Finite Group Databasebeta

← #495693

Subgroups — #495693

Order 512.

1104 conjugacy classes of subgroups.

The subgroup lattice has 565 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#135C211yesmaximal, characteristic
2C2×#2189C211yesmaximal, characteristic
3C2×#2176C211yesmaximal, characteristic, complemented
4C2×#2175C211yesmaximal, characteristic, complemented
5C4×#467C211yesmaximal, characteristic, complemented
6#13015C211yesmaximal, characteristic, complemented
7#13014C211yesmaximal, characteristic, complemented
8#12994C211yesmaximal, characteristic, complemented
9#12937C211yesmaximal, characteristic, complemented
10#12936C211yesmaximal, characteristic, complemented
11#12921C211yesmaximal, characteristic, complemented
12#12733C211yesmaximal, characteristic
13#12638C211yesmaximal, characteristic, complemented
14#12637C211yesmaximal, characteristic, complemented
15#12599C211yesmaximal, characteristic, complemented
16C22×#135C2211yescharacteristic
17C2×C4×#41—21no—
18C22×Q32C2211yescharacteristic
19C22×SD32C2211yescharacteristic
20C2×#467C2211yescharacteristic
21C4×#135—24no—
22C4×#135—24no—
23C2×C4×Q16C2211yescharacteristic
24C2×C4×Q16C2211yescharacteristic
25C2×C4×Q16C2211yescharacteristic
26C2×C4×SD16—21no—
27C2×C4×SD16—21no—
28C2×C4×D8C2211yescharacteristic
29C2×C4×D8C2211yescharacteristic
30C2×#427C2211yescharacteristic
31C2×#416—21no—
32C22×C4×C8C2211yescharacteristic
33C4×C22⋊Q8—21no—
34#2189C418yescomplemented
35#2176—24no—
36#2175—24no—
37C2×#367C2211yescharacteristic
38C2×#366C2211yescharacteristic
39C2×#358C2211yescharacteristic
40C2×#358C2211yescharacteristic
41C2×#358C2211yescharacteristic
42C2×#358C2211yescharacteristic
43C2×#358C2211yescharacteristic
44C2×#357C2211yescharacteristic
45C2×#357C2211yescharacteristic
46C2×#357C2211yescharacteristic
47C2×C22⋊C16C2211yescharacteristic
48C2×C22⋊C16C2211yescharacteristic
49C2×#346C2211yescharacteristic
50C2×#346C2211yescharacteristic
51#1969C2211yescharacteristic
52#1942C2211yescharacteristic
53#1933—21no—
54#1892—21no—
55#1808C2211yescharacteristic
56C4×#107C2211yescharacteristic
57C4×#103—21no—
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60#1380C2211yescharacteristic
61#1380C2211yescharacteristic
62#1374C2211yescharacteristic
63#1374C2211yescharacteristic
64C22×#41—21no—
65C2×#135—22no—
66C2×#135—22no—
67C2×#135—22no—
68C2×#135—22no—
69C2×#135—22no—
70C2×#135C2×C414yes—
71C22×Q16C2311yescharacteristic
72C22×Q16C2311yescharacteristic
73C22×SD16—21no—
74C22×D8C2311yescharacteristic
75C23×C8D411yescharacteristic
76C23×C8C2311yescharacteristic
77C2×C22⋊Q8—21no—
78C4×#41—28no—
79C4×#41—44no—
80C2×C4×Q8—21no—
81C2×C4×Q8—21no—
82C2×C4×Q8—21no—
83C2×C4×D4—21no—
84C2×C4×D4—21no—
85C2×C4×D4—41no—
86C2×C4×D4—41no—
87C2×#117—21no—
88C2×#117—41no—
89C22×C42—41no—
90C22×C42D411yescharacteristic
91C2×Q32—22no—
92C2×Q32—22no—
93C2×Q32C2×C414yescomplemented
94C2×SD32—22no—
95C2×SD32—22no—
96C2×SD32C2×C414yescomplemented
97C22×C16C2311yescharacteristic
98C22×C16C2311yescharacteristic
99C22×C16C2311yescharacteristic
100C22×C16C2311yescharacteristic
101#467—22no—
102#467C2×C414yescomplemented
103C4×Q16—22no—
104C4×Q16—22no—
105C4×Q16—22no—
106C4×Q16—22no—
107C4×Q16—22no—
108C4×Q16D414yes—
109C4×SD16—28no—
110C4×SD16—28no—
111C4×D8—22no—
112C4×D8—22no—
113C4×D8—22no—
114C4×D8D414yes—
115#427—22no—
116#427—22no—
117C2×#107C2311yescharacteristic
118C2×#107C2311yescharacteristic
119C2×#107C2311yescharacteristic
120C2×#107C2311yescharacteristic
121C2×#107C2311yescharacteristic
122C2×#106—21no—
123#416—28no—
124C2×#103—21no—
125C2×#103—21no—
126C2×#103—21no—
127C2×#103—21no—
128C2×#103—21no—
129C2×#102—21no—
130C2×#102—21no—
131C2×C4×C8—22no—
132C2×C4×C8—22no—
133C2×C4×C8D411yescharacteristic
134C2×C4×C8D411yescharacteristic
135C2×C4×C8C2311yescharacteristic
136C2×C4×C8C2311yescharacteristic
137#391—21no—
138#389—21no—
139#387—41no—
140#385—41no—
141#380—21no—
142C4×C4⋊C4—21no—
143C4×C4⋊C4—41no—
144C4×C22⋊C4—41no—
145#367—22no—
146#366—22no—
147#358—22no—
148#358—22no—
149#358—22no—
150#358—22no—
151#358C2×C414yescomplemented
152#357—22no—
153#357—22no—
154#357C2×C414yescomplemented
155C22⋊C16—22no—
156C22⋊C16C2×C414yescomplemented
157#346—22no—
158#346—22no—
159#340—21no—
160#340—21no—
161#340—21no—
162C2×#41—24no—
163C2×#41—24no—
164C2×#41—24no—
165C2×#41—42no—
166C2×#41—42no—
167C2×#41—42no—
168C22×Q8—21no—
169C22×Q8—21no—
170C22×D4—21no—
171C22×D4—41no—
172C23×C4—41no—
173C23×C4—41no—
174C23×C4—41no—
175C23×C4D811yescharacteristic
176C23×C4D811yescharacteristic
177C23×C4C2×D411yescharacteristic
178#135—28no—
179#135—28no—
180#135—28no—
181#135—28no—
182C2×Q16—21no—
183C2×Q16—21no—
184C2×Q16—21no—
185C2×Q16—21no—
186C2×Q16—21no—
187C2×Q16—21no—
188C2×Q16—21no—
189C2×Q16—21no—
190C2×Q16C22×C412yes—
191C2×Q16C22×C412yes—
192C2×Q16C2×D412yes—
193C2×Q16#4112yes—
194C2×SD16—24no—
195C2×SD16—24no—
196C2×SD16—24no—
197C2×D8—21no—
198C2×D8—21no—
199C2×D8—21no—
200C2×D8C22×C412yes—
201C2×D8C2×D412yes—
202C2×D8#4112yes—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—41no—
210C22×C8C22×C412yes—
211C22×C8C2×D411yescharacteristic
212C22×C8C2×D411yescharacteristic
213C22×C8C2×D411yescharacteristic
214C22×C8#4112yes—
215C22×C8C2411yescharacteristic
216C22⋊Q8—24no—
217C22⋊Q8—42no—
218C4×Q8—24no—
219C4×Q8—24no—
220C4×Q8—42no—
221C4×Q8—42no—
222C4×Q8—42no—
223C4×Q8—42no—
224C4×D4—24no—
225C4×D4—24no—
226C4×D4—42no—
227C4×D4—48no—
228C4×D4—48no—
229C4×D4—42no—
230#117—24no—
231#117—48no—
232#117—42no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—41no—
243C2×C22⋊C4—41no—
244C2×C22⋊C4—41no—
245C2×C22⋊C4—41no—
246C2×C22⋊C4—41no—
247C2×C22⋊C4—41no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—48no—
251C2×C42—41no—
252C2×C42—41no—
253C2×C42—41no—
254C2×C42—41no—
255C2×C42—41no—
256C2×C42D811yescharacteristic
257C2×C42D811yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260Q32—28no—
261SD32—28no—
262C2×C16—21no—
263C2×C16—21no—
264C2×C16—21no—
265C2×C16—21no—
266C2×C16—21no—
267C2×C16—21no—
268C2×C16—21no—
269C2×C16—21no—
270C2×C16—21no—
271C2×C16—21no—
272C2×C16C22×C412yes—
273C2×C16C22×C412yes—
274#107—21no—
275#107—21no—
276#107—21no—
277#107—21no—
278#107—21no—
279#107—21no—
280#107—21no—
281#107C22×C412yes—
282#107C2×D412yes—
283#107C2×D412yes—
284#106—24no—
285#103—24no—
286#103—24no—
287#103—24no—
288#103—42no—
289#103—42no—
290#102—24no—
291#102—24no—
292C4×C8—22no—
293C4×C8—21no—
294C4×C8—21no—
295C4×C8—22no—
296C4×C8C2×D412yes—
297C4×C8C2×D412yes—
298#95—41no—
299#95—41no—
300#95—41no—
301#95—41no—
302#95—41no—
303C24—41no—
304C24C2×D811yescharacteristic
305#41—216no—
306#41—48no—
307#41—48no—
308#41—48no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—22no—
312C2×Q8—22no—
313C2×Q8—41no—
314C2×Q8—41no—
315C2×Q8—41no—
316C2×Q8—41no—
317C2×Q8—41no—
318C2×Q8—41no—
319C2×Q8—41no—
320C2×Q8—41no—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—44no—
325C2×D4—44no—
326C2×D4—44no—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4—41no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—44no—
341C22×C4—44no—
342C22×C4—44no—
343C22×C4—41no—
344C22×C4—44no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—44no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—82no—
353C22×C4—82no—
354C22×C4—82no—
355C22×C4—82no—
356C22×C4C4×D412yes—
357C22×C4C2×D811yescharacteristic
358C22×C4C2×D811yescharacteristic
359C22×C4C2×D811yescharacteristic
360C22×C4C2×D811yescharacteristic
361C22×C4C2×D811yescharacteristic
362C22×C4#13512yes—
363C22×C4C22×D411yescharacteristic
364Q16—22no—
365Q16—22no—
366Q16—22no—
367Q16—22no—
368Q16—22no—
369Q16—41no—
370Q16—41no—
371Q16C4×D414yes—
372SD16—216no—
373D8—22no—
374D8—22no—
375D8—22no—
376D8C4×D414yes—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—44no—
389C2×C8C4×D412yes—
390C2×C8#12012yes—
391C2×C8C23×C411yescharacteristic
392C2×C8C22×D411yescharacteristic
393C2×C8C22×D411yescharacteristic
394C2×C8C2×#4111yescharacteristic
395C2×C8C2×#4111yescharacteristic
396C2×C8C2×#4111yescharacteristic
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—44no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—44no—
405C4⋊C4—44no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—82no—
411C4⋊C4—82no—
412C22⋊C4—44no—
413C22⋊C4—44no—
414C22⋊C4—44no—
415C22⋊C4—44no—
416C22⋊C4—82no—
417C42—21no—
418C42—21no—
419C42—22no—
420C42—22no—
421C42—44no—
422C42—44no—
423C42—44no—
424C42—82no—
425C42—82no—
426C42#12012yes—
427C42AGL(1,ℤ/8)12yes—
428C16—22no—
429C16—22no—
430C16—41no—
431C16—41no—
432C23—21no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—81no—
442C23—81no—
443C23—81no—
444C23—81no—
445C23C4×D812yes—
446C23C22×D811yescharacteristic
447Q8—24no—
448Q8—42no—
449Q8—42no—
450Q8—42no—
451Q8—42no—
452Q8—42no—
453Q8—81no—
454Q8—81no—
455D4—24no—
456D4—416no—
457D4—42no—
458D4—42no—
459D4—42no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—416no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—88no—
486C2×C4—88no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4C4×D812yes—
502C2×C4#44712yes—
503C2×C4#44712yes—
504C2×C4#51012yes—
505C2×C4C2×C4×D411yescharacteristic
506C2×C4C2×#12011yescharacteristic
507C2×C4#52511yescharacteristic
508C2×C4C2×#13511yescharacteristic
509C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
510C2×C4#57511yescharacteristic
511C8—22no—
512C8—22no—
513C8—41no—
514C8C2×C4×D411yescharacteristic
515C8C2×C4×D411yescharacteristic
516C8#52511yescharacteristic
517C8#52511yescharacteristic
518C22—22no—
519C22—22no—
520C22—22no—
521C22—41no—
522C22—41no—
523C22—41no—
524C22—41no—
525C22—44no—
526C22—82no—
527C22—82no—
528C22—82no—
529C22—82no—
530C22—82no—
531C22—82no—
532C22C2×C4×D811yescharacteristic
533C22C2×#44711yescharacteristic
534C22#310011yescharacteristic
535C22C2×#51011yescharacteristic
536C22#341411yescharacteristic
537C22#341411yescharacteristic
538C22#341411yescharacteristic
539C4—22no—
540C4—22no—
541C4—22no—
542C4—22no—
543C4—44no—
544C4—41no—
545C4—82no—
546C4—82no—
547C4—82no—
548C4—82no—
549C4—161no—
550C4—161no—
551C4C4×#12011yescharacteristic
552C4C4×AGL(1,ℤ/8)11yescharacteristic
553C4#300211yescharacteristic
554C4#310511yescharacteristic
555C2—42no—
556C2—84no—
557C2C4×#44711yescharacteristic
558C2#3345111yescharacteristic
559C2C4×#51011yescharacteristic
560C2#3351911yescharacteristic
561C2#3352111yescharacteristic
562C2#3367111yescharacteristic
563C2#3367611yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC22×#135C22
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#2176C2semidirect
4. C2×#2175C2semidirect
5. C4×#467C2semidirect
6. #13015C2semidirect
7. #13014C2semidirect
8. #12994C2semidirect
9. #12937C2semidirect
10. #12936C2semidirect
11. #12921C2semidirect
13. #12638C2semidirect
14. #12637C2semidirect
15. #12599C2semidirect
34. #2189C4semidirect
93. C2×Q32C2×C4semidirect
96. C2×SD32C2×C4semidirect
102. #467C2×C4semidirect
151. #358C2×C4semidirect
154. #357C2×C4semidirect
156. C22⋊C16C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1352
C2×#21892
C2×#21762
C2×#21752
C4×#4672
#130152
#130142
#129942
#129372
#129362
#129212
#127332
#126382
#126372
#125992