Finite Group Databasebeta

← #323980

Subgroups — #323980

Order 512.

584 conjugacy classes of subgroups.

The subgroup lattice has 535 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29844C211yesmaximal, characteristic
2#24710C211yesmaximal, characteristic
3#24702C211yesmaximal, characteristic
4#22098C211yesmaximal, characteristic
5#10649C211yesmaximal, characteristic, complemented
6#10617C211yesmaximal, characteristic, complemented
7#10541C211yesmaximal, characteristic, complemented
8#10277C211yesmaximal, characteristic, complemented
9#9568C211yesmaximal, characteristic, complemented
10#9561C211yesmaximal, characteristic, complemented
11#9220C211yesmaximal, characteristic, complemented
12#9216C211yesmaximal, characteristic, complemented
13#9136C211yesmaximal, characteristic
14#9130C211yesmaximal, characteristic
15#7878C211yesmaximal, characteristic
16C2×#487C2211yescharacteristic
17C2×#477C2211yescharacteristic
18C2×#475C2211yescharacteristic
19C2×C8×Q8C2211yescharacteristic
20C2×#432C2211yescharacteristic
21C2×#431C2211yescharacteristic
22C2×#431C2211yescharacteristic
23#2857C2211yescharacteristic
24#2857C2211yescharacteristic
25C4×C22⋊Q8C2211yescharacteristic
26#2077—21no—
27#2076—21no—
28#2068—21no—
29#2068—21no—
30#2036—21no—
31#2036—21no—
32#2036—21no—
33#2036—21no—
34#1990C2211yescharacteristic
35#1988C2211yescharacteristic
36#1987C2211yescharacteristic
37#1984C2211yescharacteristic
38#1956C2211yescharacteristic
39#1947C2211yescharacteristic
40#1947C2211yescharacteristic
41#1943—21no—
42#1942—21no—
43#1939—21no—
44#1939C2211yescharacteristic
45#1935—21no—
46#1933—21no—
47#1923—21no—
48#1923—21no—
49#1916—21no—
50#1845C2211yescharacteristic
51#1803C2211yescharacteristic
52#1803C2211yescharacteristic
53#1797C2211yescharacteristic
54C8×C4⋊C4C2211yescharacteristic
55C2×#333C2211yescharacteristic
56C2×#333C2211yescharacteristic
57C2×#332C2211yescharacteristic
58C2×#332C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C2×C4⋊Q8C2311yescharacteristic
68C2×C4⋊Q8C2311yescharacteristic
69C2×C22⋊Q8—21no—
70C2×C22⋊Q8—21no—
71C2×C22⋊Q8D411yescharacteristic
72C2×C4×Q8C2311yescharacteristic
73C22×C42C2311yescharacteristic
74#487—41no—
75#487—41no—
76#477—22no—
77#477—41no—
78#475—22no—
79#475—41no—
80C8×Q8—41no—
81C8×Q8—41no—
82#432—22no—
83#432—41no—
84#431—22no—
85#431—41no—
86#431—41no—
87#431—41no—
88C2×#107—21no—
89C2×#107—21no—
90C2×#107—21no—
91C2×#107—21no—
92C2×#107—21no—
93C2×#106—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8D411yescharacteristic
97C2×C4⋊C8D411yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×#103—21no—
105C2×#103—21no—
106C2×#103—21no—
107C2×#103—21no—
108C2×#103—21no—
109C2×#103—21no—
110C2×#103—21no—
111C2×#103—21no—
112C2×C22⋊C8—21no—
113C2×C22⋊C8—21no—
114C2×C22⋊C8—21no—
115C2×C22⋊C8D411yescharacteristic
116C2×C22⋊C8D411yescharacteristic
117C2×#97—21no—
118C2×C4×C8—21no—
119C2×C4×C8—21no—
120C2×C4×C8—21no—
121C2×C4×C8C2311yescharacteristic
122C2×C4×C8C2311yescharacteristic
123#391—21no—
124#391—21no—
125#391D411yescharacteristic
126#389—21no—
127#389—21no—
128#389—21no—
129#389—21no—
130#389D411yescharacteristic
131#387—21no—
132#385—21no—
133#384C2311yescharacteristic
134#384C2311yescharacteristic
135#380—21no—
136#380—21no—
137#380—21no—
138#380—21no—
139#380D411yescharacteristic
140C4×C4⋊C4—21no—
141C4×C4⋊C4C2311yescharacteristic
142C4×C22⋊C4—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146#340—21no—
147#340—21no—
148#340—21no—
149#340—21no—
150#340—21no—
151#340—21no—
152#340—21no—
153#336—21no—
154#336—21no—
155#336—21no—
156#336—21no—
157#336—21no—
158#333—22no—
159#333—41no—
160#332—22no—
161#332—41no—
162C22×Q8—21no—
163C22×Q8—21no—
164C22×Q8C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4#4111yescharacteristic
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8C2×D411yescharacteristic
175C22×C8C2×D411yescharacteristic
176C22×C8C2×D411yescharacteristic
177C22×C8C2×D411yescharacteristic
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181C4⋊Q8—21no—
182C4⋊Q8—21no—
183C4⋊Q8—21no—
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C22⋊Q8—22no—
187C22⋊Q8—22no—
188C22⋊Q8—81no—
189C22⋊Q8—81no—
190C22⋊Q8—82no—
191C4×Q8—21no—
192C4×Q8—21no—
193C4×Q8—21no—
194C4×Q8—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4C2×D411yescharacteristic
211C2×C4⋊C4C2×D411yescharacteristic
212C2×C4⋊C4C2×D411yescharacteristic
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—41no—
216C2×C22⋊C4—41no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42#4111yescharacteristic
226C2×C42C2411yescharacteristic
227#107—42no—
228#107—42no—
229#107—42no—
230#107—81no—
231#107—81no—
232#106—42no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—41no—
244C4⋊C8—41no—
245C4⋊C8—42no—
246C4⋊C8—81no—
247C4⋊C8#4112yes—
248C4⋊C8#4112yes—
249#103—42no—
250#103—42no—
251#103—42no—
252#103—42no—
253#103—42no—
254#103—42no—
255#103—81no—
256#103—81no—
257C22⋊C8—41no—
258C22⋊C8—41no—
259C22⋊C8—42no—
260C22⋊C8—42no—
261C22⋊C8—81no—
262#97—42no—
263C4×C8—21no—
264C4×C8—21no—
265C4×C8—21no—
266C4×C8—21no—
267C4×C8—42no—
268C4×C8—42no—
269C4×C8—81no—
270#95—21no—
271#95—21no—
272#95—21no—
273#95—21no—
274#95—21no—
275#95—41no—
276#95—41no—
277#95—41no—
278#95—41no—
279C24#12111yescharacteristic
280C2×Q8—21no—
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—81no—
294C2×Q8—81no—
295C2×Q8#13512yes—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4#12111yescharacteristic
321C22×C4#12111yescharacteristic
322C22×C4#12111yescharacteristic
323C22×C4C22×D411yescharacteristic
324C22×C4C22×D411yescharacteristic
325C22×C4C2×#4111yescharacteristic
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—41no—
356C4⋊C4—24no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—81no—
381C4⋊C4—81no—
382C4⋊C4—81no—
383C4⋊C4—82no—
384C4⋊C4—82no—
385C4⋊C4#13512yes—
386C22⋊C4—24no—
387C22⋊C4—81no—
388C22⋊C4—82no—
389C22⋊C4—82no—
390C42—21no—
391C42—21no—
392C42—21no—
393C42—21no—
394C42—21no—
395C42—21no—
396C42—41no—
397C42—41no—
398C42—41no—
399C42—41no—
400C42C2×#4111yescharacteristic
401C42C2×#4111yescharacteristic
402C42H(5,𝔽2)11yescharacteristic
403C42H(5,𝔽2)11yescharacteristic
404C23—21no—
405C23—21no—
406C23—21no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23C2×#12111yescharacteristic
412Q8—22no—
413Q8—22no—
414Q8—41no—
415Q8—41no—
416Q8—81no—
417Q8—81no—
418Q8—81no—
419Q8—81no—
420Q8—161no—
421Q8—161no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4#53211yescharacteristic
468C2×C4#53411yescharacteristic
469C2×C4#54611yescharacteristic
470C2×C4#54611yescharacteristic
471C2×C4#55311yescharacteristic
472C2×C4#55311yescharacteristic
473C2×C4C2×#13511yescharacteristic
474C2×C4C2×#13511yescharacteristic
475C2×C4#57611yescharacteristic
476C2×C4#57711yescharacteristic
477C2×C4#57811yescharacteristic
478C2×C4#57811yescharacteristic
479C2×C4#57811yescharacteristic
480C2×C4#57811yescharacteristic
481C8—41no—
482C8—41no—
483C8—41no—
484C8—41no—
485C8—42no—
486C8—42no—
487C8—81no—
488C8—81no—
489C8—81no—
490C8—81no—
491C22—41no—
492C22—41no—
493C22—41no—
494C22—41no—
495C22—41no—
496C22—41no—
497C22—41no—
498C22#254311yescharacteristic
499C22#304411yescharacteristic
500C22#304511yescharacteristic
501C22#305011yescharacteristic
502C22#305911yescharacteristic
503C22#306811yescharacteristic
504C22#306911yescharacteristic
505C4—21no—
506C4—21no—
507C4—41no—
508C4—42no—
509C4—41no—
510C4—41no—
511C4—42no—
512C4—81no—
513C4—81no—
514C4—161no—
515C4—161no—
516C4—161no—
517C4—161no—
518C4#317511yescharacteristic
519C4#317811yescharacteristic
520C4#320111yescharacteristic
521C4#320311yescharacteristic
522C4#330011yescharacteristic
523C4#330011yescharacteristic
524C4#330211yescharacteristic
525C4#330211yescharacteristic
526C2—81no—
527C2#1940711yescharacteristic
528C2#1941211yescharacteristic
529C2#1953111yescharacteristic
530C2#1953911yescharacteristic
531C2#2855411yescharacteristic
532C2#2859011yescharacteristic
533C2#2859611yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10649C2semidirect
6. #10617C2semidirect
7. #10541C2semidirect
8. #10277C2semidirect
9. #9568C2semidirect
10. #9561C2semidirect
11. #9220C2semidirect
12. #9216C2semidirect

Maximal subgroups

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

subgroupindex
#298442
#247102
#247022
#220982
#106492
#106172
#105412
#102772
#95682
#95612
#92202
#92162
#91362
#91302
#78782