Finite Group Databasebeta

← #6400889

Subgroups — #6400889

Order 512.

2407 conjugacy classes of subgroups.

The subgroup lattice has 543 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3391C211yesmaximal, characteristic
2#34313C211yesmaximal, characteristic, complemented
3C4×#550C211yesmaximal, characteristic, complemented
4#23188C213yesmaximal, complemented
5#23099C213yesmaximal, complemented
6#23001C213yesmaximal, complemented
7#22758C213yesmaximal, complemented
8#22738C211yesmaximal, characteristic, complemented
9#22540C211yesmaximal, characteristic, complemented
10#22540C211yesmaximal, characteristic, complemented
11#22269C213yesmaximal, complemented
12#22151C213yesmaximal, complemented
13#21503C213yesmaximal, complemented
14#21118C213yesmaximal, complemented
15#20572C211yesmaximal, characteristic, complemented
16C2×#550C2211yescharacteristic
17C2×#550C2211yescharacteristic
18C2×#520—21no—
19C2×C4×#41—21no—
20#3391C4116yescomplemented
21C2×#493C2213yes—
22C2×#459C2213yes—
23C2×#459C2213yes—
24C2×C8×Q8C2211yescharacteristic
25C2×C8×Q8C2211yescharacteristic
26C2×#442C2216yes—
27C2×C4×D8C2213yes—
28#2366C2213yes—
29#2366C2213yes—
30#2363—23no—
31#2361—23no—
32#2354C2213yes—
33Q8×C4⋊C4C2211yescharacteristic
34Q8×C4⋊C4C2211yescharacteristic
35D4×C4⋊C4C2213yes—
36#2341—21no—
37#2341—21no—
38#2324—23no—
39C4×#127C2213yes—
40C4×#127C2213yes—
41C4×#121—23no—
42#2293C2213yes—
43C42×Q8C2211yescharacteristic
44C42×D4C2213yes—
45#1982C2213yes—
46#1982C2213yes—
47#1972C2216yes—
48#1957C2213yes—
49#1957C2213yes—
50#1948C2213yes—
51#1948C2213yes—
52#1936C2216yes—
53#1926C2213yes—
54#1926C2213yes—
55#1921C2211yescharacteristic
56#1921C2213yes—
57#1879C2216yes—
58#1879C2213yes—
59#1879C2213yes—
60#1879C2216yes—
61#1879C2216yes—
62#1864C2213yes—
63#1864C2213yes—
64#1834C2213yes—
65#1802C2213yes—
66#1802C2213yes—
67C4×#106C2213yes—
68#1770C2213yes—
69#1770C2213yes—
70#1767C2213yes—
71#1762C2216yes—
72C4×#102C2213yes—
73C4×#102C2213yes—
74#1744C2213yes—
75C22×#41—21no—
76C22×#41—21no—
77C22×D8C2313yes—
78C22×D8C2316yes—
79#550—24no—
80#550—24no—
81#550C2×C418yescomplemented
82#550C2×C418yescomplemented
83C2×#127C2313yes—
84C2×#127C2313yes—
85C2×#121—23no—
86C2×#121—23no—
87#520—48no—
88C4×#41—48no—
89C2×C4×Q8—21no—
90C2×C4×Q8—21no—
91C2×C4×Q8D411yescharacteristic
92C2×C4×Q8D411yescharacteristic
93C2×C4×Q8C2311yescharacteristic
94C2×C4×D4—23no—
95C2×C4×D4—23no—
96C2×C4×D4—23no—
97C2×C4×D4—23no—
98C2×C4×D4—23no—
99C2×C4×D4—23no—
100C2×C4×D4C2313yes—
101C2×C4×D4C2313yes—
102C2×C4×D4C2313yes—
103C2×C4×D4C2313yes—
104C2×C4×D4C2316yes—
105C2×#117—23no—
106C2×#117—23no—
107C22×C4⋊C4—23no—
108C22×C42—23no—
109#493C2×C4124yescomplemented
110#459C2×C4124yescomplemented
111#459C2×C4124yescomplemented
112C8×Q8—24no—
113C8×Q8C2×C418yescomplemented
114#442—412no—
115C4×D8C2×C4124yescomplemented
116C2×#107C2311yescharacteristic
117C2×#106C2313yes—
118C2×#106C2313yes—
119C2×C4⋊C8C2313yes—
120C2×C4⋊C8C2313yes—
121C2×#102C2313yes—
122C2×#102C2316yes—
123C2×#102C2316yes—
124C2×#102C2313yes—
125C2×#102C2313yes—
126C2×#102C2313yes—
127C2×#102C2313yes—
128C2×#102C2313yes—
129C2×#97C2316yes—
130C2×C4×C8C2313yes—
131C2×C4×C8C2313yes—
132#390—23no—
133#390—23no—
134#390—23no—
135#390—23no—
136#390—23no—
137#390—23no—
138#390C2313yes—
139#390C2316yes—
140#390C2313yes—
141#389—23no—
142#389—23no—
143#389C2313yes—
144#389C2311yescharacteristic
145#388—23no—
146#388—23no—
147#388—26no—
148#386—26no—
149#386—26no—
150#385—26no—
151#384C2313yes—
152#384C2313yes—
153#380—23no—
154#380—23no—
155C4×C4⋊C4D413yes—
156C4×C4⋊C4D413yes—
157C4×C4⋊C4C2313yes—
158C4×C4⋊C4C2313yes—
159C4×C4⋊C4C2313yes—
160C4×C4⋊C4C2311yescharacteristic
161C4×C4⋊C4C2313yes—
162C4×C22⋊C4—23no—
163C4×C22⋊C4—23no—
164C4×C22⋊C4—23no—
165C4×C22⋊C4—23no—
166C43C2313yes—
167#340C2313yes—
168#340C2313yes—
169#340C2313yes—
170#340C2316yes—
171#340C2311yescharacteristic
172#340C2311yescharacteristic
173#340C2311yescharacteristic
174#336C2313yes—
175#336C2313yes—
176#336C2311yescharacteristic
177#336C2313yes—
178C2×#41—28no—
179C2×#41—28no—
180C2×#41—44no—
181C2×#41—44no—
182C2×#41—44no—
183C2×#41—44no—
184C22×Q8C2×D411yescharacteristic
185C22×D4—23no—
186C22×D4—23no—
187C22×D4—23no—
188C22×D4—23no—
189C22×D4C2413yes—
190C22×D4C2413yes—
191C23×C4—23no—
192C23×C4—23no—
193C23×C4—23no—
194C23×C4—23no—
195C23×C4—26no—
196C2×D8—46no—
197C2×D8—43no—
198C2×D8—43no—
199C2×D8—46no—
200C2×D8C22×C4112yes—
201C2×D8C22×C4124yes—
202C22×C8C2411yescharacteristic
203C22×C8C2413yes—
204C22×C8C2411yescharacteristic
205C22×C8C2413yes—
206#127—26no—
207#127—26no—
208#127C22×C4112yes—
209#127C22×C4112yes—
210#121—224no—
211#121—224no—
212C4×Q8—24no—
213C4×Q8—24no—
214C4×Q8—44no—
215C4×Q8—44no—
216C4×Q8C22×C414yes—
217C4×Q8#4114yes—
218C4×D4—26no—
219C4×D4—26no—
220C4×D4—26no—
221C4×D4—43no—
222C4×D4—43no—
223C4×D4—43no—
224C4×D4—412no—
225C4×D4—412no—
226C4×D4—46no—
227C4×D4—46no—
228C4×D4—412no—
229C4×D4—412no—
230C4×D4—86no—
231C4×D4—86no—
232C4×D4C22×C4112yes—
233C4×D4C22×C4112yes—
234#117—412no—
235#117—412no—
236C2×C4⋊C4—23no—
237C2×C4⋊C4—23no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—23no—
240C2×C4⋊C4—23no—
241C2×C4⋊C4—23no—
242C2×C4⋊C4—412no—
243C2×C4⋊C4C2×D413yes—
244C2×C4⋊C4C2×D413yes—
245C2×C4⋊C4C2×D413yes—
246C2×C4⋊C4C2413yes—
247C2×C4⋊C4C2413yes—
248C2×C4⋊C4C2411yescharacteristic
249C2×C4⋊C4C2411yescharacteristic
250C2×C4⋊C4C2411yescharacteristic
251C2×C4⋊C4C2411yescharacteristic
252C2×C22⋊C4—23no—
253C2×C22⋊C4—23no—
254C2×C22⋊C4—23no—
255C2×C22⋊C4—26no—
256C2×C22⋊C4—23no—
257C2×C22⋊C4—23no—
258C2×C22⋊C4—23no—
259C2×C22⋊C4—23no—
260C2×C22⋊C4—26no—
261C2×C22⋊C4—26no—
262C2×C22⋊C4—23no—
263C2×C42—23no—
264C2×C42—23no—
265C2×C42—23no—
266C2×C42—21no—
267C2×C42—412no—
268C2×C42C2×D413yes—
269C2×C42C2×D413yes—
270C2×C42C2×D413yes—
271C2×C42C2×D413yes—
272C2×C42C2411yescharacteristic
273C2×C42C2413yes—
274C2×C42C2413yes—
275#107C22×C414yes—
276#106—26no—
277#106—43no—
278C4⋊C8—26no—
279C4⋊C8C22×C4112yes—
280#102—26no—
281#102—43no—
282#102—46no—
283#102—43no—
284#102—43no—
285#102—46no—
286#102C22×C4112yes—
287#102C22×C4112yes—
288#97—46no—
289C4×C8—26no—
290C4×C8C22×C4112yes—
291#95—21no—
292#95—21no—
293#95—23no—
294#95—23no—
295#95—23no—
296#95—23no—
297#95C2×D416yes—
298C24—23no—
299C24—23no—
300#41—432no—
301#41—432no—
302C2×Q8C4×D414yes—
303C2×Q8AGL(1,ℤ/8)14yes—
304C2×Q8AGL(1,ℤ/8)14yes—
305C2×D4—212no—
306C2×D4—212no—
307C2×D4—212no—
308C2×D4—212no—
309C2×D4—23no—
310C2×D4—23no—
311C2×D4—23no—
312C2×D4—23no—
313C2×D4—23no—
314C2×D4—23no—
315C2×D4—23no—
316C2×D4—23no—
317C2×D4—23no—
318C2×D4—23no—
319C2×D4—46no—
320C2×D4—46no—
321C2×D4—46no—
322C2×D4—46no—
323C2×D4—83no—
324C2×D4—83no—
325C2×D4—83no—
326C2×D4—83no—
327C2×D4C23×C416yes—
328C2×D4C23×C416yes—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—212no—
333C22×C4—212no—
334C22×C4—21no—
335C22×C4—23no—
336C22×C4—23no—
337C22×C4—46no—
338C22×C4—46no—
339C22×C4—46no—
340C22×C4—46no—
341C22×C4—412no—
342C22×C4—46no—
343C22×C4—46no—
344C22×C4—46no—
345C22×C4—46no—
346C22×C4—46no—
347C22×C4—83no—
348C22×C4—86no—
349C22×C4—86no—
350C22×C4C22×D413yes—
351C22×C4C22×D413yes—
352C22×C4C22×D411yescharacteristic
353C22×C4C22×D413yes—
354C22×C4C22×D411yescharacteristic
355C22×C4C22×D413yes—
356C22×C4C2511yescharacteristic
357D8—412no—
358D8—424no—
359C2×C8—23no—
360C2×C8—23no—
361C2×C8—21no—
362C2×C8—21no—
363C2×C8—21no—
364C2×C8—23no—
365C2×C8—23no—
366C2×C8—23no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8C23×C412yes—
370C2×C8C23×C416yes—
371C4⋊C4—26no—
372C4⋊C4—26no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—23no—
377C4⋊C4—21no—
378C4⋊C4—23no—
379C4⋊C4—23no—
380C4⋊C4—23no—
381C4⋊C4—21no—
382C4⋊C4—46no—
383C4⋊C4—42no—
384C4⋊C4—46no—
385C4⋊C4—46no—
386C4⋊C4—83no—
387C4⋊C4—83no—
388C4⋊C4C4×D4112yes—
389C4⋊C4C23×C412yes—
390C4⋊C4C23×C412yes—
391C4⋊C4H(5,𝔽2)12yes—
392C22⋊C4—212no—
393C22⋊C4—212no—
394C22⋊C4—46no—
395C22⋊C4—46no—
396C22⋊C4—412no—
397C22⋊C4—46no—
398C22⋊C4—46no—
399C22⋊C4—46no—
400C22⋊C4—83no—
401C22⋊C4—86no—
402C22⋊C4—86no—
403C42—26no—
404C42—26no—
405C42—26no—
406C42—23no—
407C42—21no—
408C42—21no—
409C42—23no—
410C42—43no—
411C42—42no—
412C42—46no—
413C42—46no—
414C42—46no—
415C42C23×C416yes—
416C42C2×#4116yes—
417C23—26no—
418C23—26no—
419C23—43no—
420C23—43no—
421C23—43no—
422C23—43no—
423C23—43no—
424C23—43no—
425C23—43no—
426C23—43no—
427C23—43no—
428C23—43no—
429C23—43no—
430C23—43no—
431C23C23×D411yescharacteristic
432Q8#442116yescomplemented
433D4—26no—
434D4—26no—
435D4—26no—
436D4—26no—
437D4—424no—
438D4—424no—
439D4—43no—
440D4—43no—
441D4—43no—
442D4—43no—
443D4—812no—
444D4—812no—
445C2×C4—23no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—23no—
449C2×C4—21no—
450C2×C4—23no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—21no—
454C2×C4—23no—
455C2×C4—23no—
456C2×C4—22no—
457C2×C4—21no—
458C2×C4—23no—
459C2×C4—23no—
460C2×C4—23no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—424no—
468C2×C4—424no—
469C2×C4—41no—
470C2×C4—43no—
471C2×C4—41no—
472C2×C4—43no—
473C2×C4—41no—
474C2×C4—43no—
475C2×C4—43no—
476C2×C4—43no—
477C2×C4—43no—
478C2×C4—812no—
479C2×C4—812no—
480C2×C4—1612no—
481C2×C4C2×C4×D416yes—
482C2×C4C2×C4×D416yes—
483C2×C4C2×AGL(1,ℤ/8)16yes—
484C2×C4C2×AGL(1,ℤ/8)16yes—
485C2×C4#57612yes—
486C2×C4C24×C411yescharacteristic
487C2×C4C22×#4111yescharacteristic
488C2×C4C2×H(5,𝔽2)11yescharacteristic
489C2×C4#58511yescharacteristic
490C2×C4#58511yescharacteristic
491C2×C4#58511yescharacteristic
492C8—22no—
493C8—26no—
494C8—43no—
495C8—41no—
496C22—46no—
497C22—46no—
498C22—46no—
499C22—46no—
500C22—46no—
501C22—46no—
502C22—83no—
503C22—83no—
504C22—83no—
505C22—83no—
506C22—83no—
507C22—83no—
508C22—83no—
509C22—83no—
510C22C22×C4×D411yescharacteristic
511C22#348711yescharacteristic
512C22#348811yescharacteristic
513C22C22×AGL(1,ℤ/8)11yescharacteristic
514C22C22×AGL(1,ℤ/8)11yescharacteristic
515C22C2×#57611yescharacteristic
516C22#358411yescharacteristic
517C4—26no—
518C4—22no—
519C4—43no—
520C4—43no—
521C4—42no—
522C4—41no—
523C4—42no—
524C4—42no—
525C4—83no—
526C4—81no—
527C4—83no—
528C4C2×#442112yes—
529C4#342811yescharacteristic
530C4C4×H(5,𝔽2)11yescharacteristic
531C4#351711yescharacteristic
532C4#352411yescharacteristic
533C2—86no—
534C2—86no—
535C2#3429511yescharacteristic
536C2C22×#44211yescharacteristic
537C2C4×#57611yescharacteristic
538C2#6190311yescharacteristic
539C2#6190711yescharacteristic
540C2#6200111yescharacteristic
541C2#6200311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC2×#3391C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34313C2semidirect
3. C4×#550C2semidirect
4. #23188C2semidirect
5. #23099C2semidirect
6. #23001C2semidirect
7. #22758C2semidirect
8. #22738C2semidirect
9. #22540C2semidirect
10. #22540C2semidirect
11. #22269C2semidirect
12. #22151C2semidirect
13. #21503C2semidirect
14. #21118C2semidirect
15. #20572C2semidirect
20. #3391C4semidirect
81. #550C2×C4semidirect
82. #550C2×C4semidirect
109. #493C2×C4semidirect
110. #459C2×C4semidirect
111. #459C2×C4semidirect
113. C8×Q8C2×C4semidirect
115. C4×D8C2×C4semidirect
432. Q8#442semidirect

Maximal subgroups

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

subgroupindex
C2×#33912
#343132
C4×#5502
#231882
#230992
#230012
#227582
#227382
#225402
#222692
#221512
#215032
#211182
#205722