Finite Group Databasebeta

← #323993

Subgroups — #323993

Order 512.

631 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
1#29845C211yesmaximal, characteristic
2#24838C211yesmaximal, characteristic
3#24760C211yesmaximal, characteristic
4#22098C211yesmaximal, characteristic
5#10654C211yesmaximal, characteristic, complemented
6#10620C211yesmaximal, characteristic, complemented
7#10542C211yesmaximal, characteristic, complemented
8#10272C211yesmaximal, characteristic, complemented
9#9565C211yesmaximal, characteristic, complemented
10#9562C211yesmaximal, characteristic, complemented
11#9220C211yesmaximal, characteristic, complemented
12#9218C211yesmaximal, characteristic, complemented
13#9137C211yesmaximal, characteristic
14#9131C211yesmaximal, characteristic
15#7877C211yesmaximal, characteristic
16C2×#487C2211yescharacteristic
17C2×#475C2211yescharacteristic
18C2×#470—21no—
19C2×#462C2211yescharacteristic
20C2×C8×Q8C2211yescharacteristic
21C2×#431C2211yescharacteristic
22C2×#423C2211yescharacteristic
23C22×C4⋊C8C2211yescharacteristic
24#2857C2211yescharacteristic
25#2857C2211yescharacteristic
26C4×C22⋊Q8C2211yescharacteristic
27#2088—21no—
28#2066—21no—
29#2066—21no—
30#2066—21no—
31#2066—21no—
32#2056—21no—
33#2035—21no—
34#2035—21no—
35#1990C2211yescharacteristic
36#1988C2211yescharacteristic
37#1984C2211yescharacteristic
38#1981C2211yescharacteristic
39#1956C2211yescharacteristic
40#1948C2211yescharacteristic
41#1948C2211yescharacteristic
42#1939—21no—
43#1939C2211yescharacteristic
44#1923—21no—
45#1923—21no—
46#1916—21no—
47#1894—21no—
48#1892—21no—
49#1844C2211yescharacteristic
50#1807—21no—
51#1803C2211yescharacteristic
52#1803C2211yescharacteristic
53#1797C2211yescharacteristic
54C8×C4⋊C4C2211yescharacteristic
55C2×#332C2211yescharacteristic
56C2×#332C2211yescharacteristic
57C2×#332C2211yescharacteristic
58C2×#332C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C22×Q16—21no—
68C22×M16—21no—
69C23×C8—21no—
70C2×C4⋊Q8C2311yescharacteristic
71C2×C4⋊Q8C2311yescharacteristic
72C2×C22⋊Q8—21no—
73C2×C22⋊Q8—21no—
74C2×C22⋊Q8D411yescharacteristic
75C2×C4×Q8C2311yescharacteristic
76C22×C42C2311yescharacteristic
77#487—41no—
78#487—41no—
79#475—22no—
80#475—41no—
81#470—44no—
82#462—22no—
83#462—41no—
84C8×Q8—41no—
85C8×Q8—41no—
86#431—41no—
87#431—41no—
88C2×#106—21no—
89C2×#106—21no—
90C2×#106—21no—
91C2×#106—21no—
92C2×#106—21no—
93#423—22no—
94#423—41no—
95C2×C4⋊C8—22no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—41no—
101C2×C4⋊C8D411yescharacteristic
102C2×C4⋊C8D411yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×C4⋊C8C2311yescharacteristic
107C2×C4⋊C8C2311yescharacteristic
108C2×C4⋊C8C2311yescharacteristic
109C2×#103—21no—
110C2×#103—21no—
111C2×#103—21no—
112C2×#103—21no—
113C2×#103—21no—
114C2×#103—21no—
115C2×#103—21no—
116C2×#103—21no—
117C2×#103—21no—
118C2×#103—21no—
119C2×C22⋊C8—21no—
120C2×C22⋊C8D411yescharacteristic
121C2×C22⋊C8D411yescharacteristic
122C2×C4×C8—21no—
123C2×C4×C8—21no—
124C2×C4×C8C2311yescharacteristic
125C2×C4×C8C2311yescharacteristic
126#391—21no—
127#391—21no—
128#391D411yescharacteristic
129#389—21no—
130#389—21no—
131#389—21no—
132#389—21no—
133#389D411yescharacteristic
134#387—21no—
135#385—21no—
136#384C2311yescharacteristic
137#384C2311yescharacteristic
138#380—21no—
139#380—21no—
140#380—21no—
141#380—21no—
142#380D411yescharacteristic
143C4×C4⋊C4—21no—
144C4×C4⋊C4C2311yescharacteristic
145C4×C22⋊C4—21no—
146#340—21no—
147#340—21no—
148#340—21no—
149#340—21no—
150#340—21no—
151#340—21no—
152#340—21no—
153#340—21no—
154#336—21no—
155#336—21no—
156#336—21no—
157#336—21no—
158#336—21no—
159#332—22no—
160#332—22no—
161#332—41no—
162#332—41no—
163C22×Q8—21no—
164C22×Q8—21no—
165C22×Q8C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4#4111yescharacteristic
169C2×Q16—42no—
170C2×Q16—42no—
171C2×Q16—81no—
172C2×M16—42no—
173C2×M16—42no—
174C2×M16—81no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—42no—
182C22×C8—42no—
183C22×C8—81no—
184C22×C8C2×D411yescharacteristic
185C22×C8C2×D411yescharacteristic
186C22×C8C2×D411yescharacteristic
187C22×C8C2×D411yescharacteristic
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4⋊Q8—21no—
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196C22⋊Q8—22no—
197C22⋊Q8—42no—
198C22⋊Q8—42no—
199C22⋊Q8—81no—
200C22⋊Q8—81no—
201C22⋊Q8SD1614yes—
202C4×Q8—21no—
203C4×Q8—21no—
204C4×Q8—21no—
205C4×Q8—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4C2×D411yescharacteristic
222C2×C4⋊C4C2×D411yescharacteristic
223C2×C4⋊C4C2×D411yescharacteristic
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—41no—
227C2×C22⋊C4—41no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42#4111yescharacteristic
237C2×C42C2411yescharacteristic
238#106—42no—
239#106—42no—
240#106—81no—
241#106—81no—
242#106—81no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8—41no—
254C4⋊C8—41no—
255C4⋊C8—42no—
256C4⋊C8—42no—
257C4⋊C8—42no—
258C4⋊C8—81no—
259C4⋊C8#4112yes—
260C4⋊C8#4112yes—
261#103—42no—
262#103—42no—
263#103—42no—
264#103—42no—
265#103—42no—
266#103—42no—
267#103—81no—
268#103—81no—
269#103—81no—
270#103—81no—
271C22⋊C8—41no—
272C22⋊C8—41no—
273C22⋊C8—81no—
274C4×C8—21no—
275C4×C8—21no—
276C4×C8—21no—
277C4×C8—21no—
278C4×C8—42no—
279C4×C8—81no—
280#95—21no—
281#95—21no—
282#95—21no—
283#95—21no—
284#95—21no—
285#95—41no—
286#95—41no—
287#95—41no—
288#95—41no—
289C24#12111yescharacteristic
290C2×Q8—21no—
291C2×Q8—21no—
292C2×Q8—21no—
293C2×Q8—21no—
294C2×Q8—21no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8C2×SD1612yes—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4#12111yescharacteristic
331C22×C4#12111yescharacteristic
332C22×C4#12111yescharacteristic
333C22×C4C2×SD1612yes—
334C22×C4#13512yes—
335C22×C4C22×D411yescharacteristic
336C22×C4C22×D411yescharacteristic
337C22×C4C2×#4111yescharacteristic
338Q16—84no—
339M16—84no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—22no—
345C2×C8—22no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—84no—
371C4⋊C4—21no—
372C4⋊C4—24no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—81no—
396C4⋊C4—81no—
397C4⋊C4—81no—
398C4⋊C4—82no—
399C4⋊C4—82no—
400C4⋊C4C2×SD1612yes—
401C22⋊C4—24no—
402C22⋊C4—81no—
403C22⋊C4—82no—
404C22⋊C4—82no—
405C42—21no—
406C42—21no—
407C42—21no—
408C42—21no—
409C42—21no—
410C42—21no—
411C42—41no—
412C42—41no—
413C42—41no—
414C42—41no—
415C42C2×#4111yescharacteristic
416C42C2×#4111yescharacteristic
417C42H(5,𝔽2)11yescharacteristic
418C42H(5,𝔽2)11yescharacteristic
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23#46512yes—
426C23C2×#12111yescharacteristic
427Q8—22no—
428Q8—22no—
429Q8—41no—
430Q8—41no—
431Q8—81no—
432Q8—81no—
433Q8—81no—
434Q8—81no—
435Q8—81no—
436Q8—81no—
437Q8—81no—
438Q8—81no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4#46512yes—
484C2×C4#53211yescharacteristic
485C2×C4#53411yescharacteristic
486C2×C4#54611yescharacteristic
487C2×C4#54611yescharacteristic
488C2×C4#55311yescharacteristic
489C2×C4#55311yescharacteristic
490C2×C4C22×SD1611yescharacteristic
491C2×C4C2×#13511yescharacteristic
492C2×C4#57611yescharacteristic
493C2×C4#57711yescharacteristic
494C2×C4#57711yescharacteristic
495C2×C4#57711yescharacteristic
496C2×C4#57811yescharacteristic
497C2×C4#57811yescharacteristic
498C8—41no—
499C8—41no—
500C8—41no—
501C8—41no—
502C8—42no—
503C8—42no—
504C8—81no—
505C8—81no—
506C8—81no—
507C8—81no—
508C22—22no—
509C22—22no—
510C22—22no—
511C22—41no—
512C22—41no—
513C22—41no—
514C22—41no—
515C22#254311yescharacteristic
516C22#304311yescharacteristic
517C22#304611yescharacteristic
518C22C2×#46511yescharacteristic
519C22#306111yescharacteristic
520C22#306911yescharacteristic
521C22#306911yescharacteristic
522C4—22no—
523C4—21no—
524C4—21no—
525C4—42no—
526C4—41no—
527C4—41no—
528C4—42no—
529C4—81no—
530C4—81no—
531C4—161no—
532C4—161no—
533C4—161no—
534C4—161no—
535C4#317311yescharacteristic
536C4#317811yescharacteristic
537C4#320111yescharacteristic
538C4#320411yescharacteristic
539C4#329511yescharacteristic
540C4#329711yescharacteristic
541C4#329811yescharacteristic
542C4#330011yescharacteristic
543C2—42no—
544C2#1939711yescharacteristic
545C2#1941511yescharacteristic
546C2#1952711yescharacteristic
547C2#1955011yescharacteristic
548C2#2855511yescharacteristic
549C2#2858911yescharacteristic
550C2#2859411yescharacteristic
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. #10654C2semidirect
6. #10620C2semidirect
7. #10542C2semidirect
8. #10272C2semidirect
9. #9565C2semidirect
10. #9562C2semidirect
11. #9220C2semidirect
12. #9218C2semidirect

Maximal subgroups

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

subgroupindex
#298452
#248382
#247602
#220982
#106542
#106202
#105422
#102722
#95652
#95622
#92202
#92182
#91372
#91312
#78772