Finite Group Databasebeta

← #227437

Subgroups — #227437

Order 512.

739 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23101C211yesmaximal, characteristic
2#23095C211yesmaximal, characteristic
3#22525C211yesmaximal, characteristic, complemented
4#22525C211yesmaximal, characteristic, complemented
5C2×#1623C211yesmaximal, characteristic
6#9829C211yesmaximal, characteristic
7#9575C211yesmaximal, characteristic, complemented
8#9568C211yesmaximal, characteristic, complemented
9#9230C211yesmaximal, characteristic
10#9230C211yesmaximal, characteristic
11#9064C211yesmaximal, characteristic
12#8895C211yesmaximal, characteristic, complemented
13#8891C211yesmaximal, characteristic, complemented
14#8211C211yesmaximal, characteristic, complemented
15#8209C211yesmaximal, characteristic, complemented
16C2×Q82C2211yescharacteristic
17C2×#492C2211yescharacteristic
18C2×#460C2211yescharacteristic
19C2×#460C2211yescharacteristic
20C2×#446C2211yescharacteristic
21C2×#446C2211yescharacteristic
22#2366C2211yescharacteristic
23#2355C2211yescharacteristic
24#2355C2211yescharacteristic
25#1993—21no—
26#1993—21no—
27#1990C2211yescharacteristic
28#1990C2211yescharacteristic
29#1988—21no—
30#1988—21no—
31#1987C2211yescharacteristic
32#1987C2211yescharacteristic
33#1973—21no—
34#1968—21no—
35#1957C2211yescharacteristic
36#1957C2211yescharacteristic
37#1948C2211yescharacteristic
38#1947C2211yescharacteristic
39#1926—21no—
40#1925—21no—
41#1877—21no—
42#1877—21no—
43#1834C2211yescharacteristic
44#1805—21no—
45#1805—21no—
46#1803C2211yescharacteristic
47#1803C2211yescharacteristic
48#1770C2211yescharacteristic
49C8×C4⋊C4C2211yescharacteristic
50#1763—21no—
51#1763—21no—
52#1762—21no—
53C4×#103C2211yescharacteristic
54C4×#103C2211yescharacteristic
55C4×#102—21no—
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×Q8C2311yescharacteristic
83C2×C4×Q8C2311yescharacteristic
84C2×C4×D4—21no—
85C2×C4×D4—21no—
86#492—41no—
87#492C2×C414yes—
88#460—22no—
89#460—22no—
90#460C2×C414yes—
91#460C2×C414yes—
92#446—22no—
93#446—22no—
94#446—41no—
95#446—41no—
96C2×#107—21no—
97C2×#106—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×#103—21no—
105C2×#103—21no—
106C2×#103D411yescharacteristic
107C2×#103D411yescharacteristic
108C2×#103D411yescharacteristic
109C2×#103D411yescharacteristic
110C2×#102—21no—
111C2×#102—21no—
112C2×#102—21no—
113C2×#102—21no—
114C2×#102—21no—
115C2×#102—21no—
116C2×#97—21no—
117C2×#97—21no—
118C2×#97—21no—
119C2×C4×C8—21no—
120C2×C4×C8C2311yescharacteristic
121C2×C4×C8C2311yescharacteristic
122C2×C4×C8C2311yescharacteristic
123#391—21no—
124#391—21no—
125#390—21no—
126#390—21no—
127#387—41no—
128#387—41no—
129#386—41no—
130#384C2311yescharacteristic
131#384C2311yescharacteristic
132#382—21no—
133#382—21no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4—21no—
138C4×C4⋊C4C2311yescharacteristic
139C4×C4⋊C4C2311yescharacteristic
140#340—21no—
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—41no—
153#340D411yescharacteristic
154#340D411yescharacteristic
155#340D411yescharacteristic
156#340D411yescharacteristic
157#336—21no—
158#336—21no—
159#333C2×C414yescomplemented
160#331—41no—
161Q8⋊C8C2×C414yescomplemented
162Q8⋊C8C2×C414yescomplemented
163#322—41no—
164C22×Q8—21no—
165C22×Q8—41no—
166C22×Q8C2×D411yescharacteristic
167C22×Q8C2×D411yescharacteristic
168C22×D4—21no—
169C22×D4—21no—
170C22×D4—21no—
171C23×C4—41no—
172C2×SD16—24no—
173C2×SD16—24no—
174C2×SD16—44no—
175C2×SD16—81no—
176C2×SD16—81no—
177C2×SD16—81no—
178C2×SD16—81no—
179C2×SD16—82no—
180C2×SD16—82no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8C2×D411yescharacteristic
187C22×C8C2×D411yescharacteristic
188C22×C8C2×D411yescharacteristic
189C22×C8C2×D411yescharacteristic
190C4⋊Q8—24no—
191C4⋊Q8—24no—
192C4⋊Q8—21no—
193C4⋊Q8—21no—
194C4⋊Q8—21no—
195C4⋊Q8—42no—
196C4⋊Q8—42no—
197C4⋊Q8—44no—
198C4⋊Q8—44no—
199C4⋊Q8C22×C412yes—
200#127—21no—
201#127—21no—
202#127—21no—
203#127C22×C412yes—
204C4×Q8—21no—
205C4×Q8—21no—
206C4×Q8—21no—
207C4×Q8—21no—
208C4×Q8—24no—
209C4×Q8—24no—
210C4×Q8—42no—
211C4×Q8—42no—
212C4×Q8C22×C412yes—
213C4×Q8C22×C412yes—
214C4×Q8#4112yes—
215C4×Q8#4112yes—
216C4×D4—42no—
217C4×D4—42no—
218C4×D4—81no—
219C4×D4—81no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
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—41no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4C2×D411yescharacteristic
233C2×C4⋊C4C2×D411yescharacteristic
234C2×C4⋊C4C2×D411yescharacteristic
235C2×C4⋊C4C2×D411yescharacteristic
236C2×C4⋊C4C2×D411yescharacteristic
237C2×C4⋊C4C2×D411yescharacteristic
238C2×C22⋊C4—41no—
239C2×C22⋊C4—41no—
240C2×C22⋊C4—41no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—41no—
246C2×C42—41no—
247C2×C42—41no—
248C2×C42C2411yescharacteristic
249#107—81no—
250#106—81no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—42no—
258C4⋊C8—42no—
259C4⋊C8C22×C412yes—
260C4⋊C8C22×C412yes—
261#103—22no—
262#103—22no—
263#103—41no—
264#103—41no—
265#103—42no—
266#103—42no—
267#102—24no—
268#102—24no—
269#102—42no—
270#102—42no—
271#102—81no—
272#102—81no—
273#97—42no—
274#97—42no—
275#97—81no—
276C4×C8—21no—
277C4×C8—21no—
278C4×C8—21no—
279C4×C8—21no—
280C4×C8—21no—
281C4×C8—81no—
282C4×C8C22×C412yes—
283#95—21no—
284#95—21no—
285#95—41no—
286#95—41no—
287#95—41no—
288#95—41no—
289#95—41no—
290#95—41no—
291C24—41no—
292C2×Q8—21no—
293C2×Q8—21no—
294C2×Q8—22no—
295C2×Q8—21no—
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8—21no—
301C2×Q8—21no—
302C2×Q8—21no—
303C2×Q8—44no—
304C2×Q8—41no—
305C2×Q8—81no—
306C2×Q8—81no—
307C2×Q8—161no—
308C2×Q8—161no—
309C2×Q8C4×D412yes—
310C2×Q8C4×D412yes—
311C2×D4—22no—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—41no—
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—81no—
326C2×D4—81no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—82no—
339C22×C4—82no—
340C22×C4—161no—
341C22×C4C22×D411yescharacteristic
342C22×C4C22×D411yescharacteristic
343C22×C4H(5,𝔽2)11yescharacteristic
344SD16—84no—
345SD16—84no—
346SD16—88no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—22no—
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—41no—
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#13712yes—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—21no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—44no—
390C4⋊C4—44no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—81no—
402C4⋊C4—81no—
403C4⋊C4C4×D412yes—
404C4⋊C4C4×D412yes—
405C4⋊C4#13712yes—
406C22⋊C4—82no—
407C22⋊C4—82no—
408C22⋊C4—161no—
409C42—22no—
410C42—22no—
411C42—22no—
412C42—22no—
413C42—44no—
414C42—41no—
415C42—41no—
416C42—41no—
417C42—41no—
418C42—161no—
419C42—161no—
420C42C23×C411yescharacteristic
421C42C2×#4111yescharacteristic
422C42C2×#4111yescharacteristic
423C42H(5,𝔽2)11yescharacteristic
424C23—42no—
425C23—81no—
426C23—81no—
427C23—81no—
428C23—81no—
429C23—81no—
430C23—81no—
431C23D4211yescharacteristic
432Q8—22no—
433Q8—22no—
434Q8—22no—
435Q8—22no—
436Q8—41no—
437Q8—41no—
438Q8—41no—
439Q8—41no—
440Q8—82no—
441Q8—161no—
442Q8—164no—
443D4—42no—
444D4—42no—
445D4—42no—
446D4—42no—
447D4—82no—
448D4—81no—
449D4—81no—
450D4—81no—
451D4—81no—
452D4—161no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—41no—
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—81no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—164no—
489C2×C4C2×C4×D411yescharacteristic
490C2×C4C2×C4×D411yescharacteristic
491C2×C4#51811yescharacteristic
492C2×C4#54611yescharacteristic
493C2×C4#54611yescharacteristic
494C2×C4#55111yescharacteristic
495C2×C4C2×#13711yescharacteristic
496C2×C4C2×#13711yescharacteristic
497C2×C4#57511yescharacteristic
498C2×C4#57511yescharacteristic
499C2×C4#57711yescharacteristic
500C2×C4#57711yescharacteristic
501C2×C4#57711yescharacteristic
502C2×C4#57711yescharacteristic
503C8—22no—
504C8—42no—
505C8—41no—
506C8—42no—
507C8—42no—
508C8—41no—
509C8—41no—
510C8—81no—
511C8—81no—
512C22—82no—
513C22—82no—
514C22—82no—
515C22—161no—
516C22—161no—
517C22—161no—
518C22—161no—
519C22#235811yescharacteristic
520C22#326611yescharacteristic
521C22#327111yescharacteristic
522C22#327511yescharacteristic
523C22#327811yescharacteristic
524C22#328811yescharacteristic
525C22#328811yescharacteristic
526C4—21no—
527C4—21no—
528C4—42no—
529C4—42no—
530C4—42no—
531C4—42no—
532C4—42no—
533C4—81no—
534C4—82no—
535C4—81no—
536C4—81no—
537C4—161no—
538C4—161no—
539C4C4×#13711yescharacteristic
540C4C4×#13711yescharacteristic
541C4#294911yescharacteristic
542C4#294911yescharacteristic
543C4#332011yescharacteristic
544C4#332111yescharacteristic
545C4#332611yescharacteristic
546C4#332911yescharacteristic
547C2—162no—
548C2#2017711yescharacteristic
549C2#2018811yescharacteristic
550C2#2027311yescharacteristic
551C2#2028411yescharacteristic
552C2#2286511yescharacteristic
553C2#2288511yescharacteristic
554C2#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. #22525C2semidirect
7. #9575C2semidirect
8. #9568C2semidirect
12. #8895C2semidirect
13. #8891C2semidirect
14. #8211C2semidirect
15. #8209C2semidirect
56. #1623C4semidirect
74. Q82C2×C4semidirect
159. #333C2×C4semidirect
161. Q8⋊C8C2×C4semidirect
162. Q8⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#231012
#230952
#225252
C2×#16232
#98292
#95752
#95682
#92302
#90642
#88952
#88912
#82112
#82092