Finite Group Databasebeta

← #165077

Subgroups — #165077

Order 512.

961 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
1#36386C211yesmaximal, characteristic
2#22786C212yesmaximal, complemented
3#7710C212yesmaximal, complemented
4#7707C212yesmaximal, complemented
5#7656C212yesmaximal, complemented
6#7649C211yesmaximal, characteristic, complemented
7#7631C211yesmaximal, characteristic, complemented
8#7626C211yesmaximal, characteristic, complemented
9#7597C211yesmaximal, characteristic, complemented
10#7592C211yesmaximal, characteristic, complemented
11#7567C211yesmaximal, characteristic, complemented
12C2×#552C411yescharacteristic
13C2×#546C411yescharacteristic
14#2861C2211yescharacteristic
15#2857C2211yescharacteristic
16#2850C2211yescharacteristic
17#2566C411yescharacteristic
18#2559—22no—
19#2558—22no—
20#2545C411yescharacteristic
21#2534—22no—
22#2503C411yescharacteristic
23#2500C411yescharacteristic
24#2435C411yescharacteristic
25#2432C411yescharacteristic
26#2342—22no—
27C4×#124C2211yescharacteristic
28#2297C2211yescharacteristic
29C2×#390C2211yescharacteristic
30C42×D4C2211yescharacteristic
31#1923—22no—
32#1917—22no—
33#1870—22no—
34#1839C2212yescomplemented
35#1817C2212yes—
36#1797C2212yescomplemented
37#1753—22no—
38#1749C2212yes—
39#1744C2212yescomplemented
40#1728C2212yescomplemented
41#1272C2212yescomplemented
42#1272C2212yescomplemented
43#1272C2212yescomplemented
44#1272C2212yescomplemented
45#1270C2212yes—
46#1270C2212yes—
47#1270C2211yescharacteristic
48#1270C2211yescharacteristic
49#1270C2211yescharacteristic
50#1270C2211yescharacteristic
51C22×#41C2×C411yescharacteristic
52C23×D4C2×C411yescharacteristic
53C22×M16—22no—
54#552—44no—
55#546—44no—
56C2×C4⋊Q8C2×C411yescharacteristic
57C2×#126—21no—
58C2×#125—21no—
59C2×#125C2×C411yescharacteristic
60C2×#124C2×C411yescharacteristic
61C2×#124C2×C411yescharacteristic
62C2×#124C2×C411yescharacteristic
63C2×#123—21no—
64C2×#123—21no—
65C2×C22⋊Q8C2×C411yescharacteristic
66C2×C22⋊Q8C2×C411yescharacteristic
67C2×C22⋊Q8C2×C411yescharacteristic
68C2×#121—21no—
69C2×#121C2×C411yescharacteristic
70C2×#120—21no—
71C2×C4×Q8C2×C411yescharacteristic
72C2×C4×D4C2×C411yescharacteristic
73C2×C4×D4C2×C411yescharacteristic
74C2×C4×D4D412yes—
75C2×#117—21no—
76C2×#117C2×C411yescharacteristic
77C22×C4⋊C4C2×C411yescharacteristic
78C22×C22⋊C4—22no—
79C22×C22⋊C4—21no—
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C2×C4⋊C8—22no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×#97—22no—
87C2×C4×C8—22no—
88#400—22no—
89#396—22no—
90#396—22no—
91#394—22no—
92#394—22no—
93#394—22no—
94#391C2×C411yescharacteristic
95#391C2×C411yescharacteristic
96#391C2311yescharacteristic
97#391C2311yescharacteristic
98#390—22no—
99#390C2×C411yescharacteristic
100#390C2×C411yescharacteristic
101#390C2×C411yescharacteristic
102#390C2×C411yescharacteristic
103#390C2311yescharacteristic
104#390C2311yescharacteristic
105#389—22no—
106#388—22no—
107#388—22no—
108#388—22no—
109#388—22no—
110#388—22no—
111#388—22no—
112#386—22no—
113#386—22no—
114#385—22no—
115#385—22no—
116#384C2×C411yescharacteristic
117#382C2×C411yescharacteristic
118#382C2×C411yescharacteristic
119#381—22no—
120#380—22no—
121#379—22no—
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4D412yes—
124C4×C22⋊C4—22no—
125C4×C22⋊C4—22no—
126C4×C22⋊C4C2×C411yescharacteristic
127C4×C22⋊C4C2×C411yescharacteristic
128C4×C22⋊C4D412yes—
129C4×C22⋊C4D412yes—
130C43C2311yescharacteristic
131#336—22no—
132#336—22no—
133#336—22no—
134#336—22no—
135#336—22no—
136#336C2312yes—
137#336C2312yes—
138#336C2312yes—
139#336C2312yes—
140C25—21no—
141C2×#41—22no—
142C2×#41—41no—
143C2×#41—42no—
144C2×#41—81no—
145C22×Q8C22×C411yescharacteristic
146C22×Q8C22×C411yescharacteristic
147C22×D4—22no—
148C22×D4—21no—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4—41no—
152C22×D4—42no—
153C22×D4—81no—
154C22×D4C22⋊C411yescharacteristic
155C22×D4C22×C411yescharacteristic
156C22×D4C22×C411yescharacteristic
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4C22×C411yescharacteristic
161C23×C4C22×C411yescharacteristic
162C23×C4C2×D412yes—
163C23×C4C2×D412yes—
164C2×M16—44no—
165C2×M16—44no—
166C2×M16—82no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C4⋊Q8—81no—
172#126—44no—
173#125—42no—
174#125—44no—
175#124—22no—
176#124—41no—
177#124—42no—
178#124—81no—
179#123—44no—
180#123—44no—
181C22⋊Q8—42no—
182C22⋊Q8—42no—
183C22⋊Q8—42no—
184#121—44no—
185#121—42no—
186#120—44no—
187C4×Q8—42no—
188C4×D4—24no—
189C4×D4—42no—
190C4×D4—42no—
191C4×D4—42no—
192#117—44no—
193#117—42no—
194C2×C4⋊C4—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—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4C22⋊C411yescharacteristic
205C2×C4⋊C4C22⋊C412yes—
206C2×C4⋊C4C22×C411yescharacteristic
207C2×C4⋊C4C22×C411yescharacteristic
208C2×C4⋊C4C22×C411yescharacteristic
209C2×C4⋊C4C22×C411yescharacteristic
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4C22⋊C411yescharacteristic
230C2×C22⋊C4C22⋊C412yes—
231C2×C22⋊C4C22⋊C412yes—
232C2×C22⋊C4C22⋊C411yescharacteristic
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—81no—
236C2×C42—81no—
237C2×C42C22×C411yescharacteristic
238C2×C42C22×C411yescharacteristic
239C2×C42C22×C411yescharacteristic
240C2×C42C22×C411yescharacteristic
241C2×C42C2×D412yes—
242C2×C42C2×D412yes—
243C2×C42C2×D412yes—
244C2×C42C2×D412yes—
245C2×C42C2411yescharacteristic
246C4⋊C8—82no—
247C22⋊C8—44no—
248C22⋊C8—44no—
249C22⋊C8—82no—
250#97—82no—
251C4×C8—44no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95C22⋊C412yes—
259#95C22⋊C412yes—
260#95C22⋊C412yes—
261C24—21no—
262C24—21no—
263C24—42no—
264C24—44no—
265C24—42no—
266C24—81no—
267C24—82no—
268C24C2×C22⋊C411yescharacteristic
269C24C2×C22⋊C411yescharacteristic
270#41—84no—
271#41—84no—
272C2×Q8—21no—
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8—21no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×D4—22no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—42no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—42no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—81no—
295C2×D4—84no—
296C2×D4—84no—
297C2×D4—81no—
298C2×D4—82no—
299C2×D4#13112yes—
300C22×C4—22no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—42no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—44no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—81no—
328C22×C4—82no—
329C22×C4C2×C22⋊C412yes—
330C22×C4C2×C22⋊C412yes—
331C22×C4C2×C22⋊C412yes—
332C22×C4C2×C22⋊C411yescharacteristic
333C22×C4C2×C22⋊C411yescharacteristic
334C22×C4C2×C22⋊C411yescharacteristic
335C22×C4C2×C22⋊C411yescharacteristic
336C22×C4C23×C411yescharacteristic
337C22×C4C22×D412yes—
338M16—88no—
339C2×C8—24no—
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—
349C2×C8—42no—
350C2×C8—42no—
351C4⋊C4—22no—
352C4⋊C4—42no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—82no—
365C4⋊C4—82no—
366C22⋊C4—22no—
367C22⋊C4—42no—
368C22⋊C4—42no—
369C22⋊C4—44no—
370C22⋊C4—44no—
371C22⋊C4—44no—
372C22⋊C4—42no—
373C22⋊C4—42no—
374C22⋊C4—42no—
375C22⋊C4—42no—
376C22⋊C4—42no—
377C22⋊C4—42no—
378C22⋊C4—42no—
379C22⋊C4—42no—
380C22⋊C4—42no—
381C22⋊C4—42no—
382C22⋊C4—42no—
383C22⋊C4—42no—
384C22⋊C4—82no—
385C22⋊C4—82no—
386C22⋊C4—82no—
387C22⋊C4#10014yes—
388C42—21no—
389C42—21no—
390C42—22no—
391C42—22no—
392C42—22no—
393C42—42no—
394C42—42no—
395C42—42no—
396C42—41no—
397C42—41no—
398C42—41no—
399C42—42no—
400C42—42no—
401C23—21no—
402C23—21no—
403C23—22no—
404C23—22no—
405C23—22no—
406C23—21no—
407C23—21no—
408C23—21no—
409C23—22no—
410C23—22no—
411C23—41no—
412C23—41no—
413C23—42no—
414C23—42no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23—42no—
419C23—42no—
420C23—84no—
421C23—84no—
422C23—88no—
423C23—162no—
424C23—164no—
425C23C2×#10012yes—
426C23C22×C22⋊C411yescharacteristic
427Q8—81no—
428Q8—81no—
429Q8—81no—
430Q8—81no—
431D4—42no—
432D4—42no—
433D4—44no—
434D4—81no—
435D4—81no—
436D4—84no—
437D4—84no—
438D4—162no—
439C2×C4—21no—
440C2×C4—22no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—21no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—84no—
483C2×C4—84no—
484C2×C4—162no—
485C2×C4C2×#10012yes—
486C2×C4C2×#10012yes—
487C2×C4C2×#13111yescharacteristic
488C2×C4#56811yescharacteristic
489C2×C4#57512yes—
490C2×C4#57512yes—
491C8—44no—
492C8—82no—
493C8—82no—
494C8—82no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—42no—
499C22—42no—
500C22—41no—
501C22—42no—
502C22—41no—
503C22—41no—
504C22—42no—
505C22—44no—
506C22—42no—
507C22—81no—
508C22—81no—
509C22—82no—
510C22—81no—
511C22—81no—
512C22—81no—
513C22—82no—
514C22—82no—
515C22—168no—
516C22#288011yescharacteristic
517C22#288311yescharacteristic
518C22C22×#10011yescharacteristic
519C22#289612yes—
520C22#290112yes—
521C4—21no—
522C4—21no—
523C4—22no—
524C4—22no—
525C4—22no—
526C4—22no—
527C4—41no—
528C4—41no—
529C4—42no—
530C4—42no—
531C4—42no—
532C4—81no—
533C4—81no—
534C4—81no—
535C4—81no—
536C4—81no—
537C4—81no—
538C2—42no—
539C2—81no—
540C2—82no—
541C2—161no—
542C2#1742111yescharacteristic
543C2#1751312yes—
544C2#1762312yes—
545C2#1768511yescharacteristic
546C2#1773811yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#546C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22786C2semidirect
3. #7710C2semidirect
4. #7707C2semidirect
5. #7656C2semidirect
6. #7649C2semidirect
7. #7631C2semidirect
8. #7626C2semidirect
9. #7597C2semidirect
10. #7592C2semidirect
11. #7567C2semidirect
34. #1839C22semidirect
36. #1797C22semidirect
39. #1744C22semidirect
40. #1728C22semidirect
41. #1272C22semidirect
42. #1272C22semidirect
43. #1272C22semidirect
44. #1272C22semidirect

Maximal subgroups

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

subgroupindex
#363862
#227862
#77102
#77072
#76562
#76492
#76312
#76262
#75972
#75922
#75672