Finite Group Databasebeta

← #305749

Subgroups — #305749

Order 512.

714 conjugacy classes of subgroups.

The subgroup lattice has 541 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26111C211yesmaximal, characteristic
2#26093C211yesmaximal, characteristic, complemented
3#22523C211yesmaximal, characteristic
4D4×#103C211yesmaximal, characteristic
5#10684C211yesmaximal, characteristic, complemented
6#10602C211yesmaximal, characteristic, complemented
7#10592C211yesmaximal, characteristic, complemented
8#10353C211yesmaximal, characteristic, complemented
9#8887C211yesmaximal, characteristic, complemented
10#8886C211yesmaximal, characteristic
11#8848C211yesmaximal, characteristic, complemented
12#8843C211yesmaximal, characteristic
13#8655C211yesmaximal, characteristic, complemented
14#8644C211yesmaximal, characteristic
15#8447C211yesmaximal, characteristic
16C2×D4×Q8C2211yescharacteristic
17C2×#446C2211yescharacteristic
18C2×C8×D4C2211yescharacteristic
19C22×#103—21no—
20#2571C2211yescharacteristic
21#2355C2211yescharacteristic
22D4×C4⋊C4C2211yescharacteristic
23#2101—21no—
24#2097—21no—
25#2087—21no—
26#2068—21no—
27#1993—21no—
28#1990C2211yescharacteristic
29#1990C2211yescharacteristic
30#1988C2211yescharacteristic
31#1987—21no—
32#1981C2211yescharacteristic
33#1956C2211yescharacteristic
34#1949—21no—
35#1948C2211yescharacteristic
36#1947C2211yescharacteristic
37#1939C2211yescharacteristic
38#1939C2211yescharacteristic
39#1918—21no—
40#1892—21no—
41#1872—21no—
42#1862—21no—
43#1845C2211yescharacteristic
44#1844C2211yescharacteristic
45#1834C2211yescharacteristic
46#1804—21no—
47#1804—21no—
48#1803C2211yescharacteristic
49#1803C2211yescharacteristic
50#1802C2211yescharacteristic
51#1802C2211yescharacteristic
52#1787—21no—
53#1762—21no—
54C4×#103C2211yescharacteristic
55C4×#102C2211yescharacteristic
56C2×#335C2211yescharacteristic
57C2×#335C2211yescharacteristic
58C2×#332C2211yescharacteristic
59C2×Q8⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1273C2211yescharacteristic
67C23×Q8—21no—
68C23×C8—21no—
69D4×Q8—22no—
70D4×Q8—22no—
71D4×Q8—41no—
72D4×Q8Q814yes—
73C2×C4⋊Q8—21no—
74C2×C4⋊Q8C2311yescharacteristic
75C2×C22⋊Q8—21no—
76C2×C22⋊Q8—41no—
77C2×C4×Q8C2311yescharacteristic
78C2×C4×D4—21no—
79C2×C4×D4—21no—
80C2×C4×D4C2311yescharacteristic
81C22×C4⋊C4—21no—
82#446—22no—
83#446—41no—
84C8×D4—22no—
85C8×D4—41no—
86C2×#107—21no—
87C2×#107—21no—
88C2×#107—21no—
89C2×#106—21no—
90C2×#106—41no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×#103—21no—
99C2×#103—21no—
100C2×#103—21no—
101C2×#103—21no—
102C2×#103—21no—
103C2×#103—21no—
104C2×#103—44no—
105C2×#103D411yescharacteristic
106C2×#103D411yescharacteristic
107C2×#102—21no—
108C2×#102—21no—
109C2×#102D411yescharacteristic
110C2×#102D411yescharacteristic
111C2×C22⋊C8—21no—
112C2×#97—21no—
113C2×C4×C8C2311yescharacteristic
114C2×C4×C8C2311yescharacteristic
115C2×C4×C8C2311yescharacteristic
116#400—41no—
117#391—21no—
118#389—21no—
119#389—21no—
120#389—21no—
121#387—41no—
122#385—41no—
123#384C2311yescharacteristic
124#384C2311yescharacteristic
125#384C2311yescharacteristic
126#382—21no—
127#382—21no—
128#382—21no—
129#380—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4C2311yescharacteristic
133C4×C4⋊C4C2311yescharacteristic
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139#340—21no—
140#340—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146#340—41no—
147#340D411yescharacteristic
148#340D411yescharacteristic
149#340D411yescharacteristic
150#340D411yescharacteristic
151#336—21no—
152#336—21no—
153#335—41no—
154#335—41no—
155#332—41no—
156Q8⋊C8—22no—
157D4⋊C8—22no—
158C22×Q8—24no—
159C22×Q8—21no—
160C22×Q8—41no—
161C22×Q8—42no—
162C22×Q8—42no—
163C22×Q8—42no—
164C22×Q8—81no—
165C22×Q8—81no—
166C22×Q8C2×D411yescharacteristic
167C22×D4C2×D411yescharacteristic
168C23×C4—21no—
169C23×C4—41no—
170C23×C4—41no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—42no—
177C22×C8—42no—
178C22×C8—81no—
179C22×C8C2×D411yescharacteristic
180C22×C8C2×D411yescharacteristic
181C22×C8C2×D411yescharacteristic
182C22×C8C2×D411yescharacteristic
183C4⋊Q8—24no—
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8—42no—
188C4⋊Q8C2×Q812yes—
189C22⋊Q8—24no—
190C22⋊Q8—44no—
191C22⋊Q8—44no—
192C22⋊Q8—42no—
193C4×Q8—21no—
194C4×Q8—21no—
195C4×Q8C2×Q812yes—
196C4×Q8#4112yes—
197C4×D4—24no—
198C4×D4—21no—
199C4×D4—21no—
200C4×D4—42no—
201C4×D4—42no—
202C4×D4—81no—
203C4×D4C2×Q812yes—
204C4×D4#4112yes—
205C2×C4⋊C4—21no—
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—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—81no—
222C2×C4⋊C4C2×D411yescharacteristic
223C2×C4⋊C4C2×D411yescharacteristic
224C2×C4⋊C4C2×D411yescharacteristic
225C2×C4⋊C4C2×D411yescharacteristic
226C2×C4⋊C4C2×D411yescharacteristic
227C2×C4⋊C4C2×D411yescharacteristic
228C2×C22⋊C4—21no—
229C2×C22⋊C4—41no—
230C2×C22⋊C4—41no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—41no—
235C2×C42C2411yescharacteristic
236#107—42no—
237#107—81no—
238#107—81no—
239#106—81no—
240#106—82no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—42no—
250C4⋊C8—42no—
251C4⋊C8—81no—
252#103—22no—
253#103—41no—
254#103—42no—
255#103—42no—
256#103—42no—
257#103—81no—
258#103—81no—
259#103—81no—
260#102—22no—
261#102—22no—
262#102—42no—
263#102—42no—
264C22⋊C8—42no—
265#97—42no—
266C4×C8—21no—
267C4×C8—21no—
268C4×C8—21no—
269C4×C8—21no—
270C4×C8—21no—
271C4×C8—21no—
272#95—21no—
273#95—21no—
274#95—41no—
275#95—41no—
276#95—41no—
277#95—41no—
278#95—41no—
279#95—41no—
280#95—41no—
281C24—21no—
282C2×Q8—21no—
283C2×Q8—22no—
284C2×Q8—21no—
285C2×Q8—21no—
286C2×Q8—21no—
287C2×Q8—21no—
288C2×Q8—44no—
289C2×Q8—41no—
290C2×Q8—48no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×Q8—82no—
296C2×Q8—82no—
297C2×Q8—84no—
298C2×Q8—84no—
299C2×Q8—84no—
300C2×Q8C22⋊Q812yes—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4C22⋊Q812yes—
305C2×D4#13512yes—
306C2×D4#13712yes—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—44no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—82no—
326C22×C4—82no—
327C22×C4—82no—
328C22×C4—82no—
329C22×C4—161no—
330C22×C4C22×D411yescharacteristic
331C22×C4C22×D411yescharacteristic
332C22×C4H(5,𝔽2)11yescharacteristic
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—22no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
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×C8—41no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—84no—
360C2×C8AGL(1,ℤ/8)12yes—
361C4⋊C4—22no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—44no—
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—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—81no—
391C4⋊C4—81no—
392C4⋊C4—81no—
393C4⋊C4—82no—
394C4⋊C4—161no—
395C4⋊C4C22⋊Q812yes—
396C4⋊C4C22⋊Q812yes—
397C22⋊C4—22no—
398C22⋊C4—22no—
399C22⋊C4—44no—
400C22⋊C4—82no—
401C42—22no—
402C42—22no—
403C42—41no—
404C42—41no—
405C42—41no—
406C42—41no—
407C42—161no—
408C42C22×Q811yescharacteristic
409C42C2×#4111yescharacteristic
410C42C2×#4111yescharacteristic
411C42H(5,𝔽2)11yescharacteristic
412C23—22no—
413C23—22no—
414C23—22no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23D4211yescharacteristic
420Q8—22no—
421Q8—22no—
422Q8—41no—
423Q8—41no—
424Q8—42no—
425Q8—42no—
426Q8—88no—
427Q8—81no—
428Q8—81no—
429D4—22no—
430D4—22no—
431D4—22no—
432D4#47914yes—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
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—88no—
462C2×C4—81no—
463C2×C4—81no—
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—81no—
473C2×C4—164no—
474C2×C4C2×C22⋊Q811yescharacteristic
475C2×C4C2×C22⋊Q811yescharacteristic
476C2×C4#54311yescharacteristic
477C2×C4#54611yescharacteristic
478C2×C4#54611yescharacteristic
479C2×C4#55111yescharacteristic
480C2×C4C2×#13511yescharacteristic
481C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
482C2×C4C2×#13711yescharacteristic
483C2×C4#57511yescharacteristic
484C2×C4#57611yescharacteristic
485C2×C4#57711yescharacteristic
486C2×C4#57711yescharacteristic
487C2×C4#57811yescharacteristic
488C8—22no—
489C8—42no—
490C8—41no—
491C8—41no—
492C8—41no—
493C8—81no—
494C8—81no—
495C8—81no—
496C8—81no—
497C22—24no—
498C22—42no—
499C22—42no—
500C22—42no—
501C22—42no—
502C22—42no—
503C22—42no—
504C22#243711yescharacteristic
505C22#326611yescharacteristic
506C22#326811yescharacteristic
507C22#327211yescharacteristic
508C22#328511yescharacteristic
509C22#328811yescharacteristic
510C22#329311yescharacteristic
511C4—21no—
512C4—21no—
513C4—42no—
514C4—42no—
515C4—42no—
516C4—81no—
517C4—82no—
518C4—81no—
519C4—81no—
520C4—81no—
521C4—161no—
522C4—161no—
523C4—161no—
524C4C2×#47911yescharacteristic
525C4#307711yescharacteristic
526C4#308111yescharacteristic
527C4#308211yescharacteristic
528C4#330111yescharacteristic
529C4#331211yescharacteristic
530C4#331811yescharacteristic
531C4#332311yescharacteristic
532C2—44no—
533C2#1984111yescharacteristic
534C2#1993311yescharacteristic
535C2#2004311yescharacteristic
536C2#2017411yescharacteristic
537C2#2596511yescharacteristic
538C2#2597611yescharacteristic
539C2#2598711yescharacteristic
subgroupG/H
center, socleC23D42
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ C2×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. #26093C2semidirect
5. #10684C2semidirect
6. #10602C2semidirect
7. #10592C2semidirect
8. #10353C2semidirect
9. #8887C2semidirect
11. #8848C2semidirect
13. #8655C2semidirect

Maximal subgroups

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

subgroupindex
#261112
#260932
#225232
D4×#1032
#106842
#106022
#105922
#103532
#88872
#88862
#88482
#88432
#86552
#86442
#84472