Finite Group Databasebeta

← #411303

Subgroups — #411303

Order 512.

1338 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2314C211yesmaximal, characteristic
2C2×#1928C212yesmaximal, complemented
3#14258C211yesmaximal, characteristic
4#11324C211yesmaximal, characteristic, complemented
5#11268C211yesmaximal, characteristic, complemented
6#11096C212yesmaximal
7#11052C211yesmaximal, characteristic, complemented
8#11050C211yesmaximal, characteristic, complemented
9#10970C212yesmaximal
10#10806C212yesmaximal, complemented
11#10713C211yesmaximal, characteristic
12C22×C22⋊Q8C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#107C2212yes—
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×C22⋊C8C2212yes—
18#2314—28no—
19C2×#391C2211yescharacteristic
20C2×#391C2211yescharacteristic
21C2×#388—22no—
22C2×#387—22no—
23C2×#384C2211yescharacteristic
24C2×#382C2211yescharacteristic
25C2×#380C2212yes—
26C2×#380C2211yescharacteristic
27C2×C4×C4⋊C4C2212yes—
28#1928—28no—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2211yescharacteristic
35C2×#336C2211yescharacteristic
36C2×#336C2212yes—
37#1441—22no—
38#1440—22no—
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2211yescharacteristic
41C23×Q8C2311yescharacteristic
42C24×C4C2311yescharacteristic
43C23×C8C2312yes—
44C23×C8C2312yes—
45C2×C22⋊Q8—22no—
46C2×C22⋊Q8—22no—
47C2×C22⋊Q8—24no—
48C2×C22⋊Q8—42no—
49C2×#117—24no—
50C2×#117—24no—
51C2×#117—22no—
52C2×#117—22no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4D412yes—
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2311yescharacteristic
61C22×C22⋊C4—21no—
62C22×C22⋊C4—22no—
63C22×C22⋊C4—21no—
64C22×C22⋊C4D412yes—
65C22×C42—22no—
66C22×C42C2311yescharacteristic
67C22×C42C2311yescharacteristic
68C2×#107—24no—
69C2×#107—42no—
70C2×#107—42no—
71C2×#103—24no—
72C2×#103—24no—
73C2×#103—24no—
74C2×#103—24no—
75C2×#103—42no—
76C2×#103—42no—
77C2×C22⋊C8—24no—
78C2×C22⋊C8—24no—
79C2×C22⋊C8—42no—
80#391—24no—
81#391—22no—
82#391—22no—
83#388—216no—
84#387—216no—
85#384—24no—
86#382—24no—
87#380—24no—
88#380—24no—
89#380—42no—
90C4×C4⋊C4—24no—
91C4×C4⋊C4D418yes—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—21no—
98C2×#95—22no—
99C2×#95—22no—
100#340—24no—
101#340—24no—
102#340—24no—
103#340—42no—
104#340—42no—
105#336—24no—
106#336—22no—
107#336—22no—
108C25C2×D411yescharacteristic
109C22×Q8—21no—
110C22×Q8—22no—
111C22×Q8—22no—
112C22×Q8—21no—
113C22×Q8—21no—
114C22×Q8—22no—
115C22×Q8—22no—
116C22×Q8—21no—
117C22×Q8—21no—
118C22×Q8#4112yes—
119C23×C4—22no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—22no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4C2×D412yes—
135C23×C4C2×D412yes—
136C23×C4C2×D411yescharacteristic
137C23×C4#4114yes—
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#4114yes—
153C22⋊Q8—48no—
154C22⋊Q8—48no—
155#117—216no—
156#117—216no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—28no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—82no—
186C2×C4⋊C4—82no—
187C2×C4⋊C4C2×D414yes—
188C2×C4⋊C4C2×D414yes—
189C2×C4⋊C4#4112yes—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—24no—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—28no—
195C2×C22⋊C4—42no—
196C2×C22⋊C4—42no—
197C2×C22⋊C4—42no—
198C2×C22⋊C4—82no—
199C2×C22⋊C4—82no—
200C2×C22⋊C4—82no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—22no—
206C2×C42—28no—
207C2×C42—82no—
208C2×C42—82no—
209C2×C42C2×D414yes—
210C2×C42#4112yes—
211C2×C42#4112yes—
212#107—48no—
213#103—216no—
214#103—48no—
215C22⋊C8—48no—
216#95—28no—
217#95—28no—
218#95—28no—
219#95—42no—
220#95—44no—
221#95—44no—
222#95—82no—
223#95—82no—
224C24—21no—
225C24—21no—
226C24—21no—
227C24—21no—
228C24—21no—
229C24—21no—
230C24—22no—
231C24—22no—
232C24—22no—
233C24—22no—
234C24#12312yes—
235C24C22×D411yescharacteristic
236C2×Q8—24no—
237C2×Q8—24no—
238C2×Q8—22no—
239C2×Q8—22no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—41no—
243C2×Q8—41no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×Q8—42no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—42no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—42no—
287C22×C4—41no—
288C22×C4—42no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—42no—
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—41no—
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#12014yes—
323C22×C4#12314yes—
324C22×C4#12312yes—
325C22×C4C22×D412yes—
326C22×C4C2×#4112yes—
327C22×C4C2×#4111yescharacteristic
328C22×C4C2×#4111yescharacteristic
329C22×C4C2×#4111yescharacteristic
330C22×C4C2×#4112yes—
331C22×C4C2×#4111yescharacteristic
332C22×C4H(5,𝔽2)11yescharacteristic
333C22×C4H(5,𝔽2)12yes—
334C22×C42−1+411yescharacteristic
335C2×C8—24no—
336C2×C8—24no—
337C2×C8—24no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C4⋊C4—216no—
350C4⋊C4—24no—
351C4⋊C4—24no—
352C4⋊C4—24no—
353C4⋊C4—24no—
354C4⋊C4—24no—
355C4⋊C4—48no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—88no—
366C22⋊C4—216no—
367C22⋊C4—48no—
368C22⋊C4—88no—
369C22⋊C4—84no—
370C42—24no—
371C42—24no—
372C42—24no—
373C42—24no—
374C42—88no—
375C23—24no—
376C23—24no—
377C23—22no—
378C23—22no—
379C23—22no—
380C23—42no—
381C23—42no—
382C23—42no—
383C23—42no—
384C23—41no—
385C23—41no—
386C23—42no—
387C23—42no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—42no—
394C23—42no—
395C23—42no—
396C23—41no—
397C23—42no—
398C23C2×#12011yescharacteristic
399C23C2×#12311yescharacteristic
400C23C2×#12311yescharacteristic
401C23#52511yescharacteristic
402C23#53211yescharacteristic
403C23#53511yescharacteristic
404C23#53711yescharacteristic
405C23#57612yes—
406C23#57712yes—
407C23#57812yes—
408C23#57812yes—
409Q8—44no—
410Q8—44no—
411Q8—82no—
412Q8—82no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—42no—
432C2×C4—41no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—82no—
450C2×C4—81no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—81no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—81no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—81no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—82no—
465C2×C4—81no—
466C2×C4—82no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—82no—
472C2×C4—81no—
473C2×C4—82no—
474C2×C4#52512yes—
475C2×C4#52512yes—
476C2×C4#52512yes—
477C2×C4#53212yes—
478C2×C4#53512yes—
479C2×C4#53712yes—
480C2×C4#55112yes—
481C2×C4#55112yes—
482C2×C4#55112yes—
483C2×C4#55212yes—
484C2×C4#55211yescharacteristic
485C2×C4#55311yescharacteristic
486C2×C4#55312yes—
487C2×C4#55512yes—
488C2×C4#55511yescharacteristic
489C2×C4#55611yescharacteristic
490C8—44no—
491C8—82no—
492C22—42no—
493C22—42no—
494C22—42no—
495C22—44no—
496C22—44no—
497C22—82no—
498C22—81no—
499C22—81no—
500C22—82no—
501C22—81no—
502C22—81no—
503C22#238811yescharacteristic
504C22#248211yescharacteristic
505C22#248511yescharacteristic
506C22#249811yescharacteristic
507C22#250011yescharacteristic
508C22#261511yescharacteristic
509C22#284911yescharacteristic
510C22#301112yes—
511C22#301212yes—
512C22#309712yes—
513C22#309712yes—
514C22#309812yes—
515C22#309812yes—
516C22#312112yes—
517C22#312412yes—
518C22#317512yes—
519C22#317812yes—
520C22#322011yescharacteristic
521C22#322111yescharacteristic
522C22#322311yescharacteristic
523C22#322311yescharacteristic
524C22#325712yes—
525C22#326112yes—
526C4—42no—
527C4—44no—
528C4—44no—
529C4—82no—
530C4—84no—
531C4—82no—
532C4—84no—
533C4—161no—
534C4—161no—
535C4#259812yes—
536C4#260112yes—
537C4#260912yes—
538C4#261012yes—
539C2—82no—
540C2#1511311yescharacteristic
541C2#2358512yes—
542C2#2759511yescharacteristic
543C2#2759611yescharacteristic
544C2#2764312yes—
545C2#2787012yes—
546C2#2789911yescharacteristic
547C2#2790211yescharacteristic
548C2#2967411yescharacteristic
549C2#2967511yescharacteristic
550C2#3298912yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×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
2. C2×#1928C2semidirect
4. #11324C2semidirect
5. #11268C2semidirect
7. #11052C2semidirect
8. #11050C2semidirect
10. #10806C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23142
C2×#19282
#142582
#113242
#112682
#110962
#110522
#110502
#109702
#108062
#107132