Finite Group Databasebeta

← #231241

Subgroups — #231241

Order 512.

647 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23701C211yesmaximal, characteristic
2#23054C211yesmaximal, characteristic, complemented
3#22990C211yesmaximal, characteristic
4#9558C211yesmaximal, characteristic, complemented
5#9555C211yesmaximal, characteristic, complemented
6#9540C211yesmaximal, characteristic, complemented
7#9533C211yesmaximal, characteristic, complemented
8#8952C211yesmaximal, characteristic, complemented
9#8949C211yesmaximal, characteristic, complemented
10#8942C211yesmaximal, characteristic, complemented
11#8940C211yesmaximal, characteristic
12#8211C211yesmaximal, characteristic, complemented
13#8208C211yesmaximal, characteristic, complemented
14#8149C211yesmaximal, characteristic, complemented
15#7880C211yesmaximal, characteristic, complemented
16C2×#446C2211yescharacteristic
17C2×C4×SD16C2211yescharacteristic
18C2×#436—21no—
19C2×#436C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C2×#431C2211yescharacteristic
22C2×#423C2211yescharacteristic
23C4×C22⋊Q8C2211yescharacteristic
24C4×#121C2211yescharacteristic
25#2088C411yescharacteristic
26#2077C411yescharacteristic
27#2068C411yescharacteristic
28#2067C411yescharacteristic
29#2055C411yescharacteristic
30#2036C411yescharacteristic
31#2033C411yescharacteristic
32#2000C411yescharacteristic
33#1984C2211yescharacteristic
34#1954C2211yescharacteristic
35#1939C2211yescharacteristic
36#1939C2211yescharacteristic
37#1924—21no—
38#1924—21no—
39#1923—21no—
40#1916—21no—
41#1870—21no—
42#1851—21no—
43C4×#107C2211yescharacteristic
44#1771—21no—
45#1770C2211yescharacteristic
46C8×C4⋊C4C2211yescharacteristic
47C4×#103C2211yescharacteristic
48C4×#102C2211yescharacteristic
49C4×C22⋊C8C2211yescharacteristic
50C4×C22⋊C8C2211yescharacteristic
51C2×Q8⋊C8C2211yescharacteristic
52C2×Q8⋊C8C2211yescharacteristic
53C2×D4⋊C8C2211yescharacteristic
54C2×D4⋊C8C2211yescharacteristic
55C2×#322C2211yescharacteristic
56C2×#322C2211yescharacteristic
57#1279C2211yescharacteristic
58#1279C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1277C2211yescharacteristic
62#1277C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×SD16C2×C411yescharacteristic
68C22×M16—21no—
69C22×M16—21no—
70C2×C22⋊Q8C2×C411yescharacteristic
71C2×#121C2×C411yescharacteristic
72C2×C4×Q8C2311yescharacteristic
73C2×C4×D4—21no—
74C2×C4×D4C2311yescharacteristic
75C22×C42C2311yescharacteristic
76#446—22no—
77#446—41no—
78C4×SD16—41no—
79C4×SD16—41no—
80#436—22no—
81#436—41no—
82#436—44no—
83#431—22no—
84#431—41no—
85#431—41no—
86#431—41no—
87C2×#107C2×C411yescharacteristic
88C2×#107C2×C411yescharacteristic
89C2×#106C2×C411yescharacteristic
90#423—22no—
91#423—41no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8C2×C411yescharacteristic
97C2×C4⋊C8C2×C411yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×#103C2×C411yescharacteristic
103C2×#103C2×C411yescharacteristic
104C2×#103C2×C411yescharacteristic
105C2×#102C2×C411yescharacteristic
106C2×#102C2×C411yescharacteristic
107C2×#102C2×C411yescharacteristic
108C2×C22⋊C8—21no—
109C2×C22⋊C8—21no—
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112C2×C22⋊C8—21no—
113C2×C22⋊C8C2×C411yescharacteristic
114C2×C22⋊C8C2×C411yescharacteristic
115C2×C22⋊C8D411yescharacteristic
116C2×C22⋊C8D411yescharacteristic
117C2×#97—21no—
118C2×#97—21no—
119C2×C4×C8—21no—
120C2×C4×C8—21no—
121C2×C4×C8C2311yescharacteristic
122C2×C4×C8C2311yescharacteristic
123C2×C4×C8C2311yescharacteristic
124C2×C4×C8C2311yescharacteristic
125C2×C4×C8C2311yescharacteristic
126C2×C4×C8C2311yescharacteristic
127#391C2×C411yescharacteristic
128#390C2×C411yescharacteristic
129#389C2×C411yescharacteristic
130#389C2×C411yescharacteristic
131#388—21no—
132#387—21no—
133#386—21no—
134#385—21no—
135#380C2×C411yescharacteristic
136#380C2×C411yescharacteristic
137C4×C4⋊C4—21no—
138C4×C4⋊C4C2311yescharacteristic
139C4×C4⋊C4C2311yescharacteristic
140C4×C22⋊C4—21no—
141C4×C22⋊C4—21no—
142#340C2×C411yescharacteristic
143#340C2×C411yescharacteristic
144#340C2×C411yescharacteristic
145#340C2×C411yescharacteristic
146#340C2×C411yescharacteristic
147#340C2×C411yescharacteristic
148#336—21no—
149#336—21no—
150#336—21no—
151#336—21no—
152#336—21no—
153#336—21no—
154#336—21no—
155#336D411yescharacteristic
156#336D411yescharacteristic
157Q8⋊C8—22no—
158Q8⋊C8—41no—
159D4⋊C8—22no—
160D4⋊C8—41no—
161#322—41no—
162#322—41no—
163C22×Q8C22×C411yescharacteristic
164C22×D4—21no—
165C22×D4C22×C411yescharacteristic
166C23×C4—21no—
167C23×C4C22×C411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4#4111yescharacteristic
170C2×SD16—41no—
171C2×SD16—41no—
172C2×SD16—41no—
173C2×M16—42no—
174C2×M16—42no—
175C2×M16—42no—
176C2×M16—42no—
177C2×M16—81no—
178C2×M16—81no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8C22×C411yescharacteristic
186C22×C8C22×C411yescharacteristic
187C22×C8C22×C411yescharacteristic
188C22×C8C22×C411yescharacteristic
189C22×C8C2×D411yescharacteristic
190C22×C8C2×D411yescharacteristic
191C22×C8C2×D411yescharacteristic
192C22×C8C2×D411yescharacteristic
193C22⋊Q8—41no—
194C22⋊Q8—41no—
195#121—22no—
196#121—41no—
197C4×Q8—21no—
198C4×Q8—21no—
199C4×Q8—21no—
200C4×Q8—21no—
201C4×D4—21no—
202C4×D4—21no—
203C4×D4—21no—
204C4×D4—21no—
205C4×D4—42no—
206C4×D4—42no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4C22×C411yescharacteristic
212C2×C4⋊C4C22×C411yescharacteristic
213C2×C4⋊C4C22×C411yescharacteristic
214C2×C4⋊C4C22×C411yescharacteristic
215C2×C4⋊C4C22×C411yescharacteristic
216C2×C4⋊C4C22×C411yescharacteristic
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42C22×C411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42#4111yescharacteristic
234C2×C42C2411yescharacteristic
235#107—22no—
236#107—41no—
237#106—41no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—41no—
245C4⋊C8—41no—
246C4⋊C8—42no—
247C4⋊C8—42no—
248C4⋊C8—42no—
249C4⋊C8—42no—
250C4⋊C8#4112yes—
251C4⋊C8#4112yes—
252#103—22no—
253#103—41no—
254#103—41no—
255#102—41no—
256#102—41no—
257#102—41no—
258C22⋊C8—22no—
259C22⋊C8—41no—
260C22⋊C8—41no—
261C22⋊C8—41no—
262C22⋊C8—42no—
263C22⋊C8—42no—
264C22⋊C8—42no—
265C22⋊C8—42no—
266C22⋊C8—81no—
267#97—42no—
268#97—42no—
269C4×C8—21no—
270C4×C8—21no—
271C4×C8—21no—
272C4×C8—21no—
273C4×C8—21no—
274C4×C8—21no—
275C4×C8—21no—
276C4×C8—21no—
277C4×C8—21no—
278C4×C8—21no—
279C4×C8—21no—
280C4×C8—21no—
281C4×C8—42no—
282C4×C8—42no—
283#95—21no—
284#95—21no—
285#95—21no—
286#95—21no—
287#95—21no—
288#95—21no—
289#95—21no—
290#95—21no—
291C24—21no—
292C24C4×D411yescharacteristic
293C2×Q8—21no—
294C2×Q8—21no—
295C2×Q8—21no—
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—81no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
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—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—22no—
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—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4C4×D411yescharacteristic
338C22×C4C4×D411yescharacteristic
339C22×C4C4×D411yescharacteristic
340C22×C4C23×C411yescharacteristic
341C22×C4C22×D411yescharacteristic
342C22×C4C2×#4111yescharacteristic
343SD16—44no—
344M16—84no—
345M16—84no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—21no—
355C2×C8—22no—
356C2×C8—22no—
357C2×C8—21no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—21no—
362C2×C8—21no—
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8AGL(1,ℤ/8)12yes—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—42no—
404C4⋊C4#13112yes—
405C22⋊C4—42no—
406C22⋊C4—42no—
407C22⋊C4—42no—
408C22⋊C4—42no—
409C22⋊C4—41no—
410C22⋊C4—41no—
411C22⋊C4—42no—
412C22⋊C4—81no—
413C42—21no—
414C42—21no—
415C42—21no—
416C42—21no—
417C42—21no—
418C42—21no—
419C42—41no—
420C42—41no—
421C42—41no—
422C42—41no—
423C42—41no—
424C42—41no—
425C42C2×#4111yescharacteristic
426C42C2×#4111yescharacteristic
427C42H(5,𝔽2)11yescharacteristic
428C42H(5,𝔽2)11yescharacteristic
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23—41no—
443C23C2×C4×D411yescharacteristic
444Q8—41no—
445Q8—41no—
446Q8—41no—
447Q8—41no—
448D4—41no—
449D4—41no—
450D4—41no—
451D4—41no—
452D4—84no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—22no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—42no—
465C2×C4—42no—
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—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4#44412yes—
499C2×C4C4×#4111yescharacteristic
500C2×C4#51811yescharacteristic
501C2×C4#54611yescharacteristic
502C2×C4#54611yescharacteristic
503C2×C4#55311yescharacteristic
504C2×C4#55311yescharacteristic
505C2×C4C2×#13111yescharacteristic
506C2×C4#56811yescharacteristic
507C2×C4#56811yescharacteristic
508C2×C4#56811yescharacteristic
509C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
510C2×C4#57511yescharacteristic
511C2×C4#57711yescharacteristic
512C2×C4#57811yescharacteristic
513C8—22no—
514C8—41no—
515C8—41no—
516C8—41no—
517C8—41no—
518C8—41no—
519C8—42no—
520C8—41no—
521C8—42no—
522C8—41no—
523C8—81no—
524C8—81no—
525C8—81no—
526C8—81no—
527C22—41no—
528C22—41no—
529C22—41no—
530C22—41no—
531C22—41no—
532C22—41no—
533C22—41no—
534C22—81no—
535C22—81no—
536C22—81no—
537C22—81no—
538C22—81no—
539C22—81no—
540C22—81no—
541C22#236411yescharacteristic
542C22#293011yescharacteristic
543C22#293311yescharacteristic
544C22C4×AGL(1,ℤ/8)11yescharacteristic
545C22#294711yescharacteristic
546C22C2×#44411yescharacteristic
547C22#295711yescharacteristic
548C4—21no—
549C4—21no—
550C4—22no—
551C4—41no—
552C4—42no—
553C4—41no—
554C4—81no—
555C4—81no—
556C4—81no—
557C4—81no—
558C4—81no—
559C4—81no—
560C4#296711yescharacteristic
561C4#296911yescharacteristic
562C4#297811yescharacteristic
563C4#298111yescharacteristic
564C4#330711yescharacteristic
565C4#330811yescharacteristic
566C4#332411yescharacteristic
567C4#332711yescharacteristic
568C2—81no—
569C2—161no—
570C2#1794011yescharacteristic
571C2#1794711yescharacteristic
572C2#1800611yescharacteristic
573C2#1802411yescharacteristic
574C2#2312511yescharacteristic
575C2#2314111yescharacteristic
576C2#2314711yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#121C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×C22⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #23054C2semidirect
4. #9558C2semidirect
5. #9555C2semidirect
6. #9540C2semidirect
7. #9533C2semidirect
8. #8952C2semidirect
9. #8949C2semidirect
10. #8942C2semidirect
12. #8211C2semidirect
13. #8208C2semidirect
14. #8149C2semidirect
15. #7880C2semidirect

Maximal subgroups

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

subgroupindex
#237012
#230542
#229902
#95582
#95552
#95402
#95332
#89522
#89492
#89422
#89402
#82112
#82082
#81492
#78802