #11378C2semidirect
9. #11231C2semidirect
11. #10820C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4982
C2×#20772
C2×#19232
#151252
#113842
#113782
#112312
#110952
#108202