Finite Group Databasebeta

← #227267

Subgroups — #227267

Order 512.

860 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23093C211yesmaximal, characteristic
2#23003C211yesmaximal, characteristic, complemented
3Q8×#102C211yesmaximal, characteristic, complemented
4#22430C211yesmaximal, characteristic, complemented
5C2×#1627C211yesmaximal, characteristic
6#9822C211yesmaximal, characteristic, complemented
7#9569C211yesmaximal, characteristic, complemented
8#9566C211yesmaximal, characteristic, complemented
9#9229C211yesmaximal, characteristic
10#9222C211yesmaximal, characteristic, complemented
11#9057C211yesmaximal, characteristic, complemented
12#8897C211yesmaximal, characteristic, complemented
13#8892C211yesmaximal, characteristic, complemented
14#8206C211yesmaximal, characteristic, complemented
15C8×#103C211yesmaximal, characteristic, complemented
16C2×D4×Q8C2211yescharacteristic
17C2×#486C2211yescharacteristic
18C2×#460C2211yescharacteristic
19C2×#459C2211yescharacteristic
20C2×C8×Q8C2211yescharacteristic
21C2×#440—21no—
22C2×#436C2211yescharacteristic
23C2×#419—21no—
24#2366C2211yescharacteristic
25#2354C2211yescharacteristic
26Q8×C4⋊C4C2211yescharacteristic
27#1990—21no—
28#1990C2211yescharacteristic
29#1988—21no—
30#1988C2211yescharacteristic
31#1981C2211yescharacteristic
32#1970—21no—
33#1957C2211yescharacteristic
34#1957C2211yescharacteristic
35#1956C2211yescharacteristic
36#1947C2211yescharacteristic
37#1947C2211yescharacteristic
38#1936—21no—
39#1925—21no—
40#1894—21no—
41#1880—21no—
42#1879—21no—
43#1873—21no—
44#1862—21no—
45#1834C2211yescharacteristic
46#1803C2211yescharacteristic
47#1802—21no—
48#1802C2211yescharacteristic
49#1787—21no—
50#1770C2211yescharacteristic
51C8×C4⋊C4C2211yescharacteristic
52C4×#103C2211yescharacteristic
53C4×#102—21no—
54C4×#102—21no—
55C4×#102C2211yescharacteristic
56#1627C418yescomplemented
57C2×#332C2211yescharacteristic
58C2×#331C2211yescharacteristic
59C2×Q8⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×C82C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1277C2211yescharacteristic
67#1277C2211yescharacteristic
68C23×Q8—21no—
69C22×SD16—21no—
70C22×D8—21no—
71C22×M16—21no—
72D4×Q8—22no—
73D4×Q8—22no—
74D4×Q8—41no—
75D4×Q8C2×C414yescomplemented
76C2×C4⋊Q8—21no—
77C2×C4⋊Q8C2311yescharacteristic
78C2×#127C2311yescharacteristic
79C2×C22⋊Q8—21no—
80C2×C22⋊Q8—41no—
81C2×C4×Q8—21no—
82C2×C4×Q8C2311yescharacteristic
83C2×C4×D4—21no—
84C2×C4×D4—21no—
85C2×C4×D4—21no—
86C2×C4×D4C2311yescharacteristic
87C2×#117—21no—
88#486—41no—
89#486C2×C414yes—
90#460—22no—
91#460C2×C414yes—
92#459—22no—
93#459C2×C414yes—
94C8×Q8—22no—
95C8×Q8—41no—
96#440—44no—
97#436—22no—
98#436—41no—
99C2×#107—21no—
100C2×#107—21no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106#419—44no—
107C2×#103—21no—
108C2×#103—21no—
109C2×#103—21no—
110C2×#103—21no—
111C2×#103—21no—
112C2×#103D411yescharacteristic
113C2×#103D411yescharacteristic
114C2×#102—21no—
115C2×#102—21no—
116C2×#102—21no—
117C2×#102—21no—
118C2×#102—21no—
119C2×#102—21no—
120C2×#102—21no—
121C2×#102—41no—
122C2×#102D411yescharacteristic
123C2×#102D411yescharacteristic
124C2×C22⋊C8—21no—
125C2×#97—21no—
126C2×C4×C8—21no—
127C2×C4×C8—21no—
128C2×C4×C8C2311yescharacteristic
129C2×C4×C8C2311yescharacteristic
130C2×C4×C8C2311yescharacteristic
131#390—21no—
132#390—21no—
133#390—21no—
134#389—21no—
135#389—41no—
136#388—41no—
137#386—41no—
138#384—21no—
139#384C2311yescharacteristic
140#384C2311yescharacteristic
141#380—21no—
142C4×C4⋊C4—21no—
143C4×C4⋊C4—21no—
144C4×C4⋊C4C2311yescharacteristic
145C4×C4⋊C4C2311yescharacteristic
146#340—21no—
147#340—21no—
148#340—21no—
149#340—21no—
150#340—21no—
151#340—21no—
152#340—21no—
153#340—21no—
154#340—41no—
155#340D411yescharacteristic
156#340D411yescharacteristic
157#340D411yescharacteristic
158#340D411yescharacteristic
159#336—21no—
160#336—21no—
161#332C2×C414yescomplemented
162#331—41no—
163Q8⋊C8C2×C414yescomplemented
164D4⋊C8C2×C414yescomplemented
165C82—41no—
166C22×Q8—24no—
167C22×Q8—21no—
168C22×Q8—41no—
169C22×Q8—42no—
170C22×Q8—42no—
171C22×Q8—42no—
172C22×Q8—81no—
173C22×Q8—81no—
174C22×Q8C2×D411yescharacteristic
175C22×D4—21no—
176C22×D4—21no—
177C22×D4—21no—
178C22×D4C2×D411yescharacteristic
179C23×C4—21no—
180C23×C4—41no—
181C23×C4—41no—
182C2×SD16—24no—
183C2×SD16—42no—
184C2×SD16—42no—
185C2×D8—24no—
186C2×D8—81no—
187C2×D8—81no—
188C2×M16—42no—
189C2×M16—42no—
190C2×M16—81no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8C2×D411yescharacteristic
197C22×C8C2×D411yescharacteristic
198C22×C8C2×D411yescharacteristic
199C22×C8C2×D411yescharacteristic
200C4⋊Q8—24no—
201C4⋊Q8—21no—
202C4⋊Q8—21no—
203C4⋊Q8—21no—
204C4⋊Q8—42no—
205C4⋊Q8C22×C412yes—
206#127—21no—
207#127—21no—
208#127—21no—
209#127C22×C412yes—
210C22⋊Q8—24no—
211C22⋊Q8—44no—
212C22⋊Q8—44no—
213C22⋊Q8—42no—
214C4×Q8—21no—
215C4×Q8—21no—
216C4×Q8—42no—
217C4×Q8—81no—
218C4×Q8C22×C412yes—
219C4×Q8#4112yes—
220C4×D4—24no—
221C4×D4—21no—
222C4×D4—21no—
223C4×D4—42no—
224C4×D4—42no—
225C4×D4—42no—
226C4×D4—81no—
227C4×D4—81no—
228C4×D4C22×C412yes—
229C4×D4#4112yes—
230#117—42no—
231#117—81no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4—41no—
244C2×C4⋊C4C2×D411yescharacteristic
245C2×C4⋊C4C2×D411yescharacteristic
246C2×C4⋊C4C2×D411yescharacteristic
247C2×C4⋊C4C2×D411yescharacteristic
248C2×C4⋊C4C2×D411yescharacteristic
249C2×C4⋊C4C2×D411yescharacteristic
250C2×C22⋊C4—21no—
251C2×C22⋊C4—41no—
252C2×C22⋊C4—41no—
253C2×C22⋊C4—41no—
254C2×C22⋊C4—41no—
255C2×C22⋊C4—41no—
256C2×C22⋊C4—41no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—41no—
261C2×C42—41no—
262C2×C42C2411yescharacteristic
263#107—42no—
264#107—81no—
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—42no—
272C4⋊C8C22×C412yes—
273C4⋊C8C22×C412yes—
274#103—22no—
275#103—42no—
276#103—42no—
277#103—42no—
278#103—81no—
279#103—81no—
280#103D814yescomplemented
281#102—22no—
282#102—24no—
283#102—24no—
284#102—44no—
285#102—41no—
286#102—42no—
287#102—42no—
288#102—42no—
289#102—81no—
290#102—81no—
291C22⋊C8—42no—
292#97—42no—
293C4×C8—21no—
294C4×C8—21no—
295C4×C8—21no—
296C4×C8—21no—
297C4×C8—21no—
298C4×C8—42no—
299C4×C8—81no—
300C4×C8C22×C412yes—
301#95—21no—
302#95—21no—
303#95—41no—
304#95—41no—
305C24—21no—
306C24—41no—
307C2×Q8—21no—
308C2×Q8—21no—
309C2×Q8—21no—
310C2×Q8—22no—
311C2×Q8—44no—
312C2×Q8—41no—
313C2×Q8—48no—
314C2×Q8—41no—
315C2×Q8—41no—
316C2×Q8—41no—
317C2×Q8—41no—
318C2×Q8—82no—
319C2×Q8—82no—
320C2×Q8—84no—
321C2×Q8—84no—
322C2×Q8—84no—
323C2×Q8C4×D412yes—
324C2×Q8C2×D812yes—
325C2×Q8#13512yes—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—21no—
330C2×D4—21no—
331C2×D4—21no—
332C2×D4—21no—
333C2×D4—21no—
334C2×D4—41no—
335C2×D4—41no—
336C2×D4—41no—
337C2×D4—41no—
338C2×D4—41no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—41no—
342C2×D4—41no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—81no—
346C2×D4—81no—
347C2×D4C4×D412yes—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—44no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—82no—
366C22×C4—82no—
367C22×C4—82no—
368C22×C4—82no—
369C22×C4—161no—
370C22×C4C22×D411yescharacteristic
371C22×C4C22×D411yescharacteristic
372C22×C4H(5,𝔽2)11yescharacteristic
373SD16—48no—
374D8—84no—
375M16—84no—
376C2×C8—21no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8C2×D812yes—
402C2×C8#13712yes—
403C4⋊C4—22no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—22no—
407C4⋊C4—21no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—44no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C4⋊C4—82no—
428C4⋊C4—81no—
429C4⋊C4—81no—
430C4⋊C4—161no—
431C4⋊C4C4×D412yes—
432C4⋊C4C4×D412yes—
433C4⋊C4C2×D812yes—
434C22⋊C4—22no—
435C22⋊C4—22no—
436C22⋊C4—44no—
437C22⋊C4—82no—
438C22⋊C4—82no—
439C22⋊C4—82no—
440C22⋊C4—161no—
441C22⋊C4—161no—
442C42—22no—
443C42—22no—
444C42—41no—
445C42—41no—
446C42—41no—
447C42—41no—
448C42—82no—
449C42—82no—
450C42C23×C411yescharacteristic
451C42C2×#4111yescharacteristic
452C42C2×#4111yescharacteristic
453C42H(5,𝔽2)11yescharacteristic
454C23—22no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—42no—
462C23—81no—
463C23—81no—
464C23—81no—
465C23—81no—
466C23—81no—
467C23—81no—
468C23D4211yescharacteristic
469Q8—22no—
470Q8—22no—
471Q8—22no—
472Q8—42no—
473Q8—42no—
474Q8—88no—
475Q8—81no—
476Q8—81no—
477Q8C4×D814yes—
478D4—22no—
479D4—22no—
480D4—42no—
481D4—42no—
482D4—42no—
483D4—42no—
484D4—41no—
485D4—41no—
486D4—82no—
487D4—81no—
488D4—81no—
489D4—81no—
490D4—81no—
491D4—161no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—88no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4—81no—
524C2×C4—81no—
525C2×C4—81no—
526C2×C4—81no—
527C2×C4—81no—
528C2×C4—81no—
529C2×C4—164no—
530C2×C4C2×C4×D411yescharacteristic
531C2×C4C2×C4×D411yescharacteristic
532C2×C4#51811yescharacteristic
533C2×C4#54611yescharacteristic
534C2×C4#54611yescharacteristic
535C2×C4#55111yescharacteristic
536C2×C4C22×D811yescharacteristic
537C2×C4C2×#13511yescharacteristic
538C2×C4C2×#13711yescharacteristic
539C2×C4#57511yescharacteristic
540C2×C4#57611yescharacteristic
541C2×C4#57711yescharacteristic
542C2×C4#57711yescharacteristic
543C2×C4#57711yescharacteristic
544C8—22no—
545C8—22no—
546C8—42no—
547C8—42no—
548C8—42no—
549C8—41no—
550C8—41no—
551C8—81no—
552C8—81no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22—81no—
560C22—82no—
561C22—82no—
562C22—82no—
563C22—161no—
564C22—161no—
565C22—161no—
566C22—161no—
567C22#235811yescharacteristic
568C22#327511yescharacteristic
569C22#327811yescharacteristic
570C22D4×D811yescharacteristic
571C22#328211yescharacteristic
572C22#329011yescharacteristic
573C22#329011yescharacteristic
574C4—24no—
575C4—21no—
576C4—21no—
577C4—42no—
578C4—42no—
579C4—42no—
580C4—82no—
581C4—81no—
582C4—81no—
583C4—81no—
584C4—161no—
585C4—161no—
586C4C2×C4×D811yescharacteristic
587C4C4×#13711yescharacteristic
588C4#294611yescharacteristic
589C4#294911yescharacteristic
590C4#329411yescharacteristic
591C4#329711yescharacteristic
592C4#332011yescharacteristic
593C4#332111yescharacteristic
594C2—82no—
595C2—162no—
596C2#1988611yescharacteristic
597C2#1988911yescharacteristic
598C2#1991811yescharacteristic
599C2#1992111yescharacteristic
600C2#2286511yescharacteristic
601C2#2287011yescharacteristic
602C2#2289511yescharacteristic
subgroupG/H
center, socleC23D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#459C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #23003C2semidirect
3. Q8×#102C2semidirect
4. #22430C2semidirect
6. #9822C2semidirect
7. #9569C2semidirect
8. #9566C2semidirect
10. #9222C2semidirect
11. #9057C2semidirect
12. #8897C2semidirect
13. #8892C2semidirect
14. #8206C2semidirect
15. C8×#103C2semidirect
56. #1627C4semidirect
75. D4×Q8C2×C4semidirect
161. #332C2×C4semidirect
163. Q8⋊C8C2×C4semidirect
164. D4⋊C8C2×C4semidirect
280. #103D8semidirect

Maximal subgroups

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

subgroupindex
#230932
#230032
Q8×#1022
#224302
C2×#16272
#98222
#95692
#95662
#92292
#92222
#90572
#88972
#88922
#82062
C8×#1032