Finite Group Databasebeta

← #325495

Subgroups — #325495

Order 512.

682 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33316C211yesmaximal, characteristic, complemented
2#29139C211yesmaximal, characteristic
3#29121C211yesmaximal, characteristic
4#21682C211yesmaximal, characteristic
5#10619C211yesmaximal, characteristic, complemented
6#10619C211yesmaximal, characteristic, complemented
7#10616C211yesmaximal, characteristic, complemented
8#10615C211yesmaximal, characteristic, complemented
9#9900C211yesmaximal, characteristic, complemented
10#9898C211yesmaximal, characteristic, complemented
11#8652C211yesmaximal, characteristic, complemented
12#8650C211yesmaximal, characteristic, complemented
13#8644C211yesmaximal, characteristic, complemented
14#8642C211yesmaximal, characteristic, complemented
15#8275C211yesmaximal, characteristic, complemented
16C2×#501—21no—
17C2×#501—21no—
18C2×#436C2211yescharacteristic
19C2×#435—21no—
20C2×#435—21no—
21C2×C8×D4C2211yescharacteristic
22#2868C2211yescharacteristic
23#2859C2211yescharacteristic
24C4×#127C2211yescharacteristic
25#2099—21no—
26#2041—21no—
27#2023—21no—
28#2023—21no—
29#1957—21no—
30#1955—21no—
31#1955—21no—
32#1954C2211yescharacteristic
33#1954C2211yescharacteristic
34#1934—21no—
35#1934—21no—
36#1847—21no—
37#1847—21no—
38#1845C2211yescharacteristic
39#1845C2211yescharacteristic
40#1844C2211yescharacteristic
41#1844C2211yescharacteristic
42#1842C2211yescharacteristic
43#1834—21no—
44#1802C2211yescharacteristic
45#1802C2211yescharacteristic
46#1802C2211yescharacteristic
47#1802C2211yescharacteristic
48#1802C2211yescharacteristic
49#1802C2211yescharacteristic
50#1802C2211yescharacteristic
51#1802C2211yescharacteristic
52C4×C4⋊C8C2211yescharacteristic
53C4×C4⋊C8C2211yescharacteristic
54C4×#97C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
57#1296C2211yescharacteristic
58#1296C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C22×M16—21no—
68C23×C8—21no—
69C2×C4⋊Q8—21no—
70C2×C4⋊Q8—21no—
71C2×C4⋊Q8—21no—
72C2×#127D411yescharacteristic
73C2×#125—21no—
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76C2×C4×D4C2311yescharacteristic
77C2×C4×D4C2311yescharacteristic
78#501—44no—
79#501—44no—
80#436—22no—
81#436—22no—
82#435—44no—
83#435—44no—
84C8×D4—22no—
85C8×D4—22no—
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⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×#102—21no—
103C2×#102—21no—
104C2×#102—21no—
105C2×#102—21no—
106C2×#102—21no—
107C2×#102—21no—
108C2×#102—21no—
109C2×#102—21no—
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112C2×C22⋊C8—21no—
113C2×C22⋊C8—21no—
114C2×C22⋊C8—21no—
115C2×C22⋊C8—21no—
116C2×#97D411yescharacteristic
117C2×#97D411yescharacteristic
118C2×#97D411yescharacteristic
119C2×#97D411yescharacteristic
120C2×C4×C8—21no—
121C2×C4×C8—21no—
122C2×C4×C8C2311yescharacteristic
123C2×C4×C8C2311yescharacteristic
124#390—21no—
125#390—21no—
126#390D411yescharacteristic
127#390D411yescharacteristic
128#389—21no—
129#389—21no—
130#389—21no—
131#389—21no—
132#384—21no—
133#384—21no—
134#384—21no—
135#384D411yescharacteristic
136#384C2311yescharacteristic
137#384C2311yescharacteristic
138#384C2311yescharacteristic
139#384C2311yescharacteristic
140#382—21no—
141C43C2311yescharacteristic
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#336—21no—
151#336—21no—
152#335—22no—
153#334—22no—
154C22×Q8—41no—
155C22×D4—21no—
156C22×D4—21no—
157C22×D4C2×D411yescharacteristic
158C22×D4C2×D411yescharacteristic
159C23×C4—21no—
160C23×C4—21no—
161C2×M16—42no—
162C2×M16—42no—
163C2×M16—42no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—42no—
171C22×C8—42no—
172C22×C8—42no—
173C22×C8C2×D411yescharacteristic
174C22×C8C2×D411yescharacteristic
175C22×C8C2×D411yescharacteristic
176C22×C8C2×D411yescharacteristic
177C4⋊Q8—42no—
178C4⋊Q8—42no—
179C4⋊Q8—81no—
180C4⋊Q8—81no—
181C4⋊Q8—82no—
182#127—22no—
183#127—41no—
184#125—42no—
185#125—42no—
186C4×D4—21no—
187C4×D4—21no—
188C4×D4—21no—
189C4×D4—21no—
190C4×D4—21no—
191C4×D4—21no—
192C4×D4—21no—
193C4×D4—21no—
194C4×D4—42no—
195C4×D4—42no—
196C4×D4—42no—
197C4×D4—42no—
198C2×C4⋊C4—21no—
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—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4C2×D411yescharacteristic
218C2×C4⋊C4C2×D411yescharacteristic
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
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—42no—
237#106—42no—
238#106—42no—
239#106—42no—
240C4⋊C8—21no—
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—42no—
254#102—42no—
255#102—42no—
256#102—81no—
257#102—81no—
258#102—81no—
259#102—81no—
260#102—81no—
261#102—81no—
262C22⋊C8—42no—
263C22⋊C8—42no—
264C22⋊C8—42no—
265C22⋊C8—42no—
266C22⋊C8—42no—
267C22⋊C8—42no—
268#97—22no—
269#97—22no—
270#97D814yes—
271#97SD1614yes—
272C4×C8—21no—
273C4×C8—21no—
274C4×C8—42no—
275C4×C8—81no—
276C4×C8#4112yes—
277C4×C8#4112yes—
278#95—41no—
279#95—41no—
280C24—21no—
281C24—21no—
282C2×Q8—82no—
283C2×Q8—82no—
284C2×Q8—82no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—81no—
301C2×D4—81no—
302C2×D4#13712yes—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4#12111yescharacteristic
322C22×C4#12111yescharacteristic
323C22×C4#12111yescharacteristic
324C22×C4#12111yescharacteristic
325C22×C4C22×D411yescharacteristic
326C22×C4C22×D411yescharacteristic
327C22×C4C2×#4111yescharacteristic
328M16—84no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—21no—
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×C8—41no—
344C2×C8—41no—
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—84no—
354C2×C8#12412yes—
355C2×C8#12412yes—
356C2×C8C2×D812yes—
357C2×C8C2×D812yes—
358C2×C8C2×SD1612yes—
359C2×C8C2×SD1612yes—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
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—81no—
383C4⋊C4—81no—
384C4⋊C4—81no—
385C4⋊C4—81no—
386C4⋊C4—82no—
387C4⋊C4—82no—
388C4⋊C4—82no—
389C4⋊C4—82no—
390C4⋊C4—82no—
391C22⋊C4—41no—
392C22⋊C4—41no—
393C22⋊C4—41no—
394C22⋊C4—41no—
395C22⋊C4—42no—
396C22⋊C4—42no—
397C22⋊C4—42no—
398C22⋊C4—42no—
399C42—21no—
400C42—21no—
401C42—21no—
402C42—21no—
403C42—41no—
404C42—41no—
405C42C2×D812yes—
406C42C2×SD1612yes—
407C42AGL(1,ℤ/8)12yes—
408C42#13712yes—
409C42C2×#4111yescharacteristic
410C42C2×#4111yescharacteristic
411C422−1+411yescharacteristic
412C422−1+411yescharacteristic
413C23—22no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23C2×#12111yescharacteristic
428Q8—164no—
429D4—22no—
430D4—22no—
431D4—41no—
432D4—41no—
433D4—41no—
434D4—41no—
435D4—41no—
436D4—41no—
437D4—84no—
438D4—84no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
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—81no—
474C2×C4—81no—
475C2×C4#45912yes—
476C2×C4#45912yes—
477C2×C4#46112yes—
478C2×C4#46112yes—
479C2×C4C2×#12411yescharacteristic
480C2×C4#53611yescharacteristic
481C2×C4#54711yescharacteristic
482C2×C4#54711yescharacteristic
483C2×C4#55611yescharacteristic
484C2×C4#55611yescharacteristic
485C2×C4C22×D811yescharacteristic
486C2×C4C22×SD1611yescharacteristic
487C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
488C2×C4C2×#13711yescharacteristic
489C2×C4C2×#13711yescharacteristic
490C2×C4#57511yescharacteristic
491C2×C4#57611yescharacteristic
492C2×C4#57711yescharacteristic
493C8—22no—
494C8—22no—
495C8—22no—
496C8—81no—
497C8—81no—
498C8—81no—
499C8—81no—
500C8—81no—
501C8—81no—
502C8#48614yes—
503C22—42no—
504C22—42no—
505C22—42no—
506C22—81no—
507C22—81no—
508C22—81no—
509C22—81no—
510C22—81no—
511C22—81no—
512C22—81no—
513C22—81no—
514C22—81no—
515C22—81no—
516C22—81no—
517C22#254711yescharacteristic
518C22C2×#45911yescharacteristic
519C22C2×#46111yescharacteristic
520C22#305611yescharacteristic
521C22#306211yescharacteristic
522C22#306811yescharacteristic
523C22#306911yescharacteristic
524C4—22no—
525C4—22no—
526C4—21no—
527C4—21no—
528C4—22no—
529C4—22no—
530C4—81no—
531C4—81no—
532C4—161no—
533C4—161no—
534C4—161no—
535C4—161no—
536C4C2×#48611yescharacteristic
537C4#314111yescharacteristic
538C4#322611yescharacteristic
539C4#322711yescharacteristic
540C4#334411yescharacteristic
541C4#334611yescharacteristic
542C4#334711yescharacteristic
543C4#334811yescharacteristic
544C2—82no—
545C2—161no—
546C2#1916311yescharacteristic
547C2#1916511yescharacteristic
548C2#1923011yescharacteristic
549C2#1923211yescharacteristic
550C2#2871511yescharacteristic
551C2#2871611yescharacteristic
552C2#2873511yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127D4
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
1. #33316C2semidirect
5. #10619C2semidirect
6. #10619C2semidirect
7. #10616C2semidirect
8. #10615C2semidirect
9. #9900C2semidirect
10. #9898C2semidirect
11. #8652C2semidirect
12. #8650C2semidirect
13. #8644C2semidirect
14. #8642C2semidirect
15. #8275C2semidirect

Maximal subgroups

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

subgroupindex
#333162
#291392
#291212
#216822
#106192
#106162
#106152
#99002
#98982
#86522
#86502
#86442
#86422
#82752