Finite Group Databasebeta

← #165129

Subgroups — #165129

Order 512.

1008 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#36396C211yesmaximal, characteristic
2#22698C212yesmaximal, complemented
3#7710C212yesmaximal, complemented
4#7696C212yesmaximal, complemented
5#7664C212yesmaximal, complemented
6#7643C211yesmaximal, characteristic, complemented
7#7635C211yesmaximal, characteristic, complemented
8#7631C211yesmaximal, characteristic, complemented
9#7601C211yesmaximal, characteristic, complemented
10#7597C211yesmaximal, characteristic, complemented
11C2×#1272C211yesmaximal, characteristic, complemented
12C2×#553C411yescharacteristic
13C2×#548C411yescharacteristic
14C2×#435—22no—
15#2861C2211yescharacteristic
16#2856C2211yescharacteristic
17#2850C2211yescharacteristic
18#2565C411yescharacteristic
19#2560—22no—
20#2557—22no—
21#2544C411yescharacteristic
22#2543—22no—
23#2504C411yescharacteristic
24#2499C411yescharacteristic
25#2436C411yescharacteristic
26#2431C411yescharacteristic
27#2364—22no—
28C4×#124C2211yescharacteristic
29#2297C2211yescharacteristic
30C2×#391C2211yescharacteristic
31C42×D4C2211yescharacteristic
32#1940—22no—
33#1851—22no—
34#1850—22no—
35#1831C2212yes—
36#1797C2212yes—
37#1797C2212yes—
38#1767C2212yes—
39#1728C2212yes—
40C2×#336C2212yes—
41#1272C2212yes—
42#1272C2212yes—
43#1272C2211yescharacteristic
44#1272C2211yescharacteristic
45#1272C2211yescharacteristic
46#1272C2211yescharacteristic
47#1270C2212yes—
48#1270C2212yes—
49#1270C2212yes—
50#1270C2212yes—
51C22×#41C2×C411yescharacteristic
52C23×Q8C2×C411yescharacteristic
53C22×M16—22no—
54C23×C8—22no—
55#553—44no—
56#548—44no—
57C2×#127C2×C411yescharacteristic
58C2×#126—21no—
59C2×#125C2×C411yescharacteristic
60C2×#124—21no—
61C2×#124C2×C411yescharacteristic
62C2×#124C2×C411yescharacteristic
63C2×#124C2×C411yescharacteristic
64C2×#123—21no—
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8C2×C411yescharacteristic
67C2×#121—21no—
68C2×#121—21no—
69C2×#121—22no—
70C2×#121C2×C411yescharacteristic
71C2×#121C2×C411yescharacteristic
72C2×#121C2×C411yescharacteristic
73C2×C4×Q8C2×C411yescharacteristic
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76C2×C4×D4C2×C411yescharacteristic
77C2×C4×D4C2×C411yescharacteristic
78C2×C4×D4D412yes—
79C2×#117C2×C411yescharacteristic
80C22×C4⋊C4C2×C411yescharacteristic
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83#435—48no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×#97—22no—
91#398—22no—
92#397—22no—
93#397—22no—
94#394—22no—
95#393—22no—
96#392—22no—
97#391—22no—
98#391C2×C411yescharacteristic
99#391C2×C411yescharacteristic
100#391C2×C411yescharacteristic
101#391C2×C411yescharacteristic
102#391C2311yescharacteristic
103#391C2311yescharacteristic
104#390—22no—
105#390C2×C411yescharacteristic
106#390C2×C411yescharacteristic
107#390C2311yescharacteristic
108#390C2311yescharacteristic
109#388—22no—
110#388—22no—
111#388—22no—
112#388—22no—
113#387—22no—
114#387—22no—
115#386—22no—
116#386—22no—
117#386—22no—
118#386—22no—
119#386—22no—
120#386—22no—
121#384C2×C411yescharacteristic
122#382C2×C411yescharacteristic
123#382C2×C411yescharacteristic
124C4×C4⋊C4C2×C411yescharacteristic
125C4×C4⋊C4D412yes—
126C4×C22⋊C4—22no—
127C4×C22⋊C4—22no—
128C4×C22⋊C4C2×C411yescharacteristic
129C4×C22⋊C4C2×C411yescharacteristic
130C4×C22⋊C4D412yes—
131C4×C22⋊C4D412yes—
132C2×#95—22no—
133C43C2311yescharacteristic
134#336—22no—
135#336—22no—
136#336—22no—
137#336C2312yes—
138#336C2312yes—
139#336C2312yes—
140#336C2312yes—
141C2×#41—22no—
142C2×#41—41no—
143C2×#41—42no—
144C2×#41—81no—
145C22×Q8—21no—
146C22×Q8—22no—
147C22×Q8—41no—
148C22×Q8—42no—
149C22×Q8—81no—
150C22×Q8C22×C411yescharacteristic
151C22×Q8C22×C411yescharacteristic
152C22×D4—21no—
153C22×D4—21no—
154C22×D4—21no—
155C22×D4—21no—
156C22×D4—22no—
157C22×D4C22⋊C411yescharacteristic
158C22×D4C22×C411yescharacteristic
159C22×D4C22×C411yescharacteristic
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4C22×C411yescharacteristic
165C23×C4C22×C411yescharacteristic
166C23×C4C2×D412yes—
167C23×C4C2×D412yes—
168C2×M16—44no—
169C2×M16—44no—
170C2×M16—82no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—44no—
176C22×C8—44no—
177C22×C8—82no—
178#127—81no—
179#126—44no—
180#125—42no—
181#124—22no—
182#124—41no—
183#124—42no—
184#124—44no—
185#124—81no—
186#123—44no—
187C22⋊Q8—44no—
188C22⋊Q8—42no—
189#121—42no—
190#121—42no—
191#121—44no—
192#121—44no—
193#121—44no—
194#121—44no—
195#121—42no—
196C4×Q8—42no—
197C4×D4—24no—
198C4×D4—24no—
199C4×D4—24no—
200C4×D4—42no—
201C4×D4—44no—
202C4×D4—44no—
203#117—42no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4C22⋊C411yescharacteristic
212C2×C4⋊C4C22⋊C412yes—
213C2×C4⋊C4C22×C411yescharacteristic
214C2×C4⋊C4C22×C411yescharacteristic
215C2×C4⋊C4C22×C411yescharacteristic
216C2×C4⋊C4C22×C411yescharacteristic
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4C22⋊C411yescharacteristic
232C2×C22⋊C4C22⋊C412yes—
233C2×C22⋊C4C22⋊C412yes—
234C2×C22⋊C4C22⋊C411yescharacteristic
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—81no—
238C2×C42—81no—
239C2×C42C22×C411yescharacteristic
240C2×C42C22×C411yescharacteristic
241C2×C42C22×C411yescharacteristic
242C2×C42C22×C411yescharacteristic
243C2×C42C2×D412yes—
244C2×C42C2×D412yes—
245C2×C42C2×D412yes—
246C2×C42C2×D412yes—
247C2×C42C2411yescharacteristic
248C4⋊C8—44no—
249C4⋊C8—44no—
250C4⋊C8—82no—
251C4⋊C8—82no—
252C22⋊C8—44no—
253C22⋊C8—44no—
254#97—44no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—22no—
259#95—22no—
260#95—22no—
261#95—22no—
262#95—22no—
263#95C22⋊C412yes—
264#95C22⋊C412yes—
265#95C22⋊C412yes—
266C24—21no—
267C24—21no—
268C24C2×C22⋊C411yescharacteristic
269C24C2×C22⋊C411yescharacteristic
270#41—84no—
271#41—84no—
272C2×Q8—21no—
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—81no—
282C2×Q8—84no—
283C2×Q8—84no—
284C2×Q8#13112yes—
285C2×D4—22no—
286C2×D4—24no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—42no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—44no—
300C2×D4—44no—
301C2×D4—42no—
302C2×D4—44no—
303C2×D4—82no—
304C2×D4—82no—
305C2×D4—81no—
306C2×D4—82no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—41no—
324C22×C4—41no—
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—81no—
334C22×C4—82no—
335C22×C4—81no—
336C22×C4—82no—
337C22×C4C2×C22⋊C412yes—
338C22×C4C2×C22⋊C412yes—
339C22×C4C2×C22⋊C411yescharacteristic
340C22×C4C2×C22⋊C411yescharacteristic
341C22×C4C2×C22⋊C411yescharacteristic
342C22×C4C2×C22⋊C411yescharacteristic
343C22×C4C2×C22⋊C412yes—
344C22×C4AGL(1,ℤ/8)14yes—
345C22×C4#13714yes—
346C22×C4C23×C411yescharacteristic
347C22×C4C22×D412yes—
348M16—88no—
349C2×C8—24no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
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—88no—
362C4⋊C4—22no—
363C4⋊C4—24no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—44no—
374C22⋊C4—24no—
375C22⋊C4—22no—
376C22⋊C4—24no—
377C22⋊C4—44no—
378C22⋊C4—42no—
379C22⋊C4—42no—
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—82no—
391C22⋊C4—82no—
392C22⋊C4—82no—
393C22⋊C4#10014yes—
394C42—21no—
395C42—21no—
396C42—22no—
397C42—22no—
398C42—22no—
399C42—42no—
400C42—42no—
401C42—42no—
402C42—41no—
403C42—41no—
404C42—41no—
405C42—42no—
406C42—42no—
407C23—21no—
408C23—21no—
409C23—22no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—22no—
414C23—22no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23C2×#10012yes—
426C23#41714yes—
427C23C22×C22⋊C411yescharacteristic
428Q8—42no—
429Q8—42no—
430Q8—81no—
431Q8—81no—
432Q8—84no—
433D4—44no—
434D4—81no—
435D4—81no—
436D4—81no—
437D4—81no—
438D4—84no—
439D4—84no—
440D4—84no—
441D4—164no—
442D4—162no—
443C2×C4—21no—
444C2×C4—22no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—22no—
448C2×C4—21no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—21no—
455C2×C4—22no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—84no—
486C2×C4—84no—
487C2×C4—162no—
488C2×C4—162no—
489C2×C4C2×#10012yes—
490C2×C4C2×#10012yes—
491C2×C4#41714yes—
492C2×C4C2×#13111yescharacteristic
493C2×C4#56811yescharacteristic
494C2×C4C2×AGL(1,ℤ/8)12yes—
495C2×C4C2×#13712yes—
496C8—44no—
497C8—82no—
498C8—82no—
499C8—82no—
500C22—22no—
501C22—22no—
502C22—24no—
503C22—24no—
504C22—41no—
505C22—41no—
506C22—41no—
507C22—42no—
508C22—42no—
509C22—81no—
510C22—81no—
511C22—81no—
512C22—81no—
513C22—81no—
514C22—81no—
515C22—82no—
516C22—82no—
517C22—82no—
518C22—82no—
519C22#156014yes—
520C22#288011yescharacteristic
521C22#288311yescharacteristic
522C22C22×#10011yescharacteristic
523C22C2×#41712yes—
524C22#290112yes—
525C4—21no—
526C4—21no—
527C4—22no—
528C4—22no—
529C4—22no—
530C4—22no—
531C4—24no—
532C4—41no—
533C4—42no—
534C4—41no—
535C4—42no—
536C4—42no—
537C4—81no—
538C4—81no—
539C4—81no—
540C4—81no—
541C4—81no—
542C2—24no—
543C2—81no—
544C2—161no—
545C2—161no—
546C2#1742111yescharacteristic
547C2#1751512yes—
548C2#1762212yes—
549C2#1768511yescharacteristic
550C2C2×#156011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#36396C2
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. #22698C2semidirect
3. #7710C2semidirect
4. #7696C2semidirect
5. #7664C2semidirect
6. #7643C2semidirect
7. #7635C2semidirect
8. #7631C2semidirect
9. #7601C2semidirect
10. #7597C2semidirect
11. C2×#1272C2semidirect

Maximal subgroups

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

subgroupindex
#363962
#226982
#77102
#76962
#76642
#76432
#76352
#76312
#76012
#75972
C2×#12722