Finite Group Databasebeta

← #231708

Subgroups — #231708

Order 512.

722 conjugacy classes of subgroups.

The subgroup lattice has 549 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23217C211yesmaximal, characteristic, complemented
2#23216C211yesmaximal, characteristic
3#23190C211yesmaximal, characteristic
4#23163C211yesmaximal, characteristic, complemented
5C2×#1714C211yesmaximal, characteristic
6#9746C211yesmaximal, characteristic
7#9745C211yesmaximal, characteristic, complemented
8#9732C211yesmaximal, characteristic
9#9729C211yesmaximal, characteristic, complemented
10#9064C211yesmaximal, characteristic
11#9063C211yesmaximal, characteristic
12#9058C211yesmaximal, characteristic, complemented
13#9058C211yesmaximal, characteristic, complemented
14#8288C211yesmaximal, characteristic, complemented
15#8288C211yesmaximal, characteristic, complemented
16C2×#500C2211yescharacteristic
17C2×#498C2211yescharacteristic
18C2×#492C2211yescharacteristic
19C2×#492C2211yescharacteristic
20C2×#487C2211yescharacteristic
21C2×#486C2211yescharacteristic
22C2×#440—21no—
23#2369C2211yescharacteristic
24#2366C2211yescharacteristic
25#2365C2211yescharacteristic
26#2365C2211yescharacteristic
27#1988C2211yescharacteristic
28#1988C2211yescharacteristic
29#1987C2211yescharacteristic
30#1987C2211yescharacteristic
31#1982C2211yescharacteristic
32#1982C2211yescharacteristic
33#1981C2211yescharacteristic
34#1981C2211yescharacteristic
35#1973—21no—
36#1944—21no—
37#1934—21no—
38#1933—21no—
39#1928—21no—
40#1925—21no—
41#1920—21no—
42#1919—21no—
43#1880—21no—
44#1877—21no—
45#1863—21no—
46#1862—21no—
47#1776—21no—
48C4×#107C2211yescharacteristic
49C4×#107C2211yescharacteristic
50#1770C2211yescharacteristic
51#1770C2211yescharacteristic
52C8×C4⋊C4C2211yescharacteristic
53C8×C4⋊C4C2211yescharacteristic
54C4×#103—21no—
55C4×#102—21no—
56#1714C418yescomplemented
57C2×#334C2211yescharacteristic
58C2×#334C2211yescharacteristic
59C2×#333C2211yescharacteristic
60C2×#332C2211yescharacteristic
61C2×#332C2211yescharacteristic
62C2×#331C2211yescharacteristic
63C2×#322C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1279C2211yescharacteristic
68C22×SD16—21no—
69C22×SD16—21no—
70C22×SD16—41no—
71C2×C4⋊Q8C2311yescharacteristic
72C2×C4⋊Q8C2311yescharacteristic
73C2×C4⋊Q8C2311yescharacteristic
74C2×#127C2311yescharacteristic
75C2×C4×Q8—21no—
76C2×C4×Q8—21no—
77C2×C4×D4—21no—
78C2×C4×D4—21no—
79#500—41no—
80#500C2×C414yescomplemented
81#498—22no—
82#498C2×C414yescomplemented
83#492—22no—
84#492—41no—
85#492C2×C414yes—
86#492C2×C414yes—
87#487—22no—
88#487C2×C414yescomplemented
89#486—22no—
90#486C2×C414yes—
91#440—44no—
92C2×#107—41no—
93C2×#107D411yescharacteristic
94C2×#107D411yescharacteristic
95C2×#107D411yescharacteristic
96C2×#107D411yescharacteristic
97C2×#106—21no—
98C2×#106—21no—
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×#103—21no—
104C2×#103—21no—
105C2×#103—21no—
106C2×#103—21no—
107C2×#103—21no—
108C2×#103—21no—
109C2×#102—21no—
110C2×#102—21no—
111C2×#102—21no—
112C2×#102—21no—
113C2×#102—21no—
114C2×#102—21no—
115C2×#97—21no—
116C2×#97—21no—
117C2×C4×C8—21no—
118C2×C4×C8—21no—
119C2×C4×C8C2311yescharacteristic
120C2×C4×C8C2311yescharacteristic
121C2×C4×C8C2311yescharacteristic
122#391—21no—
123#391—21no—
124#391—41no—
125#390—21no—
126#390—21no—
127#389—21no—
128#389—21no—
129#389—21no—
130#389—21no—
131#387—41no—
132#387—41no—
133#386—41no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4—21no—
138C4×C4⋊C4C2311yescharacteristic
139C4×C4⋊C4C2311yescharacteristic
140C4×C4⋊C4C2311yescharacteristic
141C4×C4⋊C4C2311yescharacteristic
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#340D411yescharacteristic
151#340D411yescharacteristic
152#340D411yescharacteristic
153#340D411yescharacteristic
154#336—21no—
155#336—21no—
156#336—21no—
157#336—21no—
158#334—22no—
159#334—22no—
160#333—41no—
161#332—41no—
162#332—41no—
163#331—41no—
164#322C2×C414yescomplemented
165C22×Q8—21no—
166C22×Q8—21no—
167C22×Q8—21no—
168C22×D4—21no—
169C22×D4—21no—
170C22×D4—21no—
171C23×C4—41no—
172C2×SD16—24no—
173C2×SD16—24no—
174C2×SD16—42no—
175C2×SD16—42no—
176C2×SD16—44no—
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—21no—
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4⋊Q8—21no—
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196C4⋊Q8—21no—
197C4⋊Q8—21no—
198C4⋊Q8—21no—
199C4⋊Q8C22×C412yes—
200C4⋊Q8C22×C412yes—
201C4⋊Q8C22×C412yes—
202#127—21no—
203#127—21no—
204#127—21no—
205#127C22×C412yes—
206C4×Q8—42no—
207C4×Q8—42no—
208C4×Q8—81no—
209C4×Q8—81no—
210C4×D4—42no—
211C4×D4—42no—
212C4×D4—81no—
213C4×D4—81no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4C2×D411yescharacteristic
224C2×C4⋊C4C2×D411yescharacteristic
225C2×C4⋊C4C2×D411yescharacteristic
226C2×C4⋊C4C2×D411yescharacteristic
227C2×C4⋊C4C2×D411yescharacteristic
228C2×C4⋊C4C2×D411yescharacteristic
229C2×C4⋊C4C2×D411yescharacteristic
230C2×C4⋊C4C2×D411yescharacteristic
231C2×C22⋊C4—41no—
232C2×C22⋊C4—41no—
233C2×C22⋊C4—41no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—41no—
239C2×C42—41no—
240C2×C42—41no—
241C2×C42C2411yescharacteristic
242#107—22no—
243#107—22no—
244#107—44no—
245#107—41no—
246#107—41no—
247#106—24no—
248#106—24no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257#103—24no—
258#103—24no—
259#103—42no—
260#103—42no—
261#103—81no—
262#103—81no—
263#102—24no—
264#102—24no—
265#102—42no—
266#102—42no—
267#102—81no—
268#102—81no—
269#97—42no—
270#97—42no—
271C4×C8—21no—
272C4×C8—81no—
273C4×C8—81no—
274C4×C8C22×C412yes—
275C4×C8C22×C412yes—
276C4×C8C22×C412yes—
277C4×C8#4112yes—
278C4×C8#4112yes—
279#95—21no—
280#95—21no—
281#95—21no—
282#95—21no—
283#95—41no—
284#95—41no—
285#95—41no—
286#95—41no—
287#95—41no—
288#95—41no—
289C24—41no—
290C2×Q8—22no—
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8—81no—
305C2×Q8—81no—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—22no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—81no—
321C2×D4—81no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—82no—
334C22×C4—82no—
335C22×C4—161no—
336C22×C4C22×D411yescharacteristic
337C22×C4C22×D411yescharacteristic
338C22×C4H(5,𝔽2)11yescharacteristic
339SD16—48no—
340SD16—84no—
341SD16—88no—
342C2×C8—22no—
343C2×C8—22no—
344C2×C8—22no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8C4×D412yes—
366C2×C8C4×D412yes—
367C2×C8C4×D412yes—
368C2×C8C4×D412yes—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—82no—
396C4⋊C4—82no—
397C4⋊C4—161no—
398C4⋊C4AGL(1,ℤ/8)12yes—
399C4⋊C4#13712yes—
400C22⋊C4—82no—
401C22⋊C4—82no—
402C22⋊C4—161no—
403C42—41no—
404C42—41no—
405C42—41no—
406C42—41no—
407C42—41no—
408C42—41no—
409C42—41no—
410C42—41no—
411C42—82no—
412C42—82no—
413C42—161no—
414C42C23×C411yescharacteristic
415C42C2×#4111yescharacteristic
416C42C2×#4111yescharacteristic
417C42H(5,𝔽2)11yescharacteristic
418C23—42no—
419C23—81no—
420C23—81no—
421C23—81no—
422C23—81no—
423C23—81no—
424C23—81no—
425C23D4211yescharacteristic
426Q8—42no—
427Q8—42no—
428Q8—42no—
429Q8—42no—
430Q8—82no—
431Q8—81no—
432Q8—81no—
433Q8—81no—
434Q8—81no—
435Q8—161no—
436D4—42no—
437D4—42no—
438D4—42no—
439D4—42no—
440D4—82no—
441D4—81no—
442D4—81no—
443D4—81no—
444D4—81no—
445D4—161no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
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—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—164no—
482C2×C4C2×C4×D411yescharacteristic
483C2×C4C2×C4×D411yescharacteristic
484C2×C4#51811yescharacteristic
485C2×C4#55011yescharacteristic
486C2×C4#55011yescharacteristic
487C2×C4#55911yescharacteristic
488C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
489C2×C4C2×#13711yescharacteristic
490C2×C4#57511yescharacteristic
491C2×C4#57511yescharacteristic
492C2×C4#57611yescharacteristic
493C2×C4#57711yescharacteristic
494C2×C4#57711yescharacteristic
495C2×C4#57811yescharacteristic
496C8—24no—
497C8—22no—
498C8—22no—
499C8—22no—
500C8—22no—
501C8—81no—
502C8—81no—
503C8—81no—
504C8—81no—
505C22—82no—
506C22—82no—
507C22—82no—
508C22—161no—
509C22—161no—
510C22—161no—
511C22—161no—
512C22#236611yescharacteristic
513C22#326411yescharacteristic
514C22#326711yescharacteristic
515C22#327011yescharacteristic
516C22#327111yescharacteristic
517C22#329111yescharacteristic
518C22#329111yescharacteristic
519C4—21no—
520C4—21no—
521C4—42no—
522C4—42no—
523C4—81no—
524C4—81no—
525C4—81no—
526C4—81no—
527C4—82no—
528C4—82no—
529C4—82no—
530C4—81no—
531C4—81no—
532C4C4×AGL(1,ℤ/8)11yescharacteristic
533C4C4×#13711yescharacteristic
534C4#294611yescharacteristic
535C4#294711yescharacteristic
536C4#339711yescharacteristic
537C4#340011yescharacteristic
538C4#340211yescharacteristic
539C4#340311yescharacteristic
540C2—162no—
541C2#2002611yescharacteristic
542C2#2003211yescharacteristic
543C2#2010211yescharacteristic
544C2#2014711yescharacteristic
545C2#2319411yescharacteristic
546C2#2319511yescharacteristic
547C2#2319811yescharacteristic
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
1. #23217C2semidirect
4. #23163C2semidirect
7. #9745C2semidirect
9. #9729C2semidirect
12. #9058C2semidirect
13. #9058C2semidirect
14. #8288C2semidirect
15. #8288C2semidirect
56. #1714C4semidirect
80. #500C2×C4semidirect
82. #498C2×C4semidirect
88. #487C2×C4semidirect
164. #322C2×C4semidirect

Maximal subgroups

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

subgroupindex
#232172
#232162
#231902
#231632
C2×#17142
#97462
#97452
#97322
#97292
#90642
#90632
#90582
#82882