Finite Group Databasebeta

← #229934

Subgroups — #229934

Order 512.

672 conjugacy classes of subgroups.

The subgroup lattice has 555 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23698C211yesmaximal, characteristic
2#22953C211yesmaximal, characteristic
3#22952C211yesmaximal, characteristic
4C2×#1579C211yesmaximal, characteristic, complemented
5C2×#1570C211yesmaximal, characteristic
6#9664C211yesmaximal, characteristic, complemented
7#9595C211yesmaximal, characteristic, complemented
8#9547C211yesmaximal, characteristic, complemented
9#9543C211yesmaximal, characteristic, complemented
10#8942C211yesmaximal, characteristic
11#8937C211yesmaximal, characteristic
12#8846C211yesmaximal, characteristic, complemented
13#8846C211yesmaximal, characteristic, complemented
14#8288C211yesmaximal, characteristic, complemented
15#8149C211yesmaximal, characteristic
16C2×C8×Q8C2211yescharacteristic
17C2×C8×Q8C2211yescharacteristic
18C2×C4×Q16C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C2×#423C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C4×C22⋊Q8C2211yescharacteristic
23C4×C22⋊Q8C2211yescharacteristic
24#2087C411yescharacteristic
25#2076C411yescharacteristic
26#2068C411yescharacteristic
27#2068C411yescharacteristic
28#2056C411yescharacteristic
29#2036C411yescharacteristic
30#2036C411yescharacteristic
31#2001C411yescharacteristic
32#1984C2211yescharacteristic
33#1984C2211yescharacteristic
34#1940—21no—
35#1939—21no—
36#1939C2211yescharacteristic
37#1939C2211yescharacteristic
38#1916—21no—
39#1916—21no—
40#1851—21no—
41#1850—21no—
42C4×#107C2211yescharacteristic
43#1770C2211yescharacteristic
44C8×C4⋊C4C2211yescharacteristic
45C4×#103C2211yescharacteristic
46C4×#103C2211yescharacteristic
47#1753—21no—
48C8×C22⋊C4—21no—
49C4×C22⋊C8C2211yescharacteristic
50C4×C22⋊C8C2211yescharacteristic
51#1579—24no—
52#1570—24no—
53C2×#334C2211yescharacteristic
54C2×#334C2211yescharacteristic
55C2×#334C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×Q8⋊C8C2211yescharacteristic
58C2×Q8⋊C8C2211yescharacteristic
59C2×Q8⋊C8C2211yescharacteristic
60C2×Q8⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62C2×#322C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1273C2211yescharacteristic
68#1273C2211yescharacteristic
69C22×Q16C2×C411yescharacteristic
70C22×M16—21no—
71C23×C8—21no—
72C2×C22⋊Q8C2×C411yescharacteristic
73C2×C22⋊Q8C2×C411yescharacteristic
74C2×C4×Q8C2311yescharacteristic
75C2×C4×Q8C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C8×Q8—22no—
78C8×Q8—22no—
79C8×Q8—41no—
80C8×Q8—41no—
81C4×Q16—22no—
82C4×Q16—22no—
83#431—22no—
84#431—22no—
85C2×#107C2×C411yescharacteristic
86C2×#107C2×C411yescharacteristic
87C2×#107C2×C411yescharacteristic
88#423—22no—
89#423—41no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—41no—
98C2×C4⋊C8C2×C411yescharacteristic
99C2×C4⋊C8C2×C411yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×#103C2×C411yescharacteristic
105C2×#103C2×C411yescharacteristic
106C2×#103C2×C411yescharacteristic
107C2×#103C2×C411yescharacteristic
108C2×#103C2×C411yescharacteristic
109C2×#103C2×C411yescharacteristic
110C2×C22⋊C8—21no—
111C2×C22⋊C8C2×C411yescharacteristic
112C2×C22⋊C8C2×C411yescharacteristic
113C2×C22⋊C8D411yescharacteristic
114C2×C22⋊C8D411yescharacteristic
115C2×#97—21no—
116C2×C4×C8—21no—
117C2×C4×C8—21no—
118C2×C4×C8—21no—
119C2×C4×C8C2311yescharacteristic
120C2×C4×C8C2311yescharacteristic
121C2×C4×C8C2311yescharacteristic
122C2×C4×C8C2311yescharacteristic
123C2×C4×C8C2311yescharacteristic
124C2×C4×C8C2311yescharacteristic
125#391C2×C411yescharacteristic
126#391C2×C411yescharacteristic
127#389C2×C411yescharacteristic
128#389C2×C411yescharacteristic
129#387—21no—
130#387—21no—
131#385—21no—
132#385—21no—
133#380C2×C411yescharacteristic
134#380C2×C411yescharacteristic
135C4×C4⋊C4—21no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4C2311yescharacteristic
138C4×C4⋊C4C2311yescharacteristic
139C4×C22⋊C4—21no—
140C4×C22⋊C4—21no—
141#340C2×C411yescharacteristic
142#340C2×C411yescharacteristic
143#340C2×C411yescharacteristic
144#340C2×C411yescharacteristic
145#340C2×C411yescharacteristic
146#340C2×C411yescharacteristic
147#336—21no—
148#336—21no—
149#336—21no—
150#336—21no—
151#336—21no—
152#336—21no—
153#336—21no—
154#336—21no—
155#336—21no—
156#336D411yescharacteristic
157#336D411yescharacteristic
158#334—22no—
159#334—22no—
160#334—22no—
161#334D414yes—
162Q8⋊C8—22no—
163Q8⋊C8—22no—
164Q8⋊C8—41no—
165Q8⋊C8—41no—
166#322—22no—
167#322—22no—
168C22×Q8C22×C411yescharacteristic
169C22×Q8C22×C411yescharacteristic
170C23×C4C22×C411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4#4111yescharacteristic
173C2×Q16—22no—
174C2×Q16—22no—
175C2×Q16—22no—
176C2×M16—42no—
177C2×M16—42no—
178C2×M16—81no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—42no—
186C22×C8—42no—
187C22×C8—81no—
188C22×C8C22×C411yescharacteristic
189C22×C8C22×C411yescharacteristic
190C22×C8C22×C411yescharacteristic
191C22×C8C22×C411yescharacteristic
192C22×C8C2×D411yescharacteristic
193C22×C8C2×D411yescharacteristic
194C22×C8C2×D411yescharacteristic
195C22×C8C2×D411yescharacteristic
196C22⋊Q8—22no—
197C22⋊Q8—22no—
198C22⋊Q8—22no—
199C22⋊Q8—41no—
200C4×Q8—21no—
201C4×Q8—21no—
202C4×Q8—21no—
203C4×Q8—21no—
204C4×Q8—21no—
205C4×Q8—21no—
206C4×Q8—21no—
207C4×Q8—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⋊C4C22×C411yescharacteristic
215C2×C4⋊C4C22×C411yescharacteristic
216C2×C4⋊C4C22×C411yescharacteristic
217C2×C4⋊C4C22×C411yescharacteristic
218C2×C4⋊C4C22×C411yescharacteristic
219C2×C4⋊C4C22×C411yescharacteristic
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×C42—21no—
232C2×C42C22×C411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42#4111yescharacteristic
235C2×C42C2411yescharacteristic
236#107—22no—
237#107—22no—
238#107—41no—
239C4⋊C8—22no—
240C4⋊C8—22no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—42no—
248C4⋊C8—42no—
249C4⋊C8—42no—
250C4⋊C8—42no—
251C4⋊C8—81no—
252C4⋊C8—81no—
253C4⋊C8C2×D412yes—
254C4⋊C8C2×D412yes—
255#103—22no—
256#103—22no—
257#103—22no—
258#103—22no—
259#103—22no—
260#103—41no—
261C22⋊C8—22no—
262C22⋊C8—22no—
263C22⋊C8—22no—
264C22⋊C8—22no—
265C22⋊C8—42no—
266#97—81no—
267C4×C8—21no—
268C4×C8—21no—
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—42no—
278C4×C8—42no—
279C4×C8—81no—
280C4×C8C2×D412yes—
281C4×C8#4112yes—
282#95—21no—
283#95—21no—
284#95—21no—
285#95—21no—
286#95—21no—
287#95—21no—
288#95—21no—
289#95—21no—
290#95—21no—
291#95—21no—
292C24C4×D411yescharacteristic
293C2×Q8—21no—
294C2×Q8—21no—
295C2×Q8—21no—
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8—21no—
301C2×Q8—21no—
302C2×Q8—21no—
303C2×Q8#13112yes—
304C2×Q8#13112yes—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
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—21no—
324C22×C4—21no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4C4×D411yescharacteristic
328C22×C4C4×D411yescharacteristic
329C22×C4C4×D411yescharacteristic
330C22×C4#13712yes—
331C22×C4#13712yes—
332C22×C4C23×C411yescharacteristic
333C22×C4C22×D411yescharacteristic
334C22×C4C2×#4111yescharacteristic
335Q16—28no—
336M16—84no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—22no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—22no—
347C2×C8—22no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—21no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—84no—
378C2×C8C4×D412yes—
379C2×C8C4×D412yes—
380C4⋊C4—24no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—81no—
399C4⋊C4#13112yes—
400C22⋊C4—24no—
401C22⋊C4—42no—
402C22⋊C4—42no—
403C22⋊C4—81no—
404C42—21no—
405C42—21no—
406C42—21no—
407C42—21no—
408C42—21no—
409C42—21no—
410C42—41no—
411C42—41no—
412C42—41no—
413C42—41no—
414C42—41no—
415C42—41no—
416C42—41no—
417C42—41no—
418C42C22×D411yescharacteristic
419C42C2×#4111yescharacteristic
420C42H(5,𝔽2)11yescharacteristic
421C422−1+411yescharacteristic
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23—21no—
428C23#44112yes—
429C23C2×C4×D411yescharacteristic
430Q8—22no—
431Q8—22no—
432Q8—22no—
433Q8—22no—
434Q8—41no—
435Q8—41no—
436Q8—41no—
437Q8—41no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—21no—
451C2×C4—22no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
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—41no—
465C2×C4—41no—
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—81no—
481C2×C4—81no—
482C2×C4#44112yes—
483C2×C4C2×C4×D411yescharacteristic
484C2×C4#52011yescharacteristic
485C2×C4D4211yescharacteristic
486C2×C4#54711yescharacteristic
487C2×C4#54811yescharacteristic
488C2×C4#55311yescharacteristic
489C2×C4C2×#13111yescharacteristic
490C2×C4C2×#13111yescharacteristic
491C2×C4C2×#13111yescharacteristic
492C2×C4#56811yescharacteristic
493C2×C4C2×#13711yescharacteristic
494C2×C4C2×#13711yescharacteristic
495C2×C4#57711yescharacteristic
496C2×C4#57711yescharacteristic
497C8—22no—
498C8—22no—
499C8—42no—
500C8—41no—
501C8—41no—
502C8—41no—
503C8—41no—
504C8—41no—
505C8—41no—
506C8—42no—
507C8—42no—
508C8—81no—
509C8—81no—
510C8—81no—
511C22—22no—
512C22—22no—
513C22—22no—
514C22—41no—
515C22—41no—
516C22—41no—
517C22—41no—
518C22#236111yescharacteristic
519C22#292911yescharacteristic
520C22#293311yescharacteristic
521C22C2×#44111yescharacteristic
522C22#295211yescharacteristic
523C22#295511yescharacteristic
524C22#295611yescharacteristic
525C4—21no—
526C4—21no—
527C4—22no—
528C4—42no—
529C4—41no—
530C4—42no—
531C4—42no—
532C4—41no—
533C4—81no—
534C4—81no—
535C4—81no—
536C4—81no—
537C4—81no—
538C4#292911yescharacteristic
539C4#293511yescharacteristic
540C4#299211yescharacteristic
541C4#299411yescharacteristic
542C4#326611yescharacteristic
543C4#326611yescharacteristic
544C4#334411yescharacteristic
545C4#334411yescharacteristic
546C2—42no—
547C2#1790611yescharacteristic
548C2#1792711yescharacteristic
549C2#1800311yescharacteristic
550C2#1803111yescharacteristic
551C2#2299111yescharacteristic
552C2#2302111yescharacteristic
553C2#2302811yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC24C4×D4
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
4. C2×#1579C2semidirect
6. #9664C2semidirect
7. #9595C2semidirect
8. #9547C2semidirect
9. #9543C2semidirect
12. #8846C2semidirect
13. #8846C2semidirect
14. #8288C2semidirect

Maximal subgroups

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

subgroupindex
#236982
#229532
#229522
C2×#15792
C2×#15702
#96642
#95952
#95472
#95432
#89422
#89372
#88462
#82882
#81492