Finite Group Databasebeta

← #163922

Subgroups — #163922

Order 512.

1128 conjugacy classes of subgroups.

The subgroup lattice has 561 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34275C211yesmaximal, characteristic
2#22641C212yesmaximal, complemented
3#7649C211yesmaximal, characteristic, complemented
4#7640C211yesmaximal, characteristic, complemented
5#7635C211yesmaximal, characteristic, complemented
6#7626C211yesmaximal, characteristic, complemented
7#7621C212yesmaximal, complemented
8#7612C212yesmaximal, complemented
9#7603C211yesmaximal, characteristic, complemented
10#7584C212yesmaximal, complemented
11#7569C211yesmaximal, characteristic, complemented
12C2×#535C411yescharacteristic
13C2×#520C411yescharacteristic
14C2×#518C411yescharacteristic
15#2363—22no—
16#2361—22no—
17#2358—22no—
18#2358—22no—
19#2328C411yescharacteristic
20#2328C411yescharacteristic
21#2326C411yescharacteristic
22#2326C411yescharacteristic
23#2317C411yescharacteristic
24C4×#127C2211yescharacteristic
25C4×#127C2211yescharacteristic
26C4×#124C2211yescharacteristic
27C4×#124C2211yescharacteristic
28#2297C2211yescharacteristic
29C2×#390C2211yescharacteristic
30C4×#117C2211yescharacteristic
31#1940—22no—
32#1924—22no—
33#1923—22no—
34#1839C2212yescomplemented
35#1839C2212yescomplemented
36#1797C2212yes—
37#1771—22no—
38#1767C2212yescomplemented
39C4×C4⋊C8C2212yescomplemented
40#1749C2212yes—
41#1272C2212yescomplemented
42#1272C2212yescomplemented
43#1270C2211yescharacteristic
44#1270C2211yescharacteristic
45#1270C2212yes—
46#1270C2212yes—
47#1270C2211yescharacteristic
48#1270C2211yescharacteristic
49#1270C2212yescomplemented
50#1270C2212yescomplemented
51C22×#41C2×C411yescharacteristic
52C23×D4C2×C411yescharacteristic
53#535—24no—
54#535—24no—
55C2×#127C2×C411yescharacteristic
56C2×#127C2×C411yescharacteristic
57C2×#124C2×C411yescharacteristic
58C2×#124C2×C411yescharacteristic
59C2×#121—22no—
60C2×#120—22no—
61#520—44no—
62#518—44no—
63C2×C4×Q8C2×C411yescharacteristic
64C2×C4×Q8C2×C411yescharacteristic
65C2×C4×D4—21no—
66C2×C4×D4—21no—
67C2×C4×D4—21no—
68C2×C4×D4—21no—
69C2×C4×D4—22no—
70C2×C4×D4—22no—
71C2×C4×D4C2×C411yescharacteristic
72C2×C4×D4C2×C411yescharacteristic
73C2×C4×D4C2×C411yescharacteristic
74C2×C4×D4C2×C411yescharacteristic
75C2×C4×D4C2×C411yescharacteristic
76C2×C4×D4C2×C411yescharacteristic
77C2×#117—21no—
78C2×#117C2×C411yescharacteristic
79C2×#117C2×C411yescharacteristic
80C2×#117C2×C411yescharacteristic
81C22×C4⋊C4—21no—
82C22×C4⋊C4C2×C411yescharacteristic
83C22×C22⋊C4—22no—
84C22×C22⋊C4—21no—
85C22×C22⋊C4—21no—
86C22×C42C2311yescharacteristic
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×#97—22no—
93C2×#97—22no—
94C2×C4×C8—22no—
95#391C2×C411yescharacteristic
96#391C2311yescharacteristic
97#390—22no—
98#390—22no—
99#390—22no—
100#390C2×C411yescharacteristic
101#390C2×C411yescharacteristic
102#390C2×C411yescharacteristic
103#390C2311yescharacteristic
104#390C2311yescharacteristic
105#390C2311yescharacteristic
106#389—22no—
107#388—22no—
108#388—22no—
109#388—22no—
110#388—22no—
111#386—22no—
112#386—22no—
113#386—22no—
114#386—22no—
115#386—22no—
116#385—22no—
117#385—22no—
118#385—22no—
119#384C2×C411yescharacteristic
120#384C2×C411yescharacteristic
121#382C2×C411yescharacteristic
122#382C2×C411yescharacteristic
123C4×C4⋊C4D412yes—
124C4×C4⋊C4D412yes—
125C4×C22⋊C4—22no—
126C4×C22⋊C4—22no—
127C4×C22⋊C4—22no—
128C4×C22⋊C4D412yes—
129C4×C22⋊C4D412yes—
130#376C2×C411yescharacteristic
131#376C2×C411yescharacteristic
132C43C2311yescharacteristic
133C43C2311yescharacteristic
134#336—22no—
135#336—22no—
136#336—22no—
137#336—22no—
138#336C2312yes—
139#336C2312yes—
140#336C2312yes—
141#336C2312yes—
142C25—21no—
143C2×#41—22no—
144C2×#41—22no—
145C2×#41—42no—
146C2×#41—42no—
147C22×Q8C22×C411yescharacteristic
148C22×D4—21no—
149C22×D4—22no—
150C22×D4—22no—
151C22×D4—21no—
152C22×D4—22no—
153C22×D4—22no—
154C22×D4—42no—
155C22×D4—42no—
156C22×D4C22×C411yescharacteristic
157C22×D4C22×C411yescharacteristic
158C22×D4C22×C411yescharacteristic
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4C22×C411yescharacteristic
168C23×C4C2×D412yes—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173#127—22no—
174#127—22no—
175#127—22no—
176#127M1614yes—
177#124—22no—
178#124—22no—
179#124—22no—
180#124M1614yes—
181#121—216no—
182#120—216no—
183C4×Q8—42no—
184C4×Q8—42no—
185C4×D4—42no—
186C4×D4—42no—
187C4×D4—42no—
188C4×D4—42no—
189C4×D4—42no—
190C4×D4—42no—
191C4×D4—44no—
192C4×D4—44no—
193C4×D4—44no—
194C4×D4—44no—
195C4×D4—82no—
196C4×D4—82no—
197C4×D4—82no—
198C4×D4—82no—
199#117—24no—
200#117—22no—
201#117—22no—
202#117—44no—
203#117—42no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4C22⋊C411yescharacteristic
212C2×C4⋊C4C22⋊C411yescharacteristic
213C2×C4⋊C4C22⋊C412yes—
214C2×C4⋊C4C22×C411yescharacteristic
215C2×C4⋊C4C22×C411yescharacteristic
216C2×C4⋊C4C22×C411yescharacteristic
217C2×C4⋊C4C22×C411yescharacteristic
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—82no—
236C2×C22⋊C4—82no—
237C2×C22⋊C4C22⋊C411yescharacteristic
238C2×C22⋊C4C22⋊C412yes—
239C2×C22⋊C4C22⋊C411yescharacteristic
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—42no—
244C2×C42C22×C411yescharacteristic
245C2×C42C22×C411yescharacteristic
246C2×C42C22×C411yescharacteristic
247C2×C42C22×C411yescharacteristic
248C2×C42C22×C411yescharacteristic
249C2×C42C2×D412yes—
250C2×C42C2×D412yes—
251C2×C42C2×D412yes—
252C2×C42C2×D412yes—
253C2×C42C2×D412yes—
254C2×C42C2411yescharacteristic
255C4⋊C8—44no—
256C4⋊C8—44no—
257C4⋊C8—44no—
258C22⋊C8—44no—
259C22⋊C8—82no—
260#97—44no—
261#97—82no—
262C4×C8—82no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95C22⋊C412yes—
268#95C22⋊C412yes—
269#95C22⋊C412yes—
270#95C22⋊C412yes—
271C24—24no—
272C24—21no—
273C24—21no—
274C24—21no—
275C24—42no—
276C24—42no—
277C24—82no—
278C24—82no—
279C24C2×C22⋊C411yescharacteristic
280#41—48no—
281#41—48no—
282C2×Q8—21no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×Q8C2×M1612yes—
286C2×D4—24no—
287C2×D4—24no—
288C2×D4—28no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—28no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—42no—
300C2×D4—48no—
301C2×D4—48no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—82no—
306C2×D4—82no—
307C2×D4—82no—
308C2×D4—82no—
309C2×D4C2×M1612yes—
310C2×D4C2×M1612yes—
311C2×D4C2×M1612yes—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—82no—
336C22×C4—82no—
337C22×C4—82no—
338C22×C4—82no—
339C22×C4—82no—
340C22×C4—82no—
341C22×C4—82no—
342C22×C4C2×C22⋊C412yes—
343C22×C4C2×C22⋊C412yes—
344C22×C4C2×C22⋊C411yescharacteristic
345C22×C4C2×C22⋊C411yescharacteristic
346C22×C4C2×C22⋊C412yes—
347C22×C4C2×C22⋊C411yescharacteristic
348C22×C4C2×C22⋊C411yescharacteristic
349C22×C4C2×C22⋊C411yescharacteristic
350C22×C4C23×C411yescharacteristic
351C22×C4C22×D412yes—
352C2×C8—24no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C4⋊C4—24no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—82no—
375C4⋊C4—82no—
376C22⋊C4—28no—
377C22⋊C4—22no—
378C22⋊C4—28no—
379C22⋊C4—24no—
380C22⋊C4—42no—
381C22⋊C4—42no—
382C22⋊C4—42no—
383C22⋊C4—42no—
384C22⋊C4—42no—
385C22⋊C4—42no—
386C22⋊C4—42no—
387C22⋊C4—42no—
388C22⋊C4—42no—
389C22⋊C4—42no—
390C22⋊C4—42no—
391C22⋊C4—42no—
392C22⋊C4—42no—
393C22⋊C4—82no—
394C22⋊C4—82no—
395C22⋊C4—82no—
396C22⋊C4#10014yes—
397C42—21no—
398C42—21no—
399C42—22no—
400C42—22no—
401C42—22no—
402C42—22no—
403C42—24no—
404C42—22no—
405C42—22no—
406C42—42no—
407C42—42no—
408C42—41no—
409C42—41no—
410C42—42no—
411C42—42no—
412C42—42no—
413C42C2×M1612yes—
414C42C2×M1612yes—
415C23—21no—
416C23—21no—
417C23—22no—
418C23—22no—
419C23—22no—
420C23—22no—
421C23—22no—
422C23—48no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—88no—
436C23—162no—
437C23—162no—
438C23—164no—
439C23C2×#10012yes—
440C23C22×C22⋊C411yescharacteristic
441Q8—42no—
442Q8—42no—
443D4—42no—
444D4—42no—
445D4—42no—
446D4—48no—
447D4—42no—
448D4—42no—
449D4—42no—
450D4—48no—
451D4—88no—
452D4—88no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—21no—
456C2×C4—22no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—48no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—162no—
490C2×C4—162no—
491C2×C4—162no—
492C2×C4—162no—
493C2×C4—162no—
494C2×C4—162no—
495C2×C4—162no—
496C2×C4#40714yes—
497C2×C4#41012yes—
498C2×C4C2×#10012yes—
499C2×C4C2×#10012yes—
500C2×C4C22×M1611yescharacteristic
501C2×C4C22×M1611yescharacteristic
502C2×C4#57612yes—
503C2×C4#57712yes—
504C8—44no—
505C8—82no—
506C8—82no—
507C8—82no—
508C22—22no—
509C22—22no—
510C22—22no—
511C22—42no—
512C22—42no—
513C22—44no—
514C22—44no—
515C22—44no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—81no—
520C22—81no—
521C22—81no—
522C22—81no—
523C22—81no—
524C22—81no—
525C22—82no—
526C22—82no—
527C22—82no—
528C22—168no—
529C22C2×#40711yescharacteristic
530C22C2×#41011yescharacteristic
531C22C22×#10011yescharacteristic
532C22#289712yes—
533C22#290112yes—
534C4—22no—
535C4—24no—
536C4—22no—
537C4—22no—
538C4—22no—
539C4—22no—
540C4—21no—
541C4—21no—
542C4—41no—
543C4—42no—
544C4—41no—
545C4—42no—
546C4—42no—
547C4—81no—
548C4—81no—
549C4—81no—
550C4—81no—
551C2—42no—
552C2—82no—
553C2—82no—
554C2—82no—
555C2#1736411yescharacteristic
556C2#1745312yes—
557C2#1760712yes—
558C2#1768511yescharacteristic
559C2#1777011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#535C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22641C2semidirect
3. #7649C2semidirect
4. #7640C2semidirect
5. #7635C2semidirect
6. #7626C2semidirect
7. #7621C2semidirect
8. #7612C2semidirect
9. #7603C2semidirect
10. #7584C2semidirect
11. #7569C2semidirect
34. #1839C22semidirect
35. #1839C22semidirect
38. #1767C22semidirect
39. C4×C4⋊C8C22semidirect
41. #1272C22semidirect
42. #1272C22semidirect
49. #1270C22semidirect
50. #1270C22semidirect

Maximal subgroups

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

subgroupindex
#342752
#226412
#76492
#76402
#76352
#76262
#76212
#76122
#76032
#75842
#75692