Finite Group Databasebeta

← #204243

Subgroups — #204243

Order 512.

1111 conjugacy classes of subgroups.

The subgroup lattice has 601 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2604C211yesmaximal, characteristic
2#20814C211yesmaximal, characteristic, complemented
3#20469C211yesmaximal, characteristic, complemented
4C2×#1473C211yesmaximal, characteristic
5C2×#1473C211yesmaximal, characteristic
6C2×#1467C211yesmaximal, characteristic, complemented
7C2×#1461C211yesmaximal, characteristic, complemented
8#9654C212yesmaximal, complemented
9#9652C212yesmaximal
10#9651C212yesmaximal
11#9650C212yesmaximal, complemented
12C22×#126C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C2×#432C2211yescharacteristic
15C2×#432C2211yescharacteristic
16C2×#431C2211yescharacteristic
17C2×#431C2211yescharacteristic
18#2604—21no—
19#2604—21no—
20#2604—21no—
21#2604—21no—
22#2604—22no—
23#2604—22no—
24C2×#400C411yescharacteristic
25C2×#399C411yescharacteristic
26C2×#395C411yescharacteristic
27C2×#393C411yescharacteristic
28C2×#389C2212yes—
29C2×#388C412yes—
30C2×#388C2212yes—
31C2×#387C412yes—
32C2×#376C2211yescharacteristic
33#1843—21no—
34#1832—21no—
35#1797C2212yes—
36#1797C2212yes—
37#1768—22no—
38#1768—22no—
39#1749C2212yes—
40C4×C22⋊C8C2212yes—
41#1746—21no—
42#1726—21no—
43#1473—24no—
44#1473—24no—
45#1467—24no—
46#1461—24no—
47C2×#324C2211yescharacteristic
48C2×#324C2212yes—
49C2×#324C2212yes—
50C2×#324C2212yes—
51C2×#324C2212yes—
52C2×#324C2212yes—
53C2×#324C2211yescharacteristic
54C2×#323C2211yescharacteristic
55C2×#323C2212yes—
56C2×#323C2211yescharacteristic
57C23×Q8C2×C411yescharacteristic
58C2×#126—21no—
59C2×#126—21no—
60C2×#126—21no—
61C2×#126—21no—
62C2×#126—22no—
63C2×#126—22no—
64C2×#126—22no—
65C2×#126—22no—
66C2×C4×Q8—21no—
67C2×C4×Q8—21no—
68C2×C4×Q8—21no—
69C2×C4×Q8—21no—
70C2×C4×Q8—22no—
71C2×C4×Q8—22no—
72C2×C4×Q8—22no—
73C2×C4×Q8—22no—
74C22×C4⋊C4C2×C411yescharacteristic
75C22×C4⋊C4C2×C411yescharacteristic
76C22×C4⋊C4C2×C411yescharacteristic
77C22×C4⋊C4C2×C412yes—
78C22×C4⋊C4C2×C412yes—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2311yescharacteristic
81C22×C22⋊C4C2×C411yescharacteristic
82C22×C22⋊C4C2×C411yescharacteristic
83C22×C22⋊C4C2×C412yes—
84C22×C22⋊C4C2×C412yes—
85C22×C22⋊C4C2311yescharacteristic
86C22×C42C2×C412yes—
87C22×C42C2312yes—
88C22×C42C2311yescharacteristic
89#432—22no—
90#432—22no—
91#432—22no—
92#432D414yes—
93#431—22no—
94#431—22no—
95#431—22no—
96#431D414yes—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—22no—
100C2×C4⋊C8—22no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—21no—
103C2×C22⋊C8C2312yes—
104C2×C22⋊C8C2311yescharacteristic
105C2×C22⋊C8C2311yescharacteristic
106C2×C22⋊C8C2311yescharacteristic
107C2×C22⋊C8C2311yescharacteristic
108C2×C22⋊C8C2312yes—
109C2×#97—21no—
110C2×#97—21no—
111C2×#97—22no—
112C2×C4×C8—21no—
113C2×C4×C8—21no—
114C2×C4×C8—22no—
115#400—21no—
116#400—21no—
117#400—22no—
118#399—21no—
119#399—22no—
120#399—21no—
121#395—21no—
122#395—22no—
123#395—21no—
124#393—21no—
125#393—21no—
126#393—22no—
127#389—22no—
128#389—22no—
129#389—22no—
130#389—22no—
131#388—22no—
132#388—22no—
133#388—22no—
134#388—22no—
135#388—22no—
136#388—22no—
137#388—22no—
138#388—22no—
139#387—22no—
140#387—22no—
141#387—22no—
142#387—22no—
143#376D411yescharacteristic
144#376D411yescharacteristic
145#376D411yescharacteristic
146#376D411yescharacteristic
147#376D412yes—
148#376D412yes—
149C2×#95C2×C412yes—
150C2×#95C2×C412yes—
151C2×#95C2×C412yes—
152C2×#95C2×C412yes—
153C2×#95C2×C411yescharacteristic
154C2×#95C2×C411yescharacteristic
155C2×#95C2×C412yes—
156#336—22no—
157#336—22no—
158#336—22no—
159#336—22no—
160#324—24no—
161#324—24no—
162#324—24no—
163#324—24no—
164#324—24no—
165#324—22no—
166#324—22no—
167#323—24no—
168#323—22no—
169#323—22no—
170C25C22×C411yescharacteristic
171C22×Q8—21no—
172C22×Q8—21no—
173C22×Q8—22no—
174C22×Q8—22no—
175C22×Q8—24no—
176C22×Q8—42no—
177C23×C4C22×C411yescharacteristic
178C23×C4C22×C412yes—
179C23×C4C22×C412yes—
180C23×C4C22×C411yescharacteristic
181C23×C4C22×C411yescharacteristic
182C23×C4C22×C412yes—
183C23×C4C22×C411yescharacteristic
184C23×C4C22×C412yes—
185C23×C4C22×C411yescharacteristic
186C23×C4C2411yescharacteristic
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—21no—
192C22×C8—21no—
193#126—42no—
194#126—42no—
195#126—42no—
196#126—42no—
197#126—42no—
198#126—42no—
199#126—42no—
200#126—42no—
201C4×Q8—42no—
202C4×Q8—42no—
203C4×Q8—42no—
204C4×Q8—42no—
205C4×Q8—42no—
206C4×Q8—42no—
207C4×Q8—42no—
208C4×Q8—42no—
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—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—44no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—44no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—44no—
266C2×C42C2×D412yes—
267C2×C42C2×D412yes—
268C2×C42C2×D412yes—
269C2×C42C2×D412yes—
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D412yes—
273C2×C42C2×D412yes—
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42#4112yes—
277C2×C42#4112yes—
278C4⋊C8—24no—
279C4⋊C8—24no—
280C4⋊C8—44no—
281C4⋊C8—44no—
282C4⋊C8—42no—
283C4⋊C8—42no—
284C22⋊C8—21no—
285C22⋊C8—21no—
286C22⋊C8—22no—
287C22⋊C8—22no—
288C22⋊C8—21no—
289C22⋊C8—21no—
290C22⋊C8—22no—
291C22⋊C8—21no—
292C22⋊C8—21no—
293C22⋊C8C2×D412yes—
294C22⋊C8C2×D412yes—
295C22⋊C8C2×D414yes—
296#97—24no—
297#97—44no—
298#97—42no—
299C4×C8—24no—
300C4×C8—44no—
301C4×C8—42no—
302#95—22no—
303#95—22no—
304#95—22no—
305#95—22no—
306#95—22no—
307#95—22no—
308#95—22no—
309#95—22no—
310#95C22⋊C412yes—
311#95C22⋊C412yes—
312#95C22⋊C412yes—
313#95C22⋊C412yes—
314#95C22⋊C411yescharacteristic
315#95C22⋊C411yescharacteristic
316#95C22⋊C412yes—
317#95C22⋊C411yescharacteristic
318#95C22⋊C412yes—
319#95C22⋊C411yescharacteristic
320C24—21no—
321C24—21no—
322C24—21no—
323C24—22no—
324C24—22no—
325C24—42no—
326C24—41no—
327C24—41no—
328C24C23×C411yescharacteristic
329C2×Q8—28no—
330C2×Q8—44no—
331C2×Q8—44no—
332C2×Q8—48no—
333C2×Q8—44no—
334C2×Q8—42no—
335C2×Q8—42no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—42no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4C2×C22⋊C412yes—
375C22×C4C2×C22⋊C412yes—
376C22×C4C2×C22⋊C412yes—
377C22×C4C2×C22⋊C411yescharacteristic
378C22×C4C2×C22⋊C411yescharacteristic
379C22×C4C2×C22⋊C411yescharacteristic
380C22×C4C2×C22⋊C411yescharacteristic
381C22×C4C2×C22⋊C412yes—
382C22×C4C4×D412yes—
383C22×C4C4×D412yes—
384C22×C4C4×D412yes—
385C22×C4C4×D412yes—
386C22×C4#13114yes—
387C22×C4C22×D411yescharacteristic
388C22×C4C22×D411yescharacteristic
389C22×C4C22×D411yescharacteristic
390C22×C4C22×D411yescharacteristic
391C22×C4C2×#4111yescharacteristic
392C22×C4C2×#4111yescharacteristic
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—24no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—41no—
406C2×C8—41no—
407C4⋊C4—44no—
408C4⋊C4—44no—
409C4⋊C4—44no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—42no—
415C4⋊C4—41no—
416C4⋊C4—42no—
417C4⋊C4—41no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—44no—
425C22⋊C4—44no—
426C22⋊C4—44no—
427C22⋊C4—44no—
428C22⋊C4—41no—
429C22⋊C4—41no—
430C22⋊C4—41no—
431C22⋊C4—41no—
432C22⋊C4—42no—
433C22⋊C4—42no—
434C22⋊C4—42no—
435C22⋊C4—42no—
436C22⋊C4—44no—
437C42—24no—
438C42—24no—
439C42—42no—
440C42—42no—
441C42—44no—
442C42—44no—
443C42#12112yes—
444C42#12112yes—
445C42#12112yes—
446C42#12112yes—
447C42#12412yes—
448C42#12412yes—
449C42#12712yes—
450C42#12712yes—
451C23—22no—
452C23—21no—
453C23—21no—
454C23—42no—
455C23—42no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—42no—
462C23—82no—
463C23—82no—
464C23—82no—
465C23—82no—
466C23—81no—
467C23—81no—
468C23—81no—
469C23—81no—
470C23C22×C22⋊C411yescharacteristic
471C23C2×C4×D411yescharacteristic
472C23C2×C4×D411yescharacteristic
473C23C2×#13112yes—
474C23#56812yes—
475Q8—48no—
476Q8—48no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—44no—
509C2×C4—44no—
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—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
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#39012yes—
536C2×C4#39012yes—
537C2×C4#39012yes—
538C2×C4#39012yes—
539C2×C4#41014yes—
540C2×C4C2×#12111yescharacteristic
541C2×C4C2×#12111yescharacteristic
542C2×C4C2×#12411yescharacteristic
543C2×C4C2×#12711yescharacteristic
544C8—42no—
545C8—42no—
546C8—44no—
547C8—82no—
548C8—81no—
549C8—81no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—82no—
559C22—82no—
560C22—81no—
561C22—81no—
562C22—81no—
563C22—161no—
564C22—161no—
565C22—162no—
566C22C2×#39011yescharacteristic
567C22#288012yes—
568C22C2×#41012yes—
569C22#288911yescharacteristic
570C22#288911yescharacteristic
571C22#292911yescharacteristic
572C22#293011yescharacteristic
573C22#293312yes—
574C4—24no—
575C4—22no—
576C4—22no—
577C4—21no—
578C4—21no—
579C4—21no—
580C4—21no—
581C4—22no—
582C4—22no—
583C4—42no—
584C4—42no—
585C4—42no—
586C4—44no—
587C4—81no—
588C4—82no—
589C4—81no—
590C4—81no—
591C2—21no—
592C2—21no—
593C2—22no—
594C2—161no—
595C2#1741712yes—
596C2#1742012yes—
597C2#2119111yescharacteristic
598C2#2121211yescharacteristic
599C2#2122011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20814C2semidirect
3. #20469C2semidirect
6. C2×#1467C2semidirect
7. C2×#1461C2semidirect
8. #9654C2semidirect
11. #9650C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26042
#208142
#204692
C2×#14732
C2×#14672
C2×#14612
#96542
#96522
#96512
#96502