Finite Group Databasebeta

← #193270

Subgroups — #193270

Order 512.

1014 conjugacy classes of subgroups.

The subgroup lattice has 547 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34235C211yesmaximal, characteristic
2#21987C211yesmaximal, characteristic
3#21322C211yesmaximal, characteristic
4C2×#1947C211yesmaximal, characteristic
5C2×#1552C211yesmaximal, characteristic
6#8926C212yesmaximal, complemented
7#8923C212yesmaximal, complemented
8#8922C212yesmaximal, complemented
9#8922C212yesmaximal, complemented
10#7904C211yesmaximal, characteristic
11#7895C211yesmaximal, characteristic
12C2×#533C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C22×#107—21no—
15C22×C4⋊C8C2211yescharacteristic
16C2×C4×M16C2211yescharacteristic
17#2365C2211yescharacteristic
18#2365C2211yescharacteristic
19#2365C2211yescharacteristic
20#2365C2211yescharacteristic
21#2360—21no—
22#2360—21no—
23#2356—21no—
24#2351—21no—
25#2343—22no—
26#2336—22no—
27#2320—21no—
28C4×C22⋊Q8—21no—
29C4×C22⋊Q8—21no—
30C2×#384C2211yescharacteristic
31#2283C2211yescharacteristic
32#2276—21no—
33C2×C4×C4⋊C4C2211yescharacteristic
34#1992—22no—
35#1990C2212yes—
36#1990C2212yes—
37#1948C2212yes—
38#1947—21no—
39#1947—21no—
40#1947C2212yes—
41#1947C2212yes—
42#1947C2212yes—
43#1853—21no—
44#1852—21no—
45#1847—21no—
46#1836—21no—
47#1736—22no—
48C2×#340—21no—
49#1552C418yescomplemented
50C2×#332C2212yes—
51C2×#332C2212yes—
52#1279C2212yes—
53#1279C2212yes—
54#1279C2212yes—
55#1279C2212yes—
56C22×M16—21no—
57C23×C8—21no—
58#533—22no—
59#533—22no—
60#533C2×C414yes—
61#533Q814yes—
62C2×C4⋊Q8C2311yescharacteristic
63C2×C4⋊Q8C2311yescharacteristic
64C2×#125D412yes—
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8—21no—
67C2×C4×Q8—21no—
68C2×C4×Q8—21no—
69C2×C4×Q8—21no—
70C2×C4×Q8—21no—
71C2×C4×Q8—21no—
72C2×C4×Q8—21no—
73C2×#117D412yes—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4D411yescharacteristic
78C22×C4⋊C4D411yescharacteristic
79C22×C42—21no—
80C22×C42C2311yescharacteristic
81#432—22no—
82#432C2×C414yes—
83C2×#107—22no—
84C2×#107—22no—
85C2×#107—22no—
86C2×#107—44no—
87C2×#106—22no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8C2×C414yes—
91C2×C4⋊C8C2312yes—
92C2×C4⋊C8C2312yes—
93C2×C22⋊C8—21no—
94C4×M16—22no—
95C4×M16—22no—
96C2×#97—21no—
97C2×#97—21no—
98C2×C4×C8C2312yes—
99#391—21no—
100#391—21no—
101#391—21no—
102#391—21no—
103#389—22no—
104#389—22no—
105#389—21no—
106#389—21no—
107#389—21no—
108#389—21no—
109#387—22no—
110#387—22no—
111#387—22no—
112#387—22no—
113#387—22no—
114#387—22no—
115#387—22no—
116#387—22no—
117#385—22no—
118#385—22no—
119#384—21no—
120#384—21no—
121#384—21no—
122#384C2312yes—
123#383—22no—
124#383—21no—
125#382—22no—
126#382D411yescharacteristic
127#382D411yescharacteristic
128#381—21no—
129#380D411yescharacteristic
130#380D411yescharacteristic
131C4×C4⋊C4—22no—
132C4×C4⋊C4—22no—
133C4×C4⋊C4—22no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4—21no—
138C4×C4⋊C4—21no—
139C4×C4⋊C4—21no—
140C4×C4⋊C4D411yescharacteristic
141C4×C4⋊C4D411yescharacteristic
142C4×C4⋊C4C2311yescharacteristic
143C4×C4⋊C4C2311yescharacteristic
144C4×C4⋊C4C2311yescharacteristic
145C4×C4⋊C4C2311yescharacteristic
146C4×C22⋊C4—22no—
147C4×C22⋊C4—22no—
148C4×C22⋊C4—21no—
149#376—22no—
150#376—21no—
151C2×#95—21no—
152#340—22no—
153#340—22no—
154#340—22no—
155#340—22no—
156#340—24no—
157#340—24no—
158#332C2×C418yescomplemented
159#332C2×C418yescomplemented
160C22×Q8—21no—
161C22×Q8—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—41no—
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C2×M16—24no—
169C2×M16—42no—
170C2×M16—42no—
171C22×C8—24no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—42no—
176C22×C8—42no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181C4⋊Q8C22×C412yes—
182C4⋊Q8C22×C412yes—
183C4⋊Q8C2×Q812yes—
184C4⋊Q8C2×Q812yes—
185#125—24no—
186#125C4⋊C418yes—
187C22⋊Q8—24no—
188C22⋊Q8—24no—
189C22⋊Q8—24no—
190C22⋊Q8—24no—
191C4×Q8—24no—
192C4×Q8—24no—
193C4×Q8—24no—
194C4×Q8—24no—
195C4×Q8—42no—
196C4×Q8—42no—
197C4×Q8—42no—
198C4×Q8—42no—
199C4×Q8—82no—
200C4×Q8—82no—
201#117—24no—
202#117C4⋊C418yes—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—82no—
227C2×C4⋊C4C2×D411yescharacteristic
228C2×C4⋊C4C2×D411yescharacteristic
229C2×C4⋊C4C2×D411yescharacteristic
230C2×C4⋊C4C2×D411yescharacteristic
231C2×C4⋊C4C2×D411yescharacteristic
232C2×C4⋊C4C2×D411yescharacteristic
233C2×C4⋊C4C2×D412yes—
234C2×C4⋊C4C2×D412yes—
235C2×C4⋊C4C2×D411yescharacteristic
236C2×C4⋊C4C2×D411yescharacteristic
237C2×C22⋊C4—22no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—82no—
252C2×C42C22×C412yes—
253C2×C42C2×D411yescharacteristic
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C2×C42C2×Q812yes—
257C2×C42#4112yes—
258C2×C42#4112yes—
259C2×C42C2411yescharacteristic
260#107—44no—
261#107—44no—
262#107—82no—
263#106—82no—
264C4⋊C8—24no—
265C4⋊C8—22no—
266C4⋊C8—22no—
267C4⋊C8C22×C414yes—
268C4⋊C8C22×C414yes—
269C22⋊C8—24no—
270#97—24no—
271#97—42no—
272C4×C8—22no—
273C4×C8—22no—
274#95—22no—
275#95—22no—
276#95—22no—
277#95—22no—
278#95—22no—
279#95—22no—
280#95—22no—
281#95—22no—
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—
292#95—21no—
293#95—21no—
294C24#12011yescharacteristic
295C2×Q8—22no—
296C2×Q8—22no—
297C2×Q8—22no—
298C2×Q8—22no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×Q8—42no—
304C2×Q8—42no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—81no—
329C22×C4—81no—
330C22×C4—82no—
331C22×C4—82no—
332C22×C4—82no—
333C22×C4C2×C4⋊C412yes—
334C22×C4C4×D412yes—
335C22×C4C4×D412yes—
336C22×C4#12011yescharacteristic
337C22×C4#12011yescharacteristic
338C22×C4#12011yescharacteristic
339C22×C4C22⋊Q812yes—
340C22×C4C22⋊Q812yes—
341C22×C4#12312yes—
342C22×C4C22×D411yescharacteristic
343C22×C4C22×D411yescharacteristic
344C22×C4C22×D411yescharacteristic
345M16—48no—
346C2×C8—24no—
347C2×C8—24no—
348C2×C8—24no—
349C2×C8—48no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—28no—
365C4⋊C4—28no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—44no—
377C4⋊C4—44no—
378C4⋊C4—44no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—82no—
384C4⋊C4—82no—
385C4⋊C4—82no—
386C4⋊C4—82no—
387C4⋊C4—82no—
388C4⋊C4C2×C4⋊C414yes—
389C4⋊C4C2×C4⋊C414yes—
390C4⋊C4AGL(1,ℤ/8)12yes—
391C4⋊C4#13712yes—
392C22⋊C4—28no—
393C22⋊C4—42no—
394C22⋊C4—42no—
395C42—28no—
396C42—44no—
397C42—44no—
398C42—41no—
399C42—41no—
400C42—41no—
401C42—41no—
402C42—42no—
403C42—42no—
404C42—81no—
405C42—81no—
406C42—82no—
407C42—82no—
408C42C2×C4⋊C412yes—
409C42C4×D412yes—
410C42C4×D412yes—
411C42C22⋊Q812yes—
412C42C22⋊Q812yes—
413C42#12312yes—
414C42C23×C411yescharacteristic
415C42C22×Q811yescharacteristic
416C42C2×#4111yescharacteristic
417C42C2×#4111yescharacteristic
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23#38512yes—
423C23#45412yes—
424C23#45612yes—
425C23C2×#12011yescharacteristic
426Q8—42no—
427Q8—42no—
428Q8—44no—
429Q8—42no—
430Q8—42no—
431Q8—44no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—24no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—24no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—42no—
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—84no—
479C2×C4—84no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—164no—
484C2×C4#38512yes—
485C2×C4#38512yes—
486C2×C4#38512yes—
487C2×C4C22×C4⋊C411yescharacteristic
488C2×C4C2×C4×D411yescharacteristic
489C2×C4C2×C4×D411yescharacteristic
490C2×C4C2×C22⋊Q811yescharacteristic
491C2×C4C2×C22⋊Q811yescharacteristic
492C2×C4C2×#12311yescharacteristic
493C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
494C2×C4C2×#13711yescharacteristic
495C2×C4#57511yescharacteristic
496C2×C4#57511yescharacteristic
497C2×C4#57612yes—
498C2×C4#57812yes—
499C8—44no—
500C8—44no—
501C8—44no—
502C22—22no—
503C22—22no—
504C22—22no—
505C22—22no—
506C22—22no—
507C22—22no—
508C22#185714yes—
509C22C2×#38511yescharacteristic
510C22#300611yescharacteristic
511C22#300711yescharacteristic
512C22#300811yescharacteristic
513C22#301111yescharacteristic
514C22C2×#45411yescharacteristic
515C22C2×#45611yescharacteristic
516C4—21no—
517C4—21no—
518C4—22no—
519C4—22no—
520C4—22no—
521C4—44no—
522C4—44no—
523C4—42no—
524C4—42no—
525C4—81no—
526C4—81no—
527C4—81no—
528C4—81no—
529C4—81no—
530C4—81no—
531C4—162no—
532C4C4×AGL(1,ℤ/8)11yescharacteristic
533C4C4×#13711yescharacteristic
534C4#295112yes—
535C4#307911yescharacteristic
536C4#308011yescharacteristic
537C4#308312yes—
538C2—24no—
539C2#1857811yescharacteristic
540C2#1858111yescharacteristic
541C2#1858711yescharacteristic
542C2#1858811yescharacteristic
543C2C2×#185711yescharacteristic
544C2#2087511yescharacteristic
545C2#2088111yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #8926C2semidirect
7. #8923C2semidirect
8. #8922C2semidirect
9. #8922C2semidirect
49. #1552C4semidirect
158. #332C2×C4semidirect
159. #332C2×C4semidirect

Maximal subgroups

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

subgroupindex
#342352
#219872
#213222
C2×#19472
C2×#15522
#89262
#89232
#89222
#79042
#78952