Finite Group Databasebeta

← #227563

Subgroups — #227563

Order 512.

766 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#23101C211yesmaximal, characteristic
2#23003C211yesmaximal, characteristic
3#22525C211yesmaximal, characteristic, complemented
4Q8×#103C211yesmaximal, characteristic, complemented
5C2×#1623C211yesmaximal, characteristic
6#9829C211yesmaximal, characteristic
7#9576C211yesmaximal, characteristic, complemented
8#9576C211yesmaximal, characteristic, complemented
9#9230C211yesmaximal, characteristic
10#9222C211yesmaximal, characteristic
11#9064C211yesmaximal, characteristic
12#8895C211yesmaximal, characteristic, complemented
13#8892C211yesmaximal, characteristic, complemented
14#8209C211yesmaximal, characteristic, complemented
15#8203C211yesmaximal, characteristic, complemented
16C2×Q82C2211yescharacteristic
17C2×#492C2211yescharacteristic
18C2×#460C2211yescharacteristic
19C2×#460C2211yescharacteristic
20C2×#446C2211yescharacteristic
21C2×C8×Q8C2211yescharacteristic
22C2×#440—21no—
23#2366C2211yescharacteristic
24#2355C2211yescharacteristic
25Q8×C4⋊C4C2211yescharacteristic
26#1993—21no—
27#1990—21no—
28#1990C2211yescharacteristic
29#1990C2211yescharacteristic
30#1988—21no—
31#1987—21no—
32#1987C2211yescharacteristic
33#1987C2211yescharacteristic
34#1973—21no—
35#1957C2211yescharacteristic
36#1957C2211yescharacteristic
37#1947C2211yescharacteristic
38#1947C2211yescharacteristic
39#1936—21no—
40#1925—21no—
41#1880—21no—
42#1877—21no—
43#1862—21no—
44#1834C2211yescharacteristic
45#1805—21no—
46#1803—21no—
47#1803C2211yescharacteristic
48#1803C2211yescharacteristic
49#1770C2211yescharacteristic
50#1770C2211yescharacteristic
51#1763—21no—
52#1762—21no—
53C4×#103—21no—
54C4×#103C2211yescharacteristic
55C4×#103C2211yescharacteristic
56#1623C418yescomplemented
57C2×#333C2211yescharacteristic
58C2×#331C2211yescharacteristic
59C2×Q8⋊C8C2211yescharacteristic
60C2×Q8⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1277C2211yescharacteristic
67#1277C2211yescharacteristic
68C22×SD16—21no—
69C22×SD16—21no—
70C22×SD16—41no—
71Q82—22no—
72Q82—22no—
73Q82—41no—
74Q82C2×C414yescomplemented
75C2×C4⋊Q8—21no—
76C2×C4⋊Q8—21no—
77C2×C4⋊Q8—41no—
78C2×C4⋊Q8C2311yescharacteristic
79C2×#127C2311yescharacteristic
80C2×C4×Q8—21no—
81C2×C4×Q8—21no—
82C2×C4×Q8—21no—
83C2×C4×Q8C2311yescharacteristic
84C2×C4×Q8C2311yescharacteristic
85C2×C4×D4—21no—
86C2×C4×D4—21no—
87#492—41no—
88#492C2×C414yes—
89#460—22no—
90#460—22no—
91#460C2×C414yes—
92#460C2×C414yes—
93#446—22no—
94#446—41no—
95C8×Q8—22no—
96C8×Q8—41no—
97#440—44no—
98C2×#107—21no—
99C2×#107—21no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×#103—21no—
107C2×#103—21no—
108C2×#103D411yescharacteristic
109C2×#103D411yescharacteristic
110C2×#103D411yescharacteristic
111C2×#103D411yescharacteristic
112C2×#102—21no—
113C2×#102—21no—
114C2×#102—21no—
115C2×#102—21no—
116C2×#102—21no—
117C2×#102—21no—
118C2×#97—21no—
119C2×#97—21no—
120C2×#97—21no—
121C2×C4×C8—21no—
122C2×C4×C8C2311yescharacteristic
123C2×C4×C8C2311yescharacteristic
124C2×C4×C8C2311yescharacteristic
125#391—21no—
126#390—21no—
127#390—21no—
128#389—21no—
129#389—41no—
130#387—41no—
131#386—41no—
132#384—21no—
133#384C2311yescharacteristic
134#384C2311yescharacteristic
135#382—21no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4—21no—
138C4×C4⋊C4—21no—
139C4×C4⋊C4C2311yescharacteristic
140C4×C4⋊C4C2311yescharacteristic
141#340—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146#340—21no—
147#340—21no—
148#340—21no—
149#340—21no—
150#340—21no—
151#340—21no—
152#340—21no—
153#340—41no—
154#340D411yescharacteristic
155#340D411yescharacteristic
156#340D411yescharacteristic
157#340D411yescharacteristic
158#336—21no—
159#336—21no—
160#333C2×C414yescomplemented
161#331—41no—
162Q8⋊C8C2×C414yescomplemented
163Q8⋊C8C2×C414yescomplemented
164#322—41no—
165C22×Q8—21no—
166C22×Q8—41no—
167C22×Q8C2×D411yescharacteristic
168C22×Q8C2×D411yescharacteristic
169C22×D4—21no—
170C22×D4—21no—
171C22×D4—21no—
172C23×C4—41no—
173C2×SD16—24no—
174C2×SD16—24no—
175C2×SD16—44no—
176C2×SD16—42no—
177C2×SD16—42no—
178C2×SD16—81no—
179C2×SD16—81no—
180C2×SD16—82no—
181C2×SD16—82no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8C2×D411yescharacteristic
188C22×C8C2×D411yescharacteristic
189C22×C8C2×D411yescharacteristic
190C22×C8C2×D411yescharacteristic
191C4⋊Q8—24no—
192C4⋊Q8—24no—
193C4⋊Q8—21no—
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196C4⋊Q8—42no—
197C4⋊Q8—42no—
198C4⋊Q8—44no—
199C4⋊Q8—44no—
200C4⋊Q8C22×C412yes—
201#127—21no—
202#127—21no—
203#127—21no—
204#127C22×C412yes—
205C4×Q8—21no—
206C4×Q8—21no—
207C4×Q8—21no—
208C4×Q8—21no—
209C4×Q8—24no—
210C4×Q8—24no—
211C4×Q8—42no—
212C4×Q8—42no—
213C4×Q8—42no—
214C4×Q8—81no—
215C4×Q8C22×C412yes—
216C4×Q8C22×C412yes—
217C4×Q8#4112yes—
218C4×Q8#4112yes—
219C4×D4—42no—
220C4×D4—42no—
221C4×D4—81no—
222C4×D4—81no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—41no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4C2×D411yescharacteristic
238C2×C4⋊C4C2×D411yescharacteristic
239C2×C4⋊C4C2×D411yescharacteristic
240C2×C4⋊C4C2×D411yescharacteristic
241C2×C4⋊C4C2×D411yescharacteristic
242C2×C4⋊C4C2×D411yescharacteristic
243C2×C22⋊C4—41no—
244C2×C22⋊C4—41no—
245C2×C22⋊C4—41no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—41no—
251C2×C42—41no—
252C2×C42—41no—
253C2×C42C2411yescharacteristic
254#107—42no—
255#107—81no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—42no—
263C4⋊C8—42no—
264C4⋊C8C22×C412yes—
265C4⋊C8C22×C412yes—
266#103—22no—
267#103—22no—
268#103—22no—
269#103—41no—
270#103—42no—
271#103—42no—
272#102—24no—
273#102—24no—
274#102—42no—
275#102—42no—
276#102—81no—
277#102—81no—
278#97—42no—
279#97—42no—
280#97—81no—
281C4×C8—21no—
282C4×C8—21no—
283C4×C8—21no—
284C4×C8—21no—
285C4×C8—21no—
286C4×C8—42no—
287C4×C8C22×C412yes—
288#95—21no—
289#95—21no—
290#95—41no—
291#95—41no—
292#95—41no—
293#95—41no—
294C24—41no—
295C2×Q8—21no—
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—22no—
300C2×Q8—21no—
301C2×Q8—21no—
302C2×Q8—21no—
303C2×Q8—21no—
304C2×Q8—44no—
305C2×Q8—41no—
306C2×Q8—81no—
307C2×Q8—81no—
308C2×Q8—161no—
309C2×Q8—161no—
310C2×Q8C4×D412yes—
311C2×Q8C4×D412yes—
312C2×Q8AGL(1,ℤ/8)12yes—
313C2×Q8#13712yes—
314C2×D4—22no—
315C2×D4—22no—
316C2×D4—22no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4—41no—
328C2×D4—81no—
329C2×D4—81no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—82no—
342C22×C4—82no—
343C22×C4—161no—
344C22×C4C22×D411yescharacteristic
345C22×C4C22×D411yescharacteristic
346C22×C4H(5,𝔽2)11yescharacteristic
347SD16—48no—
348SD16—84no—
349SD16—88no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—22no—
353C2×C8—22no—
354C2×C8—22no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—21no—
362C2×C8—21no—
363C2×C8—21no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8#13712yes—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—44no—
394C4⋊C4—44no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—82no—
406C4⋊C4—81no—
407C4⋊C4—81no—
408C4⋊C4—161no—
409C4⋊C4C4×D412yes—
410C4⋊C4C4×D412yes—
411C22⋊C4—82no—
412C22⋊C4—82no—
413C22⋊C4—161no—
414C42—22no—
415C42—22no—
416C42—22no—
417C42—22no—
418C42—44no—
419C42—41no—
420C42—41no—
421C42—41no—
422C42—41no—
423C42—82no—
424C42—161no—
425C42C23×C411yescharacteristic
426C42C2×#4111yescharacteristic
427C42C2×#4111yescharacteristic
428C42H(5,𝔽2)11yescharacteristic
429C23—42no—
430C23—81no—
431C23—81no—
432C23—81no—
433C23—81no—
434C23—81no—
435C23—81no—
436C23D4211yescharacteristic
437Q8—22no—
438Q8—22no—
439Q8—22no—
440Q8—22no—
441Q8—22no—
442Q8—41no—
443Q8—41no—
444Q8—82no—
445Q8—161no—
446Q8—164no—
447Q8#44014yes—
448D4—42no—
449D4—42no—
450D4—42no—
451D4—42no—
452D4—82no—
453D4—81no—
454D4—81no—
455D4—81no—
456D4—81no—
457D4—161no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—42no—
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—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—164no—
494C2×C4C2×C4×D411yescharacteristic
495C2×C4C2×C4×D411yescharacteristic
496C2×C4#51811yescharacteristic
497C2×C4#54611yescharacteristic
498C2×C4#54611yescharacteristic
499C2×C4#55111yescharacteristic
500C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
501C2×C4C2×#13711yescharacteristic
502C2×C4C2×#13711yescharacteristic
503C2×C4#57511yescharacteristic
504C2×C4#57711yescharacteristic
505C2×C4#57711yescharacteristic
506C2×C4#57711yescharacteristic
507C2×C4#57811yescharacteristic
508C8—22no—
509C8—42no—
510C8—42no—
511C8—42no—
512C8—41no—
513C8—41no—
514C8—41no—
515C8—81no—
516C8—81no—
517C22—82no—
518C22—82no—
519C22—82no—
520C22—161no—
521C22—161no—
522C22—161no—
523C22—161no—
524C22#235811yescharacteristic
525C22#326511yescharacteristic
526C22#327011yescharacteristic
527C22#327511yescharacteristic
528C22#327811yescharacteristic
529C22#328811yescharacteristic
530C22#328811yescharacteristic
531C4—24no—
532C4—21no—
533C4—21no—
534C4—42no—
535C4—42no—
536C4—42no—
537C4—82no—
538C4—81no—
539C4—81no—
540C4—81no—
541C4—81no—
542C4—161no—
543C4—161no—
544C4C2×#44011yescharacteristic
545C4C4×#13711yescharacteristic
546C4#294711yescharacteristic
547C4#294911yescharacteristic
548C4#330811yescharacteristic
549C4#332011yescharacteristic
550C4#332011yescharacteristic
551C4#332111yescharacteristic
552C2—162no—
553C2#2020111yescharacteristic
554C2#2020311yescharacteristic
555C2#2021711yescharacteristic
556C2#2021911yescharacteristic
557C2#2285511yescharacteristic
558C2#2286511yescharacteristic
559C2#2289111yescharacteristic
subgroupG/H
center, socleC23D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
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
3. #22525C2semidirect
4. Q8×#103C2semidirect
7. #9576C2semidirect
8. #9576C2semidirect
12. #8895C2semidirect
13. #8892C2semidirect
14. #8209C2semidirect
15. #8203C2semidirect
56. #1623C4semidirect
74. Q82C2×C4semidirect
160. #333C2×C4semidirect
162. Q8⋊C8C2×C4semidirect
163. Q8⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#231012
#230032
#225252
Q8×#1032
C2×#16232
#98292
#95762
#92302
#92222
#90642
#88952
#88922
#82092
#82032