Finite Group Databasebeta

← #495704

Subgroups — #495704

Order 512.

1108 conjugacy classes of subgroups.

The subgroup lattice has 581 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2951C211yesmaximal, characteristic, complemented
2C2×#2186C211yesmaximal, characteristic
3C2×#2175C211yesmaximal, characteristic, complemented
4C2×C4×D16C211yesmaximal, characteristic, complemented
5#21524C211yesmaximal, characteristic, complemented
6#13013C211yesmaximal, characteristic, complemented
7#13012C211yesmaximal, characteristic, complemented
8#12991C211yesmaximal, characteristic, complemented
9#12936C211yesmaximal, characteristic, complemented
10#12935C211yesmaximal, characteristic, complemented
11#12919C211yesmaximal, characteristic, complemented
12#12733C211yesmaximal, characteristic, complemented
13#12637C211yesmaximal, characteristic, complemented
14C4×#358C211yesmaximal, characteristic, complemented
15#12600C211yesmaximal, characteristic, complemented
16C22×#135C2211yescharacteristic
17C2×#520—21no—
18C22×SD32C2211yescharacteristic
19C22×D16C2211yescharacteristic
20C2×#466C2211yescharacteristic
21#2951—24no—
22#2951—24no—
23C2×#440—21no—
24C2×#440—21no—
25C2×C4×Q16C2211yescharacteristic
26C2×C4×Q16C2211yescharacteristic
27C2×C4×D8C2211yescharacteristic
28C2×C4×D8C2211yescharacteristic
29C2×C4×D8C2211yescharacteristic
30C22×#107C2211yescharacteristic
31C2×#417—21no—
32C2×#405C2211yescharacteristic
33#2324—21no—
34#2186C418yescomplemented
35#2175—24no—
36C4×D16—24no—
37C2×#366C2211yescharacteristic
38C2×#366C2211yescharacteristic
39C2×#358C2211yescharacteristic
40C2×#358C2211yescharacteristic
41C2×#358C2211yescharacteristic
42C2×#357C2211yescharacteristic
43C2×#357C2211yescharacteristic
44C2×#357C2211yescharacteristic
45C2×#357C2211yescharacteristic
46C2×#357C2211yescharacteristic
47C2×C22⋊C16C2211yescharacteristic
48C2×C22⋊C16C2211yescharacteristic
49C2×#346C2211yescharacteristic
50C2×C4×C16C2211yescharacteristic
51#1970C2211yescharacteristic
52#1942C2211yescharacteristic
53#1936—21no—
54#1895—21no—
55#1808C2211yescharacteristic
56C4×#107C2211yescharacteristic
57#1762—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×C414yescomplemented
71C22×Q16C2311yescharacteristic
72C22×SD16—21no—
73C22×D8C2311yescharacteristic
74C22×D8C2311yescharacteristic
75C22×M16D411yescharacteristic
76C23×C8C2311yescharacteristic
77C2×#121—21no—
78#520—28no—
79#520—44no—
80C2×C4×Q8—21no—
81C2×C4×Q8—21no—
82C2×C4×D4—21no—
83C2×C4×D4—21no—
84C2×C4×D4—21no—
85C2×C4×D4—41no—
86C2×C4×D4—41no—
87C2×#117—41no—
88C2×#117D411yescharacteristic
89C22×C4⋊C4—21no—
90C22×C4⋊C4—41no—
91C2×SD32—22no—
92C2×SD32—22no—
93C2×SD32C2×C414yescomplemented
94C2×D16—22no—
95C2×D16—22no—
96C2×D16C2×C414yescomplemented
97C22×C16C2311yescharacteristic
98C22×C16C2311yescharacteristic
99C22×C16C2311yescharacteristic
100C22×C16C2311yescharacteristic
101#466—22no—
102#466C2×C414yescomplemented
103#440—28no—
104#440—28no—
105C4×Q16—22no—
106C4×Q16—22no—
107C4×Q16—22no—
108C4×Q16D414yescomplemented
109C4×D8—22no—
110C4×D8—22no—
111C4×D8—22no—
112C4×D8—22no—
113C4×D8—22no—
114C4×D8D414yescomplemented
115C2×#107—21no—
116C2×#107—22no—
117C2×#107—22no—
118C2×#107C2311yescharacteristic
119C2×#107C2311yescharacteristic
120C2×#107C2311yescharacteristic
121C2×#107C2311yescharacteristic
122C2×#107C2311yescharacteristic
123#417—28no—
124C2×#103—21no—
125C2×#103—21no—
126C2×#102—21no—
127C2×#102—21no—
128C2×#102—21no—
129C2×#102—21no—
130C2×#102—21no—
131#405—22no—
132#405—22no—
133C2×#97D411yescharacteristic
134C2×#97D411yescharacteristic
135C2×C4×C8C2311yescharacteristic
136C2×C4×C8C2311yescharacteristic
137#390—21no—
138#390—41no—
139#389—21no—
140#388—41no—
141#386—41no—
142#385—41no—
143#380—21no—
144C4×C4⋊C4—21no—
145#366—22no—
146#366—22no—
147#358—22no—
148#358—22no—
149#358C2×C414yescomplemented
150#357—22no—
151#357—22no—
152#357—22no—
153#357—22no—
154#357C2×C414yescomplemented
155C22⋊C16—22no—
156C22⋊C16C2×C414yescomplemented
157#346—22no—
158C4×C16—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×D4—21no—
170C22×D4—21no—
171C22×D4—41no—
172C22×D4—41no—
173C23×C4—41no—
174C23×C4—41no—
175C23×C4—41no—
176C23×C4—41no—
177C23×C4—41no—
178C23×C4C2×D411yescharacteristic
179#135—28no—
180#135—28no—
181#135—28no—
182#135—28no—
183C2×Q16—21no—
184C2×Q16—21no—
185C2×Q16—21no—
186C2×Q16C22×C412yes—
187C2×Q16C2×D412yes—
188C2×Q16#4112yes—
189C2×SD16—24no—
190C2×SD16—24no—
191C2×SD16—24no—
192C2×D8—21no—
193C2×D8—21no—
194C2×D8—21no—
195C2×D8—21no—
196C2×D8—21no—
197C2×D8—21no—
198C2×D8—21no—
199C2×D8—21no—
200C2×D8C22×C412yes—
201C2×D8C22×C412yes—
202C2×D8C2×D412yes—
203C2×D8#4112yes—
204C2×M16—22no—
205C2×M16—22no—
206C2×M16—41no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8C22×C412yes—
212C22×C8C2×D411yescharacteristic
213C22×C8C2×D411yescharacteristic
214C22×C8C2×D411yescharacteristic
215C22×C8#4112yes—
216C22×C8C2411yescharacteristic
217#121—24no—
218#121—42no—
219C4×Q8—24no—
220C4×Q8—24no—
221C4×Q8—42no—
222C4×Q8—42no—
223C4×D4—24no—
224C4×D4—24no—
225C4×D4—42no—
226C4×D4—48no—
227C4×D4—48no—
228C4×D4—42no—
229C4×D4—42no—
230C4×D4—42no—
231#117—22no—
232#117—22no—
233#117—48no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—48no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4—41no—
244C2×C4⋊C4—41no—
245C2×C4⋊C4—41no—
246C2×C4⋊C4D811yescharacteristic
247C2×C4⋊C4D811yescharacteristic
248C2×C22⋊C4—41no—
249C2×C22⋊C4—41no—
250C2×C22⋊C4—41no—
251C2×C22⋊C4—41no—
252C2×C22⋊C4—41no—
253C2×C22⋊C4—41no—
254C2×C22⋊C4—41no—
255C2×C22⋊C4D811yescharacteristic
256C2×C22⋊C4D811yescharacteristic
257C2×C42—41no—
258C2×C42—41no—
259C2×C42—41no—
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D411yescharacteristic
262SD32—28no—
263D16—28no—
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×C16—21no—
273C2×C16—21no—
274C2×C16C22×C412yes—
275C2×C16C22×C412yes—
276#107—24no—
277#107—21no—
278#107—21no—
279#107—21no—
280#107—21no—
281#107—21no—
282#107—21no—
283#107—21no—
284#107C22×C412yes—
285#107C2×D412yes—
286#107C2×D412yes—
287#103—24no—
288#103—24no—
289#102—24no—
290#102—24no—
291#102—24no—
292#102—42no—
293#102—42no—
294#97—22no—
295#97—22no—
296C4×C8—21no—
297C4×C8—21no—
298C4×C8C2×D412yes—
299C4×C8C2×D412yes—
300#95—41no—
301#95—41no—
302#95—41no—
303C24—41no—
304C24—41no—
305C24C2×D811yescharacteristic
306#41—216no—
307#41—48no—
308#41—48no—
309#41—48no—
310C2×Q8—22no—
311C2×Q8—22no—
312C2×Q8—22no—
313C2×Q8—41no—
314C2×Q8—41no—
315C2×Q8—41no—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—22no—
319C2×D4—22no—
320C2×D4—44no—
321C2×D4—44no—
322C2×D4—44no—
323C2×D4—44no—
324C2×D4—41no—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—161no—
333C2×D4—161no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—44no—
339C22×C4—44no—
340C22×C4—44no—
341C22×C4—41no—
342C22×C4—44no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—44no—
347C22×C4—44no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—82no—
351C22×C4—161no—
352C22×C4—161no—
353C22×C4—161no—
354C22×C4—161no—
355C22×C4—161no—
356C22×C4—161no—
357C22×C4—161no—
358C22×C4—161no—
359C22×C4C4×D412yes—
360C22×C4C2×D811yescharacteristic
361C22×C4C2×D811yescharacteristic
362C22×C4C2×D811yescharacteristic
363C22×C4C2×D811yescharacteristic
364C22×C4C2×D811yescharacteristic
365C22×C4#13512yes—
366C22×C4C22×D411yescharacteristic
367Q16—22no—
368Q16—22no—
369Q16—22no—
370Q16C4×D414yes—
371SD16—216no—
372D8—22no—
373D8—22no—
374D8—22no—
375D8—22no—
376D8—22no—
377D8—41no—
378D8—41no—
379D8C4×D414yes—
380M16—44no—
381C2×C8—21no—
382C2×C8—21no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8C4×D412yes—
393C2×C8#12012yes—
394C2×C8C23×C411yescharacteristic
395C2×C8C22×D411yescharacteristic
396C2×C8C22×D411yescharacteristic
397C2×C8C2×#4111yescharacteristic
398C2×C8C2×#4111yescharacteristic
399C2×C8C2×#4111yescharacteristic
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—44no—
406C4⋊C4—44no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—44no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—161no—
416C22⋊C4—22no—
417C22⋊C4—22no—
418C22⋊C4—44no—
419C22⋊C4—44no—
420C22⋊C4—44no—
421C22⋊C4—82no—
422C22⋊C4—161no—
423C22⋊C4—161no—
424C22⋊C4—161no—
425C42—21no—
426C42—21no—
427C42—44no—
428C42—44no—
429C42—161no—
430C42#12012yes—
431C42AGL(1,ℤ/8)12yes—
432C16—22no—
433C16—22no—
434C16—41no—
435C16—41no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—81no—
447C23—81no—
448C23—81no—
449C23—81no—
450C23—81no—
451C23—81no—
452C23—81no—
453C23—81no—
454C23—81no—
455C23—81no—
456C23C4×D812yes—
457C23C22×D811yescharacteristic
458Q8—24no—
459Q8—42no—
460Q8—42no—
461Q8—42no—
462D4—24no—
463D4—416no—
464D4—42no—
465D4—42no—
466D4—42no—
467D4—42no—
468D4—42no—
469D4—81no—
470D4—81no—
471D4—164no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—416no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4—81no—
506C2×C4—81no—
507C2×C4—164no—
508C2×C4—164no—
509C2×C4—164no—
510C2×C4—164no—
511C2×C4C4×D812yes—
512C2×C4#44712yes—
513C2×C4#44712yes—
514C2×C4C2×C4×D411yescharacteristic
515C2×C4C2×#12011yescharacteristic
516C2×C4#52511yescharacteristic
517C2×C4C2×#13511yescharacteristic
518C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
519C2×C4#57511yescharacteristic
520C8—22no—
521C8—22no—
522C8—41no—
523C8C2×C4×D411yescharacteristic
524C8C2×C4×D411yescharacteristic
525C8#52511yescharacteristic
526C8#52511yescharacteristic
527C22—22no—
528C22—22no—
529C22—22no—
530C22—41no—
531C22—41no—
532C22—41no—
533C22—41no—
534C22—44no—
535C22—82no—
536C22—82no—
537C22—82no—
538C22—82no—
539C22—82no—
540C22—82no—
541C22—82no—
542C22—82no—
543C22—82no—
544C22—161no—
545C22—161no—
546C22—161no—
547C22—161no—
548C22C2×C4×D811yescharacteristic
549C22C2×#44711yescharacteristic
550C22#310011yescharacteristic
551C22#341511yescharacteristic
552C22#341511yescharacteristic
553C22#341611yescharacteristic
554C22#341711yescharacteristic
555C4—22no—
556C4—22no—
557C4—22no—
558C4—44no—
559C4—41no—
560C4—41no—
561C4—82no—
562C4—82no—
563C4—82no—
564C4—161no—
565C4—161no—
566C4C4×#12011yescharacteristic
567C4C4×AGL(1,ℤ/8)11yescharacteristic
568C4#300211yescharacteristic
569C4#310511yescharacteristic
570C2—42no—
571C2—84no—
572C2—162no—
573C2C4×#44711yescharacteristic
574C2#3345411yescharacteristic
575C2#3345911yescharacteristic
576C2#3352211yescharacteristic
577C2#3352711yescharacteristic
578C2#3367911yescharacteristic
579C2#3368411yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC2×#2186C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#405 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2951C2semidirect
3. C2×#2175C2semidirect
4. C2×C4×D16C2semidirect
5. #21524C2semidirect
6. #13013C2semidirect
7. #13012C2semidirect
8. #12991C2semidirect
9. #12936C2semidirect
10. #12935C2semidirect
11. #12919C2semidirect
12. #12733C2semidirect
13. #12637C2semidirect
14. C4×#358C2semidirect
15. #12600C2semidirect
34. #2186C4semidirect
70. C2×#135C2×C4semidirect
93. C2×SD32C2×C4semidirect
96. C2×D16C2×C4semidirect
102. #466C2×C4semidirect
108. C4×Q16D4semidirect
114. C4×D8D4semidirect
149. #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×#29512
C2×#21862
C2×#21752
C2×C4×D162
#215242
#130132
#130122
#129912
#129362
#129352
#129192
#127332
#126372
C4×#3582
#126002