Finite Group Databasebeta

← #467100

Subgroups — #467100

Order 512.

1330 conjugacy classes of subgroups.

The subgroup lattice has 565 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2861C211yesmaximal, characteristic
2C2×#2026C211yesmaximal, characteristic, complemented
3C2×#2025C211yesmaximal, characteristic
4#14709C211yesmaximal, characteristic, complemented
5#11385C212yesmaximal, complemented
6#11190C212yesmaximal, complemented
7#11189C212yesmaximal
8#11052C211yesmaximal, characteristic, complemented
9#11051C211yesmaximal, characteristic
10#11009C212yesmaximal
11#10795C211yesmaximal, characteristic, complemented
12C22×#125C2211yescharacteristic
13C22×#125C2211yescharacteristic
14C22×#124C2211yescharacteristic
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#102C2212yes—
18C22×#102C2212yes—
19#2861—24no—
20#2861—24no—
21C2×#391C2211yescharacteristic
22C2×#391C2211yescharacteristic
23C2×#390C2211yescharacteristic
24C2×#390C2211yescharacteristic
25C2×#388—22no—
26C2×#388—22no—
27C2×#382C2211yescharacteristic
28C2×C43C2211yescharacteristic
29#2026—24no—
30#2025—24no—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2212yes—
36C2×#336C2211yescharacteristic
37C2×#336C2211yescharacteristic
38C2×#336C2212yes—
39#1436—22no—
40#1436—22no—
41C4⋊#95C2212yes—
42C4⋊#95C2212yes—
43C23×Q8C2311yescharacteristic
44C23×D4C2311yescharacteristic
45C23×C8C2312yes—
46C23×C8C2312yes—
47C2×#125—22no—
48C2×#125—22no—
49C2×#125—24no—
50C2×#125—22no—
51C2×#125—22no—
52C2×#125—24no—
53C2×#125—42no—
54C2×#125—42no—
55C2×#124—22no—
56C2×#124—22no—
57C2×#124—24no—
58C2×#124—42no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2312yes—
66C22×C22⋊C4—21no—
67C22×C22⋊C4—21no—
68C22×C22⋊C4—22no—
69C22×C22⋊C4—22no—
70C22×C42D412yes—
71C22×C42D412yes—
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75C2×#103—24no—
76C2×#103—24no—
77C2×#103—42no—
78C2×#103—42no—
79C2×#103—42no—
80C2×#103—42no—
81C2×#102—24no—
82C2×#102—24no—
83C2×#102—42no—
84C2×#102—42no—
85C2×#102—42no—
86C2×#102—42no—
87#391—22no—
88#391—22no—
89#391—22no—
90#391D414yes—
91#390—22no—
92#390—22no—
93#390—22no—
94#390D414yes—
95#388—216no—
96#388—216no—
97#382—42no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C43—22no—
106C43—22no—
107#340—24no—
108#340—24no—
109#340—42no—
110#340—42no—
111#336—22no—
112#336—22no—
113#336—24no—
114#336—42no—
115C25—21no—
116C22×Q8—21no—
117C22×Q8—22no—
118C22×Q8—22no—
119C22×Q8—22no—
120C22×Q8—22no—
121C22×Q8—21no—
122C22×Q8—21no—
123C22×Q8—21no—
124C22×Q8—21no—
125C22×Q8C2×D412yes—
126C22×D4—21no—
127C22×D4—22no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—21no—
131C22×D4—22no—
132C22×D4—21no—
133C22×D4—21no—
134C22×D4—21no—
135C22×D4C2×D412yes—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4C2×D412yes—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D412yes—
145C23×C4C2411yescharacteristic
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160#125—48no—
161#125—48no—
162#125—48no—
163#125—48no—
164#124—48no—
165#124—48no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—28no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
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—44no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4C2×D412yes—
193C2×C4⋊C4C2×D412yes—
194C2×C4⋊C4#4114yes—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—28no—
198C2×C22⋊C4—28no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4—44no—
201C2×C22⋊C4—44no—
202C2×C22⋊C4—42no—
203C2×C22⋊C4—42no—
204C2×C22⋊C4—42no—
205C2×C22⋊C4—42no—
206C2×C22⋊C4—44no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—24no—
213C2×C42—24no—
214C2×C42—24no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—42no—
222C2×C42—42no—
223C2×C42—42no—
224C2×C42C2×D412yes—
225C2×C42C2×D412yes—
226C2×C42#4114yes—
227#103—48no—
228#103—48no—
229#102—48no—
230#102—48no—
231#95—28no—
232#95—28no—
233#95—44no—
234#95—44no—
235#95—44no—
236#95—44no—
237#95—44no—
238C24—22no—
239C24—41no—
240C24—41no—
241C24—41no—
242C24—41no—
243C24—41no—
244C24—41no—
245C24—42no—
246C24—42no—
247C24—42no—
248C24—42no—
249C24C22×D411yescharacteristic
250C2×Q8—22no—
251C2×Q8—22no—
252C2×Q8—24no—
253C2×Q8—24no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×D4—22no—
267C2×D4—22no—
268C2×D4—24no—
269C2×D4—24no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—42no—
281C2×D4—42no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—22no—
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—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—42no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
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—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4#12014yes—
334C22×C4#12412yes—
335C22×C4#12412yes—
336C22×C4#13514yes—
337C22×C4C22×D411yescharacteristic
338C22×C4C22×D411yescharacteristic
339C22×C4C2×#4112yes—
340C22×C4C2×#4112yes—
341C22×C4H(5,𝔽2)11yescharacteristic
342C22×C4H(5,𝔽2)12yes—
343C22×C4H(5,𝔽2)12yes—
344C22×C42−1+411yescharacteristic
345C22×C42−1+411yescharacteristic
346C22×C42−1+411yescharacteristic
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—416no—
366C4⋊C4—48no—
367C4⋊C4—48no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C22⋊C4—416no—
377C22⋊C4—416no—
378C22⋊C4—48no—
379C22⋊C4—48no—
380C42—24no—
381C42—24no—
382C42—24no—
383C42—24no—
384C42—48no—
385C42—48no—
386C42—41no—
387C42—41no—
388C42—41no—
389C42—41no—
390C23—42no—
391C23—42no—
392C23—42no—
393C23—44no—
394C23—44no—
395C23—81no—
396C23—82no—
397C23—81no—
398C23—82no—
399C23—81no—
400C23—81no—
401C23—81no—
402C23—81no—
403C23—81no—
404C23—81no—
405C23—82no—
406C23—82no—
407C23—82no—
408C23—82no—
409C23—82no—
410C23—82no—
411C23—82no—
412C23—82no—
413C23C2×#12011yescharacteristic
414C23#52511yescharacteristic
415C23C2×#12411yescharacteristic
416C23#53511yescharacteristic
417C23#53511yescharacteristic
418C23#53611yescharacteristic
419C23#53711yescharacteristic
420C23C2×#13512yes—
421C23#57512yes—
422C23#57512yes—
423C23#57512yes—
424Q8—44no—
425Q8—44no—
426Q8—82no—
427Q8—82no—
428D4—44no—
429D4—44no—
430D4—82no—
431D4—82no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—24no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
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—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4#53611yescharacteristic
488C2×C4#53611yescharacteristic
489C2×C4#53711yescharacteristic
490C2×C4#53711yescharacteristic
491C2×C4#54212yes—
492C2×C4#54212yes—
493C2×C4#54212yes—
494C2×C4#54212yes—
495C2×C4#54612yes—
496C2×C4#54612yes—
497C2×C4#54612yes—
498C2×C4#54612yes—
499C2×C4#55612yes—
500C2×C4#55912yes—
501C2×C4H(3,𝔽4)12yes—
502C2×C4#56212yes—
503C8—82no—
504C8—82no—
505C22—82no—
506C22—82no—
507C22—84no—
508C22—82no—
509C22—84no—
510C22—161no—
511C22—162no—
512C22—162no—
513C22—161no—
514C22—161no—
515C22—161no—
516C22#240411yescharacteristic
517C22#243511yescharacteristic
518C22#259211yescharacteristic
519C22#261611yescharacteristic
520C22#263211yescharacteristic
521C22#264011yescharacteristic
522C22#281911yescharacteristic
523C22#300212yes—
524C22#300512yes—
525C22#310212yes—
526C22#311112yes—
527C22#313111yescharacteristic
528C22#313511yescharacteristic
529C22#314211yescharacteristic
530C22#314211yescharacteristic
531C22#321212yes—
532C22#321212yes—
533C22#321912yes—
534C22#321912yes—
535C22#323212yes—
536C22#324012yes—
537C22#325012yes—
538C22#325612yes—
539C4—42no—
540C4—42no—
541C4—44no—
542C4—44no—
543C4—82no—
544C4—82no—
545C4—84no—
546C4—82no—
547C4—162no—
548C4#270012yes—
549C4#270012yes—
550C4#271212yes—
551C4#271212yes—
552C2—162no—
553C2#1655911yescharacteristic
554C2#2393812yes—
555C2#2585811yescharacteristic
556C2#2586211yescharacteristic
557C2#2935111yescharacteristic
558C2#2935311yescharacteristic
559C2#2969312yes—
560C2#3006512yes—
561C2#3020711yescharacteristic
562C2#3021711yescharacteristic
563C2#3280912yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2026C2semidirect
4. #14709C2semidirect
5. #11385C2semidirect
6. #11190C2semidirect
8. #11052C2semidirect
11. #10795C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28612
C2×#20262
C2×#20252
#147092
#113852
#111902
#111892
#110522
#110512
#110092
#107952