Finite Group Databasebeta

← #468858

Subgroups — #468858

Order 512.

1238 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2647C211yesmaximal, characteristic, complemented
2C2×#2042C212yesmaximal
3#14273C211yesmaximal, characteristic
4#11207C212yesmaximal, complemented
5#11067C212yesmaximal, complemented
6#11066C212yesmaximal
7#11052C211yesmaximal, characteristic, complemented
8#11051C211yesmaximal, characteristic
9#11050C211yesmaximal, characteristic, complemented
10#11049C211yesmaximal, characteristic
11#10796C211yesmaximal, characteristic, complemented
12C22×#124C2211yescharacteristic
13C22×#103C2212yes—
14C22×#103C2212yes—
15C22×#102C2212yes—
16C22×#102C2212yes—
17#2647—28no—
18C2×#400—22no—
19C2×#398—22no—
20C2×#391C2211yescharacteristic
21C2×#391C2211yescharacteristic
22C2×#390C2211yescharacteristic
23C2×#390C2211yescharacteristic
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#384C2211yescharacteristic
27C2×#384C2211yescharacteristic
28C2×#382C2211yescharacteristic
29C2×#376C2211yescharacteristic
30#2042—28no—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2211yescharacteristic
36C2×#336C2211yescharacteristic
37C2×#336C2212yes—
38C2×#336C2212yes—
39#1439—22no—
40#1439—22no—
41C23×Q8C2311yescharacteristic
42C23×D4C2311yescharacteristic
43C23×C8C2312yes—
44C23×C8C2312yes—
45C2×#124—22no—
46C2×#124—22no—
47C2×#124—24no—
48C2×#124—42no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2312yes—
58C22×C22⋊C4—21no—
59C22×C22⋊C4—21no—
60C22×C22⋊C4—22no—
61C22×C22⋊C4—22no—
62C22×C42C2311yescharacteristic
63C22×C42C2311yescharacteristic
64C22×C42C2311yescharacteristic
65C2×#103—24no—
66C2×#103—24no—
67C2×#103—24no—
68C2×#103—42no—
69C2×#103—42no—
70C2×#103—42no—
71C2×#102—24no—
72C2×#102—24no—
73C2×#102—24no—
74C2×#102—42no—
75C2×#102—42no—
76C2×#102—42no—
77#400—216no—
78#398—216no—
79#391—22no—
80#391—22no—
81#391—42no—
82#390—22no—
83#390—22no—
84#390—42no—
85#389—24no—
86#389—24no—
87#389—24no—
88#389D418yes—
89#384—24no—
90#384—42no—
91#382—24no—
92#376—24no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95D412yes—
99C2×#95D412yes—
100#340—42no—
101#340—42no—
102#340—42no—
103#340—42no—
104#336—24no—
105#336—22no—
106#336—22no—
107#336—42no—
108C25—21no—
109C22×Q8—21no—
110C22×Q8—22no—
111C22×Q8—21no—
112C22×Q8—21no—
113C22×Q8—22no—
114C22×Q8—22no—
115C22×Q8—21no—
116C22×Q8—22no—
117C22×Q8—21no—
118C22×Q8#4112yes—
119C22×D4—22no—
120C22×D4—21no—
121C22×D4—21no—
122C22×D4—21no—
123C22×D4—22no—
124C22×D4—22no—
125C22×D4—21no—
126C22×D4—21no—
127C22×D4—22no—
128C22×D4#4112yes—
129C23×C4—21no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4C2×D412yes—
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D412yes—
138C23×C4C2411yescharacteristic
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153#124—48no—
154#124—48no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—28no—
174C2×C4⋊C4—28no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—82no—
179C2×C4⋊C4—82no—
180C2×C4⋊C4—82no—
181C2×C4⋊C4—82no—
182C2×C4⋊C4—82no—
183C2×C4⋊C4—82no—
184C2×C4⋊C4—82no—
185C2×C4⋊C4—82no—
186C2×C4⋊C4C2×D414yes—
187C2×C4⋊C4C2×D414yes—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—24no—
190C2×C22⋊C4—42no—
191C2×C22⋊C4—42no—
192C2×C22⋊C4—42no—
193C2×C22⋊C4—42no—
194C2×C22⋊C4—44no—
195C2×C22⋊C4—44no—
196C2×C22⋊C4—82no—
197C2×C22⋊C4—82no—
198C2×C22⋊C4—82no—
199C2×C22⋊C4—82no—
200C2×C42—22no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—21no—
207C2×C42—22no—
208C2×C42—21no—
209C2×C42C2×D414yes—
210C2×C42#4112yes—
211C2×C42#4112yes—
212#103—48no—
213#103—48no—
214#102—48no—
215#102—48no—
216#95—28no—
217#95—28no—
218#95—28no—
219#95—24no—
220#95—24no—
221#95—44no—
222#95—44no—
223C24—22no—
224C24—41no—
225C24—41no—
226C24—41no—
227C24—41no—
228C24—41no—
229C24—41no—
230C24—42no—
231C24—42no—
232C24—42no—
233C24—42no—
234C24C22×D411yescharacteristic
235C2×Q8—24no—
236C2×Q8—24no—
237C2×Q8—22no—
238C2×Q8—22no—
239C2×Q8—42no—
240C2×Q8—42no—
241C2×Q8—41no—
242C2×Q8—41no—
243C2×Q8—42no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×Q8—42no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×D4—24no—
252C2×D4—24no—
253C2×D4—22no—
254C2×D4—22no—
255C2×D4—42no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—42no—
266C2×D4—42no—
267C22×C4—22no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—41no—
292C22×C4—42no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—41no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4#12012yes—
320C22×C4#12012yes—
321C22×C4#12414yes—
322C22×C4C22×D412yes—
323C22×C4C2×#4111yescharacteristic
324C22×C4C2×#4111yescharacteristic
325C22×C4C2×#4111yescharacteristic
326C22×C4C2×#4111yescharacteristic
327C22×C4H(5,𝔽2)11yescharacteristic
328C22×C4H(5,𝔽2)12yes—
329C22×C4H(5,𝔽2)12yes—
330C22×C42−1+411yescharacteristic
331C22×C42−1+412yes—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C4⋊C4—24no—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—24no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—88no—
359C4⋊C4—88no—
360C4⋊C4—88no—
361C4⋊C4—84no—
362C4⋊C4—84no—
363C22⋊C4—48no—
364C22⋊C4—48no—
365C22⋊C4—88no—
366C22⋊C4—88no—
367C42—24no—
368C42—24no—
369C42—24no—
370C42—24no—
371C42—42no—
372C42—42no—
373C23—42no—
374C23—44no—
375C23—44no—
376C23—42no—
377C23—42no—
378C23—81no—
379C23—81no—
380C23—82no—
381C23—82no—
382C23—81no—
383C23—81no—
384C23—81no—
385C23—81no—
386C23—81no—
387C23—81no—
388C23—82no—
389C23—82no—
390C23—82no—
391C23—82no—
392C23—82no—
393C23—82no—
394C23—82no—
395C23—82no—
396C23C2×#12011yescharacteristic
397C23#52511yescharacteristic
398C23C2×#12411yescharacteristic
399C23#53511yescharacteristic
400C23#53511yescharacteristic
401C23#53611yescharacteristic
402C23#53711yescharacteristic
403C23#57612yes—
404C23#57712yes—
405C23#57712yes—
406C23#57812yes—
407Q8—44no—
408Q8—44no—
409Q8—82no—
410Q8—82no—
411D4—44no—
412D4—44no—
413D4—82no—
414D4—82no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—44no—
448C2×C4—81no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4#53512yes—
471C2×C4#53712yes—
472C2×C4#53912yes—
473C2×C4#53912yes—
474C2×C4#54012yes—
475C2×C4#54012yes—
476C2×C4#54612yes—
477C2×C4#54612yes—
478C2×C4#54812yes—
479C2×C4#54812yes—
480C2×C4#55211yescharacteristic
481C2×C4#55211yescharacteristic
482C2×C4#55311yescharacteristic
483C2×C4#55311yescharacteristic
484C2×C4#56212yes—
485C2×C4#56312yes—
486C8—82no—
487C8—82no—
488C22—82no—
489C22—82no—
490C22—84no—
491C22—84no—
492C22—82no—
493C22—161no—
494C22—161no—
495C22—162no—
496C22—161no—
497C22—161no—
498C22—162no—
499C22#240411yescharacteristic
500C22#243611yescharacteristic
501C22#256211yescharacteristic
502C22#261511yescharacteristic
503C22#263311yescharacteristic
504C22#264111yescharacteristic
505C22#282011yescharacteristic
506C22#301212yes—
507C22#301312yes—
508C22#311912yes—
509C22#312012yes—
510C22#313612yes—
511C22#313712yes—
512C22#322011yescharacteristic
513C22#322111yescharacteristic
514C22#322211yescharacteristic
515C22#322311yescharacteristic
516C22#322412yes—
517C22#322512yes—
518C22#324312yes—
519C22#324412yes—
520C22#326212yes—
521C22#326312yes—
522C4—44no—
523C4—42no—
524C4—42no—
525C4—84no—
526C4—82no—
527C4—84no—
528C4—82no—
529C4—161no—
530C4—161no—
531C4#270712yes—
532C4#270712yes—
533C4#271112yes—
534C4#271112yes—
535C2—162no—
536C2#1663311yescharacteristic
537C2#2394111yescharacteristic
538C2#2394411yescharacteristic
539C2#2591612yes—
540C2#2913412yes—
541C2#2967712yes—
542C2#3008812yes—
543C2#3022211yescharacteristic
544C2#3022311yescharacteristic
545C2#3282911yescharacteristic
546C2#3283011yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2647C2semidirect
4. #11207C2semidirect
5. #11067C2semidirect
7. #11052C2semidirect
9. #11050C2semidirect
11. #10796C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26472
C2×#20422
#142732
#112072
#110672
#110662
#110522
#110512
#110502
#110492
#107962