Finite Group Databasebeta

← #6484965

Subgroups — #6484965

Order 512.

1478 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35332C211yesmaximal, characteristic
2#35283C211yesmaximal, characteristic, complemented
3#33057C212yesmaximal, complemented
4#33051C212yesmaximal, complemented
5#30195C214yesmaximal, complemented
6#29359C212yesmaximal, complemented
7#29358C212yesmaximal, complemented
8#29223C212yesmaximal, complemented
9#29148C212yesmaximal, complemented
10#29134C214yesmaximal, complemented
11#28754C212yesmaximal, complemented
12#28727C212yesmaximal, complemented
13#21898C212yesmaximal, complemented
14#21897C212yesmaximal
15#20427C211yesmaximal, characteristic, complemented
16C2×#541C2211yescharacteristic
17C2×#540C2211yescharacteristic
18C2×#489C2212yes—
19C2×#489C2212yes—
20C2×#424C2212yes—
21C2×C4×M16C2211yescharacteristic
22C22×#97C2211yescharacteristic
23#2868C2211yescharacteristic
24#2862C2211yescharacteristic
25#2859C2211yescharacteristic
26#2858C2211yescharacteristic
27#2857C2212yes—
28#2850C2212yes—
29#2647C2212yes—
30#2642C2211yescharacteristic
31#2641C2211yescharacteristic
32#2628C2212yes—
33#2620C2212yes—
34#2559—22no—
35#2559—22no—
36#2558—22no—
37#2558—22no—
38#2511—22no—
39#2492—22no—
40#2451—22no—
41#2423—22no—
42#2331C2212yes—
43#2326C2212yes—
44#2314C2212yes—
45C4×C4⋊Q8C2212yes—
46C4×#124C2212yes—
47C4×#117C2211yescharacteristic
48#2100C2214yes—
49#2099C2214yes—
50#2074C2214yes—
51#2073C2214yes—
52#2042C2214yes—
53#2042C2214yes—
54#2024C2214yes—
55#2023C2214yes—
56#1993C2212yes—
57#1993C2212yes—
58#1989C2214yes—
59#1989C2214yes—
60#1982C2214yes—
61#1981C2214yes—
62#1959C2212yes—
63#1959C2212yes—
64#1936C2214yes—
65#1935C2214yes—
66#1934C2214yes—
67#1933C2214yes—
68#1895C2214yes—
69#1894C2214yes—
70#1893C2214yes—
71#1892C2214yes—
72#1836C2212yes—
73#1836C2212yes—
74#1805C2214yes—
75#1804C2214yes—
76#1803C2214yes—
77#1802C2214yes—
78#1768C2212yes—
79#1768C2212yes—
80#1767C2212yes—
81#1767C2212yes—
82#1729C2212yes—
83C22×M16C2312yes—
84C23×C8C2312yes—
85#541—28no—
86#540—28no—
87C2×C4⋊Q8—22no—
88C2×C4⋊Q8C2311yescharacteristic
89C2×C4⋊Q8C2312yes—
90C2×C4⋊Q8C2312yes—
91C2×#127C2311yescharacteristic
92C2×#126—22no—
93C2×#125—22no—
94C2×#125C2311yescharacteristic
95C2×#125C2311yescharacteristic
96C2×#124C2312yes—
97C2×#124C2312yes—
98C2×#123—22no—
99C2×C22⋊Q8—22no—
100C2×C22⋊Q8C2312yes—
101C2×#121—22no—
102C2×#121C2312yes—
103C2×C4×Q8—22no—
104C2×C4×Q8C2312yes—
105C2×C4×D4C2312yes—
106C2×#117C2311yescharacteristic
107C2×#117C2311yescharacteristic
108C22×C42C2311yescharacteristic
109#489—44no—
110#489—44no—
111#424—28no—
112C2×C4⋊C8C2314yes—
113C2×C4⋊C8C2314yes—
114C2×#103C2314yes—
115C2×#103C2314yes—
116C2×#103C2314yes—
117C2×#103C2314yes—
118C2×#102C2314yes—
119C2×#102C2314yes—
120C2×#102C2314yes—
121C2×#102C2314yes—
122C4×M16—42no—
123C2×#97—42no—
124C2×#97C2312yes—
125C2×#97C2312yes—
126C2×#97C2312yes—
127C2×C4×C8C2312yes—
128#400—22no—
129#400—22no—
130#400—22no—
131#400—24no—
132#399—22no—
133#398—24no—
134#396—22no—
135#396—22no—
136#395—22no—
137#394—22no—
138#394—22no—
139#394—22no—
140#394—22no—
141#393—22no—
142#391—22no—
143#391—22no—
144#391—22no—
145#391—22no—
146#391—24no—
147#391C2312yes—
148#391C2312yes—
149#390—22no—
150#390—22no—
151#390C2312yes—
152#390C2312yes—
153#389—22no—
154#389—22no—
155#389—22no—
156#389—22no—
157#389C2314yes—
158#389C2314yes—
159#388—24no—
160#388—22no—
161#388—22no—
162#388—22no—
163#388—22no—
164#388—22no—
165#388—22no—
166#387—24no—
167#387—24no—
168#386—22no—
169#386—22no—
170#385—22no—
171#385—22no—
172#384—22no—
173#384C2312yes—
174#384C2312yes—
175#384C2312yes—
176#382C2312yes—
177#382C2312yes—
178#382C2312yes—
179#380—22no—
180#380C2314yes—
181C4×C4⋊C4—22no—
182C4×C4⋊C4D412yes—
183C4×C4⋊C4D412yes—
184C4×C4⋊C4C2314yes—
185C4×C22⋊C4—22no—
186C4×C22⋊C4—22no—
187C4×C22⋊C4D412yes—
188C4×C22⋊C4D412yes—
189#376C2311yescharacteristic
190#376C2311yescharacteristic
191C43C2311yescharacteristic
192C43C2311yescharacteristic
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#336C2314yes—
202#336C2314yes—
203C22×Q8—22no—
204C22×Q8—22no—
205C22×Q8C2412yes—
206C22×D4—22no—
207C22×D4—22no—
208C22×D4C2412yes—
209C23×C4—22no—
210C23×C4C2×D412yes—
211C23×C4C2411yescharacteristic
212C2×M16—24no—
213C2×M16—42no—
214C2×M16—42no—
215C22×C8—24no—
216C22×C8—42no—
217C22×C8—42no—
218C22×C8C2414yes—
219C22×C8C2414yes—
220C4⋊Q8—24no—
221C4⋊Q8—24no—
222C4⋊Q8—24no—
223C4⋊Q8—44no—
224C4⋊Q8—84no—
225#127—24no—
226#126—48no—
227#125—24no—
228#125—24no—
229#125—48no—
230#124—24no—
231#124—24no—
232#124—44no—
233#123—48no—
234C22⋊Q8—28no—
235C22⋊Q8—84no—
236#121—28no—
237#121—48no—
238C4×Q8—28no—
239C4×Q8—84no—
240C4×D4—28no—
241#117—42no—
242#117#4118yes—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4C2×D411yescharacteristic
258C2×C4⋊C4C2×D411yescharacteristic
259C2×C4⋊C4C2×D412yes—
260C2×C4⋊C4C2412yes—
261C2×C4⋊C4C2412yes—
262C2×C4⋊C4C2414yes—
263C2×C4⋊C4C2414yes—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4C2×D411yescharacteristic
277C2×C22⋊C4C2×D411yescharacteristic
278C2×C22⋊C4C2×D412yes—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—24no—
283C2×C42C2×D412yes—
284C2×C42C2×D412yes—
285C2×C42C2×D412yes—
286C2×C42C2×D412yes—
287C2×C42C2×D412yes—
288C2×C42C2411yescharacteristic
289C2×C42C2411yescharacteristic
290C2×C42C2411yescharacteristic
291C2×C42C2411yescharacteristic
292C2×C42C2411yescharacteristic
293C2×C42C2411yescharacteristic
294C4⋊C8—28no—
295C4⋊C8—28no—
296#103—44no—
297#103—44no—
298#103—44no—
299#103—44no—
300#102—44no—
301#102—44no—
302#102—44no—
303#102—44no—
304#97—24no—
305#97—24no—
306#97—42no—
307C4×C8—42no—
308#95—22no—
309#95—22no—
310#95—22no—
311#95—22no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—22no—
317#95—24no—
318#95—22no—
319#95—22no—
320#95—22no—
321#95C2×D412yes—
322#95C2×D412yes—
323#95C2×D412yes—
324#95C2×D412yes—
325C24—22no—
326C24C22×D411yescharacteristic
327C2×Q8—24no—
328C2×Q8—22no—
329C2×Q8—22no—
330C2×Q8—24no—
331C2×Q8—44no—
332C2×Q8—44no—
333C2×Q8—44no—
334C2×Q8—44no—
335C2×Q8—84no—
336C2×D4—24no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—24no—
340C2×D4—44no—
341C2×D4—44no—
342C2×D4—82no—
343C2×D4—82no—
344C2×D4—82no—
345C2×D4—82no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—44no—
358C22×C4—82no—
359C22×C4—82no—
360C22×C4C22×D412yes—
361C22×C4C22×D411yescharacteristic
362C22×C4C22×D411yescharacteristic
363C22×C4C22×D411yescharacteristic
364C22×C4C22×D411yescharacteristic
365C22×C4C22×D411yescharacteristic
366C22×C4C22×D412yes—
367C22×C4C22×D412yes—
368C22×C4C22×D412yes—
369C22×C4C2×#4114yes—
370C22×C4C2511yescharacteristic
371M16—48no—
372C2×C8—24no—
373C2×C8—24no—
374C2×C8—24no—
375C2×C8—24no—
376C2×C8—24no—
377C2×C8—24no—
378C2×C8—48no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—24no—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—24no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—44no—
388C4⋊C4—44no—
389C4⋊C4—44no—
390C4⋊C4—44no—
391C4⋊C4—44no—
392C4⋊C4—44no—
393C4⋊C4—44no—
394C4⋊C4—44no—
395C4⋊C4—44no—
396C4⋊C4—82no—
397C4⋊C4—82no—
398C4⋊C4—84no—
399C4⋊C4—84no—
400C4⋊C4—84no—
401C4⋊C4#12418yes—
402C22⋊C4—41no—
403C22⋊C4—41no—
404C22⋊C4—44no—
405C22⋊C4—44no—
406C22⋊C4—44no—
407C22⋊C4—44no—
408C22⋊C4—44no—
409C22⋊C4—44no—
410C22⋊C4—44no—
411C22⋊C4—82no—
412C22⋊C4—82no—
413C22⋊C4—82no—
414C22⋊C4—82no—
415C22⋊C4—84no—
416C22⋊C4#12418yes—
417C42—24no—
418C42—21no—
419C42—21no—
420C42—22no—
421C42—22no—
422C42—42no—
423C42—42no—
424C42—42no—
425C42—42no—
426C42—44no—
427C42—84no—
428C42C2×#4114yes—
429C42C2×#4114yes—
430C42H(5,𝔽2)12yes—
431C422−1+412yes—
432C23—22no—
433C23—22no—
434C23—21no—
435C23—21no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23C2×#12412yes—
444C23C23×D411yescharacteristic
445Q8—44no—
446Q8—44no—
447Q8—88no—
448Q8—164no—
449D4—44no—
450D4—44no—
451D4—164no—
452D4—164no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—84no—
490C2×C4—164no—
491C2×C4C2×#12414yes—
492C2×C4C2×#12414yes—
493C2×C4C2×#12412yes—
494C2×C4#53714yes—
495C2×C4C22×#4112yes—
496C2×C4C2×H(5,𝔽2)11yescharacteristic
497C2×C4C2×2−1+411yescharacteristic
498C2×C4#58512yes—
499C8—44no—
500C8—44no—
501C22—22no—
502C22—24no—
503C22—41no—
504C22—41no—
505C22—42no—
506C22—82no—
507C22—82no—
508C22—82no—
509C22—82no—
510C22—82no—
511C22—82no—
512C22—82no—
513C22C22×#12411yescharacteristic
514C22C2×#53711yescharacteristic
515C22#344911yescharacteristic
516C22#358512yes—
517C22#358612yes—
518C4—22no—
519C4—24no—
520C4—24no—
521C4—24no—
522C4—41no—
523C4—41no—
524C4—41no—
525C4—41no—
526C4—42no—
527C4—84no—
528C4—82no—
529C4—82no—
530C4—82no—
531C4—84no—
532C4#350612yes—
533C4#350712yes—
534C2—42no—
535C2—162no—
536C2#3518011yescharacteristic
537C2#6062512yes—
538C2#6091912yes—
539C2#6101911yescharacteristic
540C2#6102011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#540C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #35283C2semidirect
3. #33057C2semidirect
4. #33051C2semidirect
5. #30195C2semidirect
6. #29359C2semidirect
7. #29358C2semidirect
8. #29223C2semidirect
9. #29148C2semidirect
10. #29134C2semidirect
11. #28754C2semidirect
12. #28727C2semidirect
13. #21898C2semidirect
15. #20427C2semidirect

Maximal subgroups

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

subgroupindex
#353322
#352832
#330572
#330512
#301952
#293592
#293582
#292232
#291482
#291342
#287542
#287272
#218982
#218972
#204272