Finite Group Databasebeta

← #276135

Subgroups — #276135

Order 512.

595 conjugacy classes of subgroups.

The subgroup lattice has 523 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29868C211yesmaximal, characteristic
2#28689C211yesmaximal, characteristic, complemented
3#22796C211yesmaximal, characteristic
4#22111C211yesmaximal, characteristic
5#10481C211yesmaximal, characteristic
6#10480C211yesmaximal, characteristic
7#10278C211yesmaximal, characteristic
8#10278C211yesmaximal, characteristic
9#9739C211yesmaximal, characteristic, complemented
10#9730C211yesmaximal, characteristic, complemented
11#9691C211yesmaximal, characteristic, complemented
12#9603C211yesmaximal, characteristic, complemented
13#9575C211yesmaximal, characteristic, complemented
14#9567C211yesmaximal, characteristic, complemented
15#8590C211yesmaximal, characteristic, complemented
16C2×#500C2211yescharacteristic
17C2×#500C2211yescharacteristic
18C2×#476C2211yescharacteristic
19C2×#476C2211yescharacteristic
20C2×#476C2211yescharacteristic
21C2×#433—21no—
22#2868C2211yescharacteristic
23C4×C4⋊Q8C2211yescharacteristic
24#2293C2211yescharacteristic
25#2084—21no—
26#2084—21no—
27#2083—21no—
28#2083—21no—
29#2071—21no—
30#2071—21no—
31#1990C2211yescharacteristic
32#1990C2211yescharacteristic
33#1987C2211yescharacteristic
34#1987C2211yescharacteristic
35#1987C2211yescharacteristic
36#1954C2211yescharacteristic
37#1954C2211yescharacteristic
38#1948C2211yescharacteristic
39#1947—21no—
40#1947—21no—
41#1941—21no—
42#1934—21no—
43#1926—21no—
44#1924—21no—
45#1923—21no—
46#1919—21no—
47#1842C2211yescharacteristic
48#1831C2211yescharacteristic
49#1802C2211yescharacteristic
50#1798—21no—
51#1770C2211yescharacteristic
52C8×C4⋊C4C2211yescharacteristic
53C4×C4⋊C8C2211yescharacteristic
54C4×C4⋊C8C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#333C2211yescharacteristic
58C2×#333C2211yescharacteristic
59C2×#333C2211yescharacteristic
60C2×#333C2211yescharacteristic
61C2×#333C2211yescharacteristic
62C2×#333C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C2×C4⋊Q8—21no—
68C2×C4⋊Q8—21no—
69C2×C4⋊Q8C2311yescharacteristic
70C2×C4⋊Q8C2311yescharacteristic
71C2×C4⋊Q8C2311yescharacteristic
72C2×C4×Q8—21no—
73C2×C4×D4C2311yescharacteristic
74C2×#117—21no—
75#500—22no—
76#500—22no—
77#500—41no—
78#500—41no—
79#476—22no—
80#476—22no—
81#476—41no—
82#476—41no—
83#476—41no—
84#476—41no—
85#433—44no—
86C2×#107—21no—
87C2×#107—21no—
88C2×#107—21no—
89C2×#107—21no—
90C2×#106—21no—
91C2×#106—21no—
92C2×#106—21no—
93C2×#106—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8D411yescharacteristic
97C2×C4⋊C8D411yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×#102—21no—
105C2×#102—21no—
106C2×#102—21no—
107C2×#102—21no—
108C2×C22⋊C8—21no—
109C2×C22⋊C8—21no—
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112C2×#97—21no—
113C2×C4×C8—21no—
114C2×C4×C8—21no—
115C2×C4×C8C2311yescharacteristic
116C2×C4×C8C2311yescharacteristic
117#391—21no—
118#390D411yescharacteristic
119#390D411yescharacteristic
120#389—21no—
121#389—21no—
122#389D411yescharacteristic
123#389D411yescharacteristic
124#388—21no—
125#388—21no—
126#384—21no—
127#384—21no—
128#384—21no—
129#384—21no—
130#384C2311yescharacteristic
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4C2311yescharacteristic
134C4×C22⋊C4—21no—
135C43C2311yescharacteristic
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—21no—
147#340—21no—
148#336—21no—
149#336—21no—
150#336—21no—
151#336—21no—
152#336—21no—
153#336D411yescharacteristic
154#336D411yescharacteristic
155#335—41no—
156#334—41no—
157#333—22no—
158#333—22no—
159#333—41no—
160#333—41no—
161#333—41no—
162#333—41no—
163C22×Q8—21no—
164C22×Q8—21no—
165C22×Q8—21no—
166C22×D4C2×D411yescharacteristic
167C23×C4—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8C2×D411yescharacteristic
175C22×C8C2×D411yescharacteristic
176C22×C8C2×D411yescharacteristic
177C22×C8C2×D411yescharacteristic
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181C4⋊Q8—21no—
182C4⋊Q8—21no—
183C4⋊Q8—21no—
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—82no—
190C4⋊Q8—82no—
191C4⋊Q8#4112yes—
192C4×Q8—82no—
193C4×D4—21no—
194C4×D4—21no—
195C4×D4—21no—
196C4×D4—21no—
197#117—42no—
198#117—42no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4C2×D411yescharacteristic
216C2×C4⋊C4C2×D411yescharacteristic
217C2×C4⋊C4C2×D411yescharacteristic
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C42—21no—
224C2×C42—41no—
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42#4111yescharacteristic
230C2×C42#4111yescharacteristic
231C2×C42C2411yescharacteristic
232#107—42no—
233#107—42no—
234#107—42no—
235#107—42no—
236#106—24no—
237#106—42no—
238#106—42no—
239#106—81no—
240C4⋊C8—22no—
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—21no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8—42no—
253C4⋊C8—81no—
254C4⋊C8SD1614yes—
255C4⋊C8#4112yes—
256#102—24no—
257#102—42no—
258#102—42no—
259#102—81no—
260C22⋊C8—42no—
261C22⋊C8—42no—
262C22⋊C8—81no—
263C22⋊C8—81no—
264#97—42no—
265C4×C8—21no—
266C4×C8—21no—
267C4×C8—21no—
268C4×C8—21no—
269C4×C8—42no—
270C4×C8—81no—
271#95—21no—
272#95—21no—
273#95—21no—
274#95—41no—
275#95—41no—
276C24—21no—
277C2×Q8—22no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8—81no—
284C2×Q8—81no—
285C2×Q8—81no—
286C2×Q8—81no—
287C2×Q8—81no—
288C2×Q8—81no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4#12111yescharacteristic
309C22×C4#12111yescharacteristic
310C22×C4#12111yescharacteristic
311C22×C4#12111yescharacteristic
312C22×C4C22×D411yescharacteristic
313C22×C4C22×D411yescharacteristic
314C22×C4C2×#4111yescharacteristic
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—22no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8C2×SD1612yes—
344C2×C8C2×SD1612yes—
345C4⋊C4—22no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—22no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—81no—
370C4⋊C4—82no—
371C4⋊C4—82no—
372C4⋊C4—82no—
373C4⋊C4—82no—
374C4⋊C4—82no—
375C22⋊C4—41no—
376C22⋊C4—41no—
377C22⋊C4—42no—
378C22⋊C4—42no—
379C22⋊C4—42no—
380C22⋊C4—81no—
381C42—21no—
382C42—21no—
383C42—21no—
384C42—21no—
385C42—21no—
386C42—21no—
387C42—22no—
388C42—41no—
389C42—41no—
390C42—41no—
391C42—82no—
392C42C2×SD1612yes—
393C42#13712yes—
394C42C2×#4111yescharacteristic
395C42C2×#4111yescharacteristic
396C42H(5,𝔽2)11yescharacteristic
397C42H(5,𝔽2)11yescharacteristic
398C23—41no—
399C23—41no—
400C23—41no—
401C23—41no—
402C23—41no—
403C23—41no—
404C23—41no—
405C23C2×#12111yescharacteristic
406Q8—82no—
407Q8—82no—
408Q8—161no—
409Q8—161no—
410Q8—161no—
411Q8—161no—
412D4—41no—
413D4—41no—
414D4—41no—
415D4—41no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4#46112yes—
453C2×C4#46112yes—
454C2×C4#47212yes—
455C2×C4#52511yescharacteristic
456C2×C4#53511yescharacteristic
457C2×C4#54611yescharacteristic
458C2×C4#55011yescharacteristic
459C2×C4#55311yescharacteristic
460C2×C4#55511yescharacteristic
461C2×C4C22×SD1611yescharacteristic
462C2×C4C2×#13711yescharacteristic
463C2×C4#57711yescharacteristic
464C2×C4#57711yescharacteristic
465C2×C4#57811yescharacteristic
466C2×C4#57811yescharacteristic
467C2×C4#57811yescharacteristic
468C2×C4#57811yescharacteristic
469C8—22no—
470C8—22no—
471C8—41no—
472C8—41no—
473C8—42no—
474C8—81no—
475C8—81no—
476C8—81no—
477C8—81no—
478C8—81no—
479C22—81no—
480C22—81no—
481C22—81no—
482C22—81no—
483C22—81no—
484C22—81no—
485C22—81no—
486C22#241411yescharacteristic
487C22C2×#46111yescharacteristic
488C22#304611yescharacteristic
489C22#305911yescharacteristic
490C22#306311yescharacteristic
491C22C2×#47211yescharacteristic
492C22#306811yescharacteristic
493C4—22no—
494C4—21no—
495C4—21no—
496C4—22no—
497C4—22no—
498C4—41no—
499C4—81no—
500C4—81no—
501C4—82no—
502C4—81no—
503C4—161no—
504C4—161no—
505C4—161no—
506C4#312011yescharacteristic
507C4#312711yescharacteristic
508C4#322311yescharacteristic
509C4#322511yescharacteristic
510C4#329811yescharacteristic
511C4#332911yescharacteristic
512C4#339211yescharacteristic
513C4#339811yescharacteristic
514C2—161no—
515C2#1919311yescharacteristic
516C2#1923311yescharacteristic
517C2#1954711yescharacteristic
518C2#1959111yescharacteristic
519C2#2463011yescharacteristic
520C2#2466411yescharacteristic
521C2#2467011yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #28689C2semidirect
9. #9739C2semidirect
10. #9730C2semidirect
11. #9691C2semidirect
12. #9603C2semidirect
13. #9575C2semidirect
14. #9567C2semidirect
15. #8590C2semidirect

Maximal subgroups

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

subgroupindex
#298682
#286892
#227962
#221112
#104812
#104802
#102782
#97392
#97302
#96912
#96032
#95752
#95672
#85902