Finite Group Databasebeta

← #230209

Subgroups — #230209

Order 512.

721 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23190C211yesmaximal, characteristic
2#23003C211yesmaximal, characteristic
3#22964C211yesmaximal, characteristic, complemented
4#22740C211yesmaximal, characteristic, complemented
5C2×#1666C211yesmaximal, characteristic
6#9829C211yesmaximal, characteristic
7#9622C211yesmaximal, characteristic, complemented
8#9574C211yesmaximal, characteristic, complemented
9#9563C211yesmaximal, characteristic
10#8983C211yesmaximal, characteristic, complemented
11#8979C211yesmaximal, characteristic
12#8922C211yesmaximal, characteristic, complemented
13#8921C211yesmaximal, characteristic
14#8288C211yesmaximal, characteristic, complemented
15#8205C211yesmaximal, characteristic, complemented
16C2×#492C2211yescharacteristic
17C2×#475C2211yescharacteristic
18C2×#460C2211yescharacteristic
19C2×C8×Q8C2211yescharacteristic
20C2×#440—21no—
21C2×#440—21no—
22#2366C2211yescharacteristic
23#2365C2211yescharacteristic
24Q8×C4⋊C4C2211yescharacteristic
25Q8×C4⋊C4C2211yescharacteristic
26#1990C2211yescharacteristic
27#1990C2211yescharacteristic
28#1982C2211yescharacteristic
29#1981C2211yescharacteristic
30#1973—21no—
31#1957C2211yescharacteristic
32#1957C2211yescharacteristic
33#1948C2211yescharacteristic
34#1947—21no—
35#1947—21no—
36#1947C2211yescharacteristic
37#1947C2211yescharacteristic
38#1947C2211yescharacteristic
39#1941—21no—
40#1936—21no—
41#1920—21no—
42#1880—21no—
43#1880—21no—
44#1877—21no—
45#1865—21no—
46#1862—21no—
47#1834—21no—
48#1834C2211yescharacteristic
49C4×#107—21no—
50C4×#107C2211yescharacteristic
51#1770C2211yescharacteristic
52#1770C2211yescharacteristic
53C8×C4⋊C4C2211yescharacteristic
54C4×#103—21no—
55C4×#103C2211yescharacteristic
56#1666C418yescomplemented
57C2×#334C2211yescharacteristic
58C2×#332C2211yescharacteristic
59C2×#331C2211yescharacteristic
60C2×Q8⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1277C2211yescharacteristic
68C22×SD16—21no—
69C22×SD16—21no—
70C22×SD16—21no—
71C2×C4⋊Q8C2311yescharacteristic
72C2×#127C2311yescharacteristic
73C2×C4×Q8—21no—
74C2×C4×Q8—21no—
75C2×C4×Q8—21no—
76C2×C4×Q8C2311yescharacteristic
77C2×C4×D4—21no—
78C2×C4×D4—21no—
79#492—22no—
80#492C2×C414yes—
81#475—22no—
82#475C2×C414yescomplemented
83#460—22no—
84#460C2×C414yes—
85C8×Q8—22no—
86C8×Q8C2×C414yescomplemented
87#440—44no—
88#440—44no—
89C2×#107—21no—
90C2×#107—21no—
91C2×#107—21no—
92C2×#107D411yescharacteristic
93C2×#107D411yescharacteristic
94C2×#106—21no—
95C2×#106—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×#103—21no—
102C2×#103—21no—
103C2×#103—21no—
104C2×#103D411yescharacteristic
105C2×#103D411yescharacteristic
106C2×#102—21no—
107C2×#102—21no—
108C2×#102—21no—
109C2×#102—21no—
110C2×#102—21no—
111C2×#102—41no—
112C2×#97—21no—
113C2×#97—21no—
114C2×C4×C8—21no—
115C2×C4×C8—21no—
116C2×C4×C8C2311yescharacteristic
117C2×C4×C8C2311yescharacteristic
118C2×C4×C8C2311yescharacteristic
119#391—21no—
120#390—21no—
121#390—21no—
122#389—21no—
123#389—21no—
124#389—21no—
125#389—41no—
126#389—41no—
127#387—41no—
128#386—41no—
129#384—21no—
130#384—21no—
131#384C2311yescharacteristic
132#384C2311yescharacteristic
133C4×C4⋊C4—21no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4C2311yescharacteristic
137C4×C4⋊C4C2311yescharacteristic
138C4×C4⋊C4C2311yescharacteristic
139#340—21no—
140#340—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146#340—21no—
147#340—41no—
148#340D411yescharacteristic
149#340D411yescharacteristic
150#340D411yescharacteristic
151#340D411yescharacteristic
152#336—21no—
153#336—21no—
154#336—21no—
155#334C2×C414yescomplemented
156#332C2×C414yescomplemented
157#331C2×C414yes—
158Q8⋊C8—22no—
159#322—22no—
160C22×Q8—21no—
161C22×Q8C2×D411yescharacteristic
162C22×D4—21no—
163C22×D4—21no—
164C22×D4—21no—
165C23×C4—41no—
166C2×SD16—24no—
167C2×SD16—24no—
168C2×SD16—24no—
169C2×SD16—42no—
170C2×SD16—42no—
171C2×SD16—42no—
172C2×SD16—42no—
173C2×SD16—81no—
174C2×SD16—81no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8C2×D411yescharacteristic
181C22×C8C2×D411yescharacteristic
182C22×C8C2×D411yescharacteristic
183C22×C8C2×D411yescharacteristic
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8C22×C412yes—
188#127—21no—
189#127—21no—
190#127—21no—
191#127C22×C412yes—
192C4×Q8—21no—
193C4×Q8—21no—
194C4×Q8—42no—
195C4×Q8—42no—
196C4×Q8—42no—
197C4×Q8—81no—
198C4×Q8—81no—
199C4×Q8—81no—
200C4×Q8C22×C412yes—
201C4×Q8#4112yes—
202C4×D4—42no—
203C4×D4—42no—
204C4×D4—81no—
205C4×D4—81no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
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—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4C2×D411yescharacteristic
221C2×C4⋊C4C2×D411yescharacteristic
222C2×C4⋊C4C2×D411yescharacteristic
223C2×C4⋊C4C2×D411yescharacteristic
224C2×C4⋊C4C2×D411yescharacteristic
225C2×C4⋊C4C2×D411yescharacteristic
226C2×C4⋊C4C2×D411yescharacteristic
227C2×C22⋊C4—41no—
228C2×C22⋊C4—41no—
229C2×C22⋊C4—41no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—41no—
235C2×C42—41no—
236C2×C42—41no—
237C2×C42C2411yescharacteristic
238#107—22no—
239#107—41no—
240#107—42no—
241#107—42no—
242#107—42no—
243#106—24no—
244#106—81no—
245C4⋊C8—24no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8C22×C412yes—
252C4⋊C8C22×C412yes—
253C4⋊C8C22×C412yes—
254#103—24no—
255#103—22no—
256#103—22no—
257#103—42no—
258#103—81no—
259#102—24no—
260#102—42no—
261#102—42no—
262#102—81no—
263#102—81no—
264#102—82no—
265#97—42no—
266#97—42no—
267C4×C8—24no—
268C4×C8—21no—
269C4×C8—21no—
270C4×C8—21no—
271C4×C8—21no—
272C4×C8—81no—
273C4×C8C22×C412yes—
274C4×C8#4112yes—
275#95—21no—
276#95—21no—
277#95—21no—
278#95—41no—
279#95—41no—
280#95—41no—
281#95—41no—
282C24—41no—
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8—21no—
286C2×Q8—22no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—41no—
292C2×Q8C4×D412yes—
293C2×Q8AGL(1,ℤ/8)12yes—
294C2×Q8#13712yes—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—81no—
310C2×D4—81no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—82no—
323C22×C4—82no—
324C22×C4—161no—
325C22×C4C22×D411yescharacteristic
326C22×C4C22×D411yescharacteristic
327C22×C4H(5,𝔽2)11yescharacteristic
328SD16—48no—
329SD16—48no—
330SD16—84no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—22no—
336C2×C8—22no—
337C2×C8—22no—
338C2×C8—22no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8C4×D412yes—
357C2×C8C4×D412yes—
358C4⋊C4—22no—
359C4⋊C4—21no—
360C4⋊C4—22no—
361C4⋊C4—21no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—82no—
385C4⋊C4—82no—
386C4⋊C4—82no—
387C4⋊C4—81no—
388C4⋊C4—81no—
389C4⋊C4—161no—
390C4⋊C4—161no—
391C4⋊C4C4×D412yes—
392C4⋊C4AGL(1,ℤ/8)12yes—
393C22⋊C4—82no—
394C22⋊C4—82no—
395C22⋊C4—161no—
396C42—22no—
397C42—22no—
398C42—41no—
399C42—41no—
400C42—41no—
401C42—41no—
402C42—41no—
403C42—41no—
404C42—82no—
405C42—82no—
406C42—82no—
407C42C23×C411yescharacteristic
408C42C2×#4111yescharacteristic
409C42C2×#4111yescharacteristic
410C422−1+411yescharacteristic
411C23—42no—
412C23—81no—
413C23—81no—
414C23—81no—
415C23—81no—
416C23—81no—
417C23—81no—
418C23D4211yescharacteristic
419Q8—22no—
420Q8—22no—
421Q8—22no—
422Q8—42no—
423Q8—42no—
424Q8—81no—
425Q8—81no—
426Q8#44014yescomplemented
427D4—42no—
428D4—42no—
429D4—42no—
430D4—42no—
431D4—82no—
432D4—81no—
433D4—81no—
434D4—81no—
435D4—81no—
436D4—161no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—164no—
473C2×C4C2×C4×D411yescharacteristic
474C2×C4C2×C4×D411yescharacteristic
475C2×C4#52011yescharacteristic
476C2×C4#54711yescharacteristic
477C2×C4#54811yescharacteristic
478C2×C4#55311yescharacteristic
479C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
480C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
481C2×C4C2×#13711yescharacteristic
482C2×C4#57511yescharacteristic
483C2×C4#57611yescharacteristic
484C2×C4#57711yescharacteristic
485C2×C4#57711yescharacteristic
486C2×C4#57811yescharacteristic
487C8—22no—
488C8—22no—
489C8—42no—
490C8—41no—
491C8—41no—
492C8—42no—
493C8—42no—
494C8—42no—
495C8—81no—
496C22—82no—
497C22—82no—
498C22—82no—
499C22—161no—
500C22—161no—
501C22—161no—
502C22—161no—
503C22#236111yescharacteristic
504C22#326411yescharacteristic
505C22#326511yescharacteristic
506C22#327011yescharacteristic
507C22#327111yescharacteristic
508C22#329011yescharacteristic
509C22#329111yescharacteristic
510C4—21no—
511C4—21no—
512C4—24no—
513C4—42no—
514C4—42no—
515C4—81no—
516C4—81no—
517C4—81no—
518C4—81no—
519C4—81no—
520C4—82no—
521C4—161no—
522C4—161no—
523C4C2×#44011yescharacteristic
524C4C4×AGL(1,ℤ/8)11yescharacteristic
525C4#294911yescharacteristic
526C4#295111yescharacteristic
527C4#334311yescharacteristic
528C4#334611yescharacteristic
529C4#336811yescharacteristic
530C4#337111yescharacteristic
531C2—162no—
532C2#2003011yescharacteristic
533C2#2003711yescharacteristic
534C2#2008711yescharacteristic
535C2#2012111yescharacteristic
536C2#2301011yescharacteristic
537C2#2303211yescharacteristic
538C2#2304411yescharacteristic
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. #22964C2semidirect
4. #22740C2semidirect
7. #9622C2semidirect
8. #9574C2semidirect
10. #8983C2semidirect
12. #8922C2semidirect
14. #8288C2semidirect
15. #8205C2semidirect
56. #1666C4semidirect
82. #475C2×C4semidirect
86. C8×Q8C2×C4semidirect
155. #334C2×C4semidirect
156. #332C2×C4semidirect
426. Q8#440semidirect

Maximal subgroups

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

subgroupindex
#231902
#230032
#229642
#227402
C2×#16662
#98292
#96222
#95742
#95632
#89832
#89792
#89222
#89212
#82882
#82052