Finite Group Databasebeta

← #208438

Subgroups — #208438

Order 512.

1066 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35014C211yesmaximal, characteristic
2#21906C211yesmaximal, characteristic
3#13802C211yesmaximal, characteristic
4#9912C211yesmaximal, characteristic, complemented
5#9909C211yesmaximal, characteristic, complemented
6#9714C211yesmaximal, characteristic, complemented
7#9711C211yesmaximal, characteristic, complemented
8#9151C212yesmaximal
9#9143C212yesmaximal
10#8219C212yesmaximal, complemented
11#8217C212yesmaximal, complemented
12#2857C2211yescharacteristic
13#2857C2211yescharacteristic
14#2848C2212yes—
15#2646—21no—
16#2626—21no—
17#2589—22no—
18#2587—22no—
19#2497C412yescomplemented
20#2492C412yescomplemented
21#2429C412yescomplemented
22#2423C412yescomplemented
23#2329—21no—
24#2312—21no—
25C4×C22⋊Q8C2212yes—
26C2×#380C2211yescharacteristic
27C2×C4×C22⋊C4—21no—
28C2×C4×C22⋊C4C2211yescharacteristic
29C22×#95C2211yescharacteristic
30#1924—22no—
31#1924—22no—
32#1797C2211yescharacteristic
33#1797C2211yescharacteristic
34#1779C2212yes—
35#1749C2212yes—
36C2×#328C2212yes—
37C2×#328C2212yes—
38C2×#328C2212yes—
39C2×#328C2212yes—
40#1436—21no—
41#1436—21no—
42#1434C2211yescharacteristic
43#1434C2211yescharacteristic
44C22⋊#95—21no—
45C4×#95C2211yescharacteristic
46C4×#95C2211yescharacteristic
47#1294C2212yes—
48#1294C2212yes—
49#1294C2212yes—
50#1294C2212yes—
51C24×C4—21no—
52C24×C4C2311yescharacteristic
53C2×C4⋊Q8C2×C411yescharacteristic, complemented
54C2×C4⋊Q8C2×C411yescharacteristic, complemented
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8C2×C412yes—
57C2×C22⋊Q8C2×C412yes—
58C2×C22⋊Q8C2312yes—
59C2×C4×Q8C2×C412yescomplemented
60C22×C4⋊C4—21no—
61C22×C22⋊C4—21no—
62C22×C22⋊C4D411yescharacteristic
63C22×C22⋊C4D411yescharacteristic
64C22×C42—21no—
65C22×C42—21no—
66C22×C42C2311yescharacteristic
67C22×C42C2311yescharacteristic
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8—22no—
72C2×C22⋊C8C2312yes—
73C2×C22⋊C8C2312yes—
74C2×#97—22no—
75#400—22no—
76#400—22no—
77#399—22no—
78#398—22no—
79#398—22no—
80#396—22no—
81#396—22no—
82#393—22no—
83#391—21no—
84#391—22no—
85#391C2×C412yescomplemented
86#391C2×C412yescomplemented
87#389—21no—
88#389—21no—
89#389—21no—
90#389—22no—
91#389C2×C412yescomplemented
92#389C2×C412yescomplemented
93#388—22no—
94#388—22no—
95#388—22no—
96#388—22no—
97#387—22no—
98#387—22no—
99#387—22no—
100#385—22no—
101#385—22no—
102#385—22no—
103#385—22no—
104#384C2×C411yescharacteristic, complemented
105#384C2×C411yescharacteristic, complemented
106#380—21no—
107#380—21no—
108#380—21no—
109#380—22no—
110#380—22no—
111#380C2×C412yes—
112#380C2×C412yes—
113#380C2312yes—
114#379—22no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4C2×C412yescomplemented
117C4×C22⋊C4—22no—
118C4×C22⋊C4—44no—
119C4×C22⋊C4C2×C412yes—
120C4×C22⋊C4C2×C412yes—
121C4×C22⋊C4D412yes—
122C4×C22⋊C4D412yes—
123C2×#95—21no—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—21no—
129C2×#95—21no—
130C2×#95—21no—
131C2×#95—21no—
132C2×#95D411yescharacteristic
133C2×#95D411yescharacteristic
134C2×#95C2311yescharacteristic
135C2×#95C2311yescharacteristic
136C2×#95C2311yescharacteristic
137C2×#95C2311yescharacteristic
138#336—21no—
139#336—21no—
140#336—22no—
141#328—24no—
142#328—24no—
143#328—24no—
144#328—24no—
145C25C2×D411yescharacteristic
146C22×Q8—21no—
147C22×Q8C22×C412yes—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—24no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4—81no—
162C23×C4—81no—
163C23×C4—81no—
164C23×C4—81no—
165C23×C4C22×C412yes—
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D412yes—
171C23×C4C2×D411yescharacteristic
172C23×C4C2411yescharacteristic
173C22×C8—22no—
174C22×C8—22no—
175C4⋊Q8—81no—
176C4⋊Q8—81no—
177C22⋊Q8—22no—
178C22⋊Q8—22no—
179C22⋊Q8—22no—
180C22⋊Q8—22no—
181C22⋊Q8—42no—
182C22⋊Q8—42no—
183C22⋊Q8—82no—
184C22⋊Q8—82no—
185C4×Q8—82no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—81no—
201C2×C4⋊C4—81no—
202C2×C4⋊C4C22×C412yes—
203C2×C4⋊C4C22×C412yes—
204C2×C4⋊C4C22×C412yes—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—42no—
217C2×C22⋊C4—42no—
218C2×C22⋊C4C22⋊C412yes—
219C2×C22⋊C4C22⋊C412yes—
220C2×C22⋊C4C22⋊C412yes—
221C2×C22⋊C4C22⋊C412yes—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—42no—
225C2×C42—42no—
226C2×C42—42no—
227C2×C42—42no—
228C2×C42—81no—
229C2×C42—81no—
230C2×C42—81no—
231C2×C42—81no—
232C2×C42C22×C412yes—
233C2×C42C22×C411yescharacteristic
234C2×C42C22×C411yescharacteristic
235C2×C42C2×D412yes—
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C4⋊C8—81no—
239C4⋊C8—81no—
240C22⋊C8—22no—
241C22⋊C8—22no—
242C22⋊C8—22no—
243C22⋊C8—22no—
244C22⋊C8—82no—
245C22⋊C8—82no—
246#97—82no—
247#95—24no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—21no—
262#95—42no—
263#95—42no—
264#95—42no—
265#95—42no—
266#95—81no—
267#95—81no—
268#95C22⋊C412yes—
269#95C22⋊C412yes—
270#95C22⋊C412yes—
271#95C22⋊C412yes—
272#95#4112yes—
273#95#4112yes—
274#95#4112yes—
275#95#4112yes—
276C24—21no—
277C24—21no—
278C24—21no—
279C24—21no—
280C24—41no—
281C24—41no—
282C24—41no—
283C24—41no—
284C24C2×C22⋊C412yes—
285C24#13712yes—
286C24#13712yes—
287C24C22×D411yescharacteristic
288C2×Q8—22no—
289C2×Q8—22no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—81no—
294C2×Q8—81no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—48no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—48no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—82no—
358C22×C4—82no—
359C22×C4—82no—
360C22×C4—82no—
361C22×C4—84no—
362C22×C4—84no—
363C22×C4—84no—
364C22×C4—84no—
365C22×C4—84no—
366C22×C4C2×C22⋊C412yes—
367C22×C4C2×C22⋊C412yes—
368C22×C4C2×C22⋊C411yescharacteristic
369C22×C4C2×C22⋊C411yescharacteristic
370C22×C4C2×C22⋊C412yes—
371C22×C4C2×C22⋊C411yescharacteristic
372C22×C4C2×C22⋊C411yescharacteristic
373C22×C4AGL(1,ℤ/8)12yes—
374C22×C4#13712yes—
375C22×C4C23×C411yescharacteristic
376C22×C4C22×D411yescharacteristic
377C22×C4C2×#4111yescharacteristic
378C22×C4C2×#4111yescharacteristic
379C22×C4C2×#4111yescharacteristic
380C22×C4C2×#4111yescharacteristic
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—44no—
395C4⋊C4—81no—
396C4⋊C4—82no—
397C4⋊C4—81no—
398C4⋊C4—81no—
399C4⋊C4—82no—
400C4⋊C4—82no—
401C4⋊C4—82no—
402C4⋊C4—81no—
403C4⋊C4—81no—
404C4⋊C4—81no—
405C4⋊C4—84no—
406C22⋊C4—24no—
407C22⋊C4—24no—
408C22⋊C4—24no—
409C22⋊C4—24no—
410C22⋊C4—48no—
411C22⋊C4—42no—
412C22⋊C4—42no—
413C22⋊C4—42no—
414C22⋊C4—42no—
415C22⋊C4—82no—
416C22⋊C4—82no—
417C22⋊C4—82no—
418C22⋊C4—82no—
419C42—42no—
420C42—41no—
421C42—41no—
422C42—42no—
423C42—41no—
424C42—41no—
425C42—82no—
426C42—82no—
427C42—84no—
428C42—84no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—22no—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23#41914yes—
456C23C22×C22⋊C411yescharacteristic
457C23#52511yescharacteristic
458C23#52511yescharacteristic
459C23C2×AGL(1,ℤ/8)11yescharacteristic
460C23C2×#13711yescharacteristic
461C23C2×#13711yescharacteristic
462C23C2×#13711yescharacteristic
463Q8—82no—
464Q8—82no—
465Q8—162no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—24no—
475C2×C4—22no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—42no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4—816no—
536C2×C4—81no—
537C2×C4—81no—
538C2×C4—81no—
539C2×C4—81no—
540C2×C4—81no—
541C2×C4—81no—
542C2×C4—81no—
543C2×C4—81no—
544C2×C4—81no—
545C2×C4—81no—
546C2×C4—81no—
547C2×C4—81no—
548C2×C4#41712yes—
549C2×C4#41712yes—
550C2×C4C4×#4111yescharacteristic
551C2×C4C4×#4111yescharacteristic
552C2×C4#53211yescharacteristic
553C2×C4#53211yescharacteristic
554C8—82no—
555C8—82no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—21no—
565C22—21no—
566C22—21no—
567C22—21no—
568C22—24no—
569C22—24no—
570C22—24no—
571C22—44no—
572C22—44no—
573C22—44no—
574C22—44no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22#229711yescharacteristic
580C22#288311yescharacteristic
581C22#288311yescharacteristic
582C22C2×#41911yescharacteristic
583C22C2×#41711yescharacteristic
584C22#290111yescharacteristic
585C22#290111yescharacteristic
586C22#311011yescharacteristic
587C22#311011yescharacteristic
588C22#311311yescharacteristic
589C22#311311yescharacteristic
590C4—21no—
591C4—21no—
592C4—21no—
593C4—21no—
594C4—22no—
595C4—22no—
596C4—44no—
597C4—42no—
598C4—42no—
599C4—41no—
600C4—41no—
601C4—42no—
602C4—82no—
603C4—82no—
604C4—82no—
605C4—82no—
606C4—82no—
607C4—82no—
608C4—82no—
609C2—21no—
610C2—21no—
611C2—21no—
612C2—21no—
613C2—44no—
614C2#1762111yescharacteristic
615C2#1762111yescharacteristic
616C2#1762211yescharacteristic
617C2#1763011yescharacteristic
618C2#2139411yescharacteristic
619C2#2141911yescharacteristic
620C2#2142211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9912C2semidirect
5. #9909C2semidirect
6. #9714C2semidirect
7. #9711C2semidirect
10. #8219C2semidirect
11. #8217C2semidirect
19. #2497C4semidirect
20. #2492C4semidirect
21. #2429C4semidirect
22. #2423C4semidirect
53. C2×C4⋊Q8C2×C4semidirect
54. C2×C4⋊Q8C2×C4semidirect
59. C2×C4×Q8C2×C4semidirect
85. #391C2×C4semidirect
86. #391C2×C4semidirect
91. #389C2×C4semidirect
92. #389C2×C4semidirect
104. #384C2×C4semidirect
105. #384C2×C4semidirect
116. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#350142
#219062
#138022
#99122
#99092
#97142
#97112
#91512
#91432
#82192
#82172