Finite Group Databasebeta

← #6399858

Subgroups — #6399858

Order 512.

2164 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×D8C211yesmaximal, characteristic
2#34311C211yesmaximal, characteristic, complemented
3C4×D4×Q8C211yesmaximal, characteristic, complemented
4#23157C213yesmaximal, complemented
5#23060C213yesmaximal, complemented
6#22962C213yesmaximal, complemented
7#22766C213yesmaximal
8#22740C211yesmaximal, characteristic, complemented
9Q8×#102C211yesmaximal, characteristic, complemented
10Q8×#102C211yesmaximal, characteristic, complemented
11#22247C213yesmaximal, complemented
12#22110C213yesmaximal, complemented
13C4×#474C213yesmaximal, complemented
14#21113C213yesmaximal
15#20569C211yesmaximal, characteristic, complemented
16C2×D4×Q8C2211yescharacteristic
17C2×D4×Q8C2211yescharacteristic
18C2×#519—21no—
19C22×C4×Q8—21no—
20Q8×D8C4116yescomplemented
21C2×#500C2213yes—
22C2×#474C2213yes—
23C2×#474C2213yes—
24C2×C8×Q8C2211yescharacteristic
25C2×C8×Q8C2211yescharacteristic
26C2×C4×D8C2213yes—
27#2365C2213yes—
28#2365C2213yes—
29#2362—23no—
30#2360—23no—
31#2354C2213yes—
32Q8×C4⋊C4C2211yescharacteristic
33Q8×C4⋊C4C2211yescharacteristic
34D4×C4⋊C4C2213yes—
35Q8×C22⋊C4—21no—
36Q8×C22⋊C4—21no—
37#2323—23no—
38C4×C4⋊Q8C2213yes—
39C4×C4⋊Q8C2213yes—
40C4×C22⋊Q8—23no—
41#2293C2213yes—
42C42×Q8C2211yescharacteristic
43C42×D4C2213yes—
44#1990C2213yes—
45#1990C2213yes—
46#1988C2213yes—
47#1988C2213yes—
48#1947C2213yes—
49#1947C2213yes—
50#1942C2216yes—
51#1934C2216yes—
52#1925C2213yes—
53#1925C2213yes—
54#1918C2216yes—
55#1918C2216yes—
56#1918C2213yes—
57#1918C2216yes—
58#1918C2213yes—
59#1879C2211yescharacteristic
60#1879C2213yes—
61#1862C2213yes—
62#1862C2213yes—
63#1834C2213yes—
64#1802C2213yes—
65#1802C2213yes—
66C4×#107C2216yes—
67C4×#107C2213yes—
68C8×C4⋊C4C2213yes—
69C8×C4⋊C4C2213yes—
70C4×C4⋊C8C2213yes—
71C4×#102C2213yes—
72C4×#102C2216yes—
73C4×#102C2213yes—
74#1744C2213yes—
75C23×Q8—21no—
76C23×Q8—21no—
77C22×D8C2311yescharacteristic
78D4×Q8—24no—
79D4×Q8—24no—
80D4×Q8C2×C418yes—
81D4×Q8C2×C418yescomplemented
82C2×C4⋊Q8C2313yes—
83C2×C4⋊Q8C2313yes—
84C2×C22⋊Q8—23no—
85C2×C22⋊Q8—23no—
86#519—48no—
87C2×C4×Q8—21no—
88C2×C4×Q8—21no—
89C2×C4×Q8—23no—
90C2×C4×Q8—23no—
91C2×C4×Q8—23no—
92C2×C4×Q8—23no—
93C2×C4×Q8—48no—
94C2×C4×Q8D411yescharacteristic
95C2×C4×Q8D411yescharacteristic
96C2×C4×Q8C2311yescharacteristic
97C2×C4×D4C2311yescharacteristic
98C2×C4×D4C2313yes—
99C2×C4×D4C2313yes—
100C2×C4×D4C2313yes—
101C2×#117—23no—
102C2×#117—23no—
103C22×C4⋊C4—23no—
104C22×C42—23no—
105#500C2×C4124yescomplemented
106#474C2×C4124yes—
107#474C2×C4124yescomplemented
108C8×Q8—24no—
109C8×Q8C2×C418yescomplemented
110C4×D8C2×C4124yes—
111C2×#107C2313yes—
112C2×#107C2313yes—
113C2×#107C2313yes—
114C2×#107C2316yes—
115C2×C4⋊C8C2313yes—
116C2×C4⋊C8C2313yes—
117C2×#102C2313yes—
118C2×#102C2311yescharacteristic
119C2×#102C2311yescharacteristic
120C2×#102C2313yes—
121C2×#102C2313yes—
122C2×#102C2313yes—
123C2×C4×C8C2313yes—
124C2×C4×C8C2316yes—
125C2×C4×C8C2313yes—
126#391—23no—
127#391—23no—
128#391—23no—
129#391—23no—
130#390C2313yes—
131#390C2311yescharacteristic
132#389—23no—
133#389—23no—
134#389—23no—
135#389—23no—
136#389C2313yes—
137#389C2316yes—
138#389C2313yes—
139#388—23no—
140#388—23no—
141#388—26no—
142#387—26no—
143#387—26no—
144#385—26no—
145#384C2313yes—
146#384C2313yes—
147#380—23no—
148#380—23no—
149C4×C4⋊C4—23no—
150C4×C4⋊C4—23no—
151C4×C4⋊C4D413yes—
152C4×C4⋊C4D413yes—
153C4×C4⋊C4C2313yes—
154C4×C4⋊C4C2316yes—
155C4×C4⋊C4C2313yes—
156C4×C4⋊C4C2313yes—
157C4×C4⋊C4C2313yes—
158C4×C4⋊C4C2313yes—
159C4×C22⋊C4—23no—
160C4×C22⋊C4—23no—
161C4×C22⋊C4—23no—
162C4×C22⋊C4—23no—
163C43C2313yes—
164#340C2316yes—
165#340C2316yes—
166#340C2313yes—
167#340C2316yes—
168#340C2313yes—
169#340C2313yes—
170#340C2311yescharacteristic
171#340C2313yes—
172#340C2313yes—
173#336C2313yes—
174#336C2313yes—
175#336C2311yescharacteristic
176#336C2313yes—
177C22×Q8—28no—
178C22×Q8—28no—
179C22×Q8—23no—
180C22×Q8—23no—
181C22×Q8—44no—
182C22×Q8—44no—
183C22×Q8—44no—
184C22×Q8—44no—
185C22×Q8C2×D411yescharacteristic
186C22×D4C2411yescharacteristic
187C22×D4C2411yescharacteristic
188C23×C4—23no—
189C23×C4—23no—
190C23×C4—21no—
191C23×C4—23no—
192C2×D8—22no—
193C2×D8—22no—
194C2×D8C22×C414yes—
195C22×C8C2411yescharacteristic
196C22×C8C2413yes—
197C22×C8C2411yescharacteristic
198C22×C8C2413yes—
199C4⋊Q8—26no—
200C4⋊Q8—26no—
201C4⋊Q8C22×C4112yes—
202C4⋊Q8C22×C4112yes—
203C22⋊Q8—224no—
204C22⋊Q8—224no—
205C4×Q8—24no—
206C4×Q8—44no—
207C4×Q8—44no—
208C4×Q8—412no—
209C4×Q8—412no—
210C4×Q8—412no—
211C4×Q8—412no—
212C4×Q8D818yescomplemented
213C4×Q8C22×C414yes—
214C4×Q8#4114yes—
215C4×D4—22no—
216C4×D4—22no—
217C4×D4—26no—
218C4×D4—26no—
219C4×D4—26no—
220C4×D4—43no—
221C4×D4C22×C4112yes—
222C4×D4C22×C4112yes—
223#117—412no—
224#117—412no—
225C2×C4⋊C4—23no—
226C2×C4⋊C4—23no—
227C2×C4⋊C4—23no—
228C2×C4⋊C4—26no—
229C2×C4⋊C4—23no—
230C2×C4⋊C4—23no—
231C2×C4⋊C4—23no—
232C2×C4⋊C4—23no—
233C2×C4⋊C4—23no—
234C2×C4⋊C4—412no—
235C2×C4⋊C4C2×D413yes—
236C2×C4⋊C4C2×D413yes—
237C2×C4⋊C4C2×D413yes—
238C2×C4⋊C4C2413yes—
239C2×C4⋊C4C2413yes—
240C2×C4⋊C4C2411yescharacteristic
241C2×C4⋊C4C2413yes—
242C2×C4⋊C4C2413yes—
243C2×C4⋊C4C2411yescharacteristic
244C2×C22⋊C4—23no—
245C2×C22⋊C4—23no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—23no—
249C2×C22⋊C4—23no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—23no—
252C2×C42—23no—
253C2×C42—26no—
254C2×C42—23no—
255C2×C42—23no—
256C2×C42—23no—
257C2×C42—412no—
258C2×C42C2×D413yes—
259C2×C42C2×D413yes—
260C2×C42C2×D413yes—
261C2×C42C2×D413yes—
262C2×C42C2411yescharacteristic
263C2×C42C2413yes—
264C2×C42C2413yes—
265#107—26no—
266#107—43no—
267#107C22×C4112yes—
268#107C22×C4124yes—
269C4⋊C8—26no—
270C4⋊C8C22×C4112yes—
271#102—26no—
272#102—22no—
273#102—22no—
274#102—43no—
275#102C22×C4112yes—
276#102C22×C4112yes—
277C4×C8—26no—
278C4×C8—46no—
279C4×C8C22×C4112yes—
280#95—23no—
281#95—23no—
282#95—23no—
283#95—23no—
284#95—23no—
285#95—26no—
286#95—26no—
287#95—23no—
288#95C2×D416yes—
289C24—21no—
290C24—21no—
291C2×Q8—212no—
292C2×Q8—212no—
293C2×Q8—432no—
294C2×Q8—432no—
295C2×Q8—46no—
296C2×Q8—46no—
297C2×Q8—46no—
298C2×Q8—46no—
299C2×Q8C4×D414yes—
300C2×Q8C2×D814yes—
301C2×Q8#13514yes—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—21no—
309C2×D4—21no—
310C2×D4C23×C412yes—
311C2×D4C23×C412yes—
312C2×D42−1+412yes—
313C2×D42−1+412yes—
314C22×C4—24no—
315C22×C4—21no—
316C22×C4—23no—
317C22×C4—23no—
318C22×C4—212no—
319C22×C4—212no—
320C22×C4—21no—
321C22×C4—23no—
322C22×C4—23no—
323C22×C4—42no—
324C22×C4—46no—
325C22×C4—42no—
326C22×C4—46no—
327C22×C4—46no—
328C22×C4—46no—
329C22×C4—46no—
330C22×C4—46no—
331C22×C4—83no—
332C22×C4C22×D413yes—
333C22×C4C22×D411yescharacteristic
334C22×C4C22×D413yes—
335C22×C4C22×D413yes—
336C22×C4C22×D411yescharacteristic
337C22×C4C22×D413yes—
338C22×C4C2511yescharacteristic
339D8—28no—
340C2×C8—23no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—23no—
344C2×C8—21no—
345C2×C8—23no—
346C2×C8—23no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—23no—
350C2×C8C23×C412yes—
351C2×C8C23×C416yes—
352C4⋊C4—212no—
353C4⋊C4—212no—
354C4⋊C4—26no—
355C4⋊C4—23no—
356C4⋊C4—23no—
357C4⋊C4—21no—
358C4⋊C4—23no—
359C4⋊C4—21no—
360C4⋊C4—23no—
361C4⋊C4—23no—
362C4⋊C4—23no—
363C4⋊C4—21no—
364C4⋊C4—412no—
365C4⋊C4—46no—
366C4⋊C4—46no—
367C4⋊C4—46no—
368C4⋊C4—46no—
369C4⋊C4—83no—
370C4⋊C4—83no—
371C4⋊C4C4×D4112yes—
372C4⋊C4C2×D8112yes—
373C4⋊C4C23×C416yes—
374C4⋊C4C23×C416yes—
375C4⋊C42−1+412yes—
376C22⋊C4—24no—
377C22⋊C4—212no—
378C22⋊C4—212no—
379C22⋊C4—46no—
380C22⋊C4—42no—
381C22⋊C4—42no—
382C22⋊C4—46no—
383C22⋊C4—83no—
384C42—26no—
385C42—21no—
386C42—21no—
387C42—23no—
388C42—23no—
389C42—43no—
390C42—43no—
391C42—46no—
392C42—46no—
393C42—46no—
394C42—412no—
395C42—46no—
396C42C2×D8112yes—
397C42C23×C416yes—
398C42C2×#4116yes—
399C23—22no—
400C23—22no—
401C23—22no—
402C23—22no—
403C23—41no—
404C23—41no—
405C23—41no—
406C23—41no—
407C23—41no—
408C23—41no—
409C23—41no—
410C23—41no—
411C23—41no—
412C23—41no—
413C23C23×D411yescharacteristic
414Q8—424no—
415Q8—424no—
416Q8C4×D8116yes—
417D4—22no—
418D4—22no—
419D4—22no—
420D4—22no—
421D4—22no—
422D4—41no—
423D4—41no—
424D4#51914yescomplemented
425C2×C4—21no—
426C2×C4—23no—
427C2×C4—23no—
428C2×C4—23no—
429C2×C4—21no—
430C2×C4—23no—
431C2×C4—21no—
432C2×C4—23no—
433C2×C4—26no—
434C2×C4—26no—
435C2×C4—21no—
436C2×C4—22no—
437C2×C4—23no—
438C2×C4—21no—
439C2×C4—23no—
440C2×C4—48no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—43no—
444C2×C4—43no—
445C2×C4—43no—
446C2×C4—43no—
447C2×C4—424no—
448C2×C4—424no—
449C2×C4—41no—
450C2×C4—43no—
451C2×C4—43no—
452C2×C4—43no—
453C2×C4—43no—
454C2×C4—41no—
455C2×C4—43no—
456C2×C4—41no—
457C2×C4—43no—
458C2×C4—812no—
459C2×C4C2×C4×D416yes—
460C2×C4C2×C4×D416yes—
461C2×C4C22×D812yes—
462C2×C4C22×D816yes—
463C2×C4C22×D816yes—
464C2×C4C2×#13516yes—
465C2×C4C24×C411yescharacteristic
466C2×C4C22×#4111yescharacteristic
467C2×C4C2×2−1+411yescharacteristic
468C2×C4C2×2−1+411yescharacteristic
469C2×C4C2×2−1+411yescharacteristic
470C2×C4#58511yescharacteristic
471C8—22no—
472C8—26no—
473C8—43no—
474C8—41no—
475C22—24no—
476C22—42no—
477C22—42no—
478C22—42no—
479C22—42no—
480C22—42no—
481C22—42no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22—81no—
486C22—81no—
487C22—81no—
488C22—81no—
489C22C22×C4×D411yescharacteristic
490C22#348611yescharacteristic
491C22#348911yescharacteristic
492C22C23×D811yescharacteristic
493C22C22×#13511yescharacteristic
494C22#358611yescharacteristic
495C22#358611yescharacteristic
496C4—26no—
497C4—22no—
498C4—26no—
499C4—42no—
500C4—41no—
501C4—43no—
502C4—46no—
503C4—46no—
504C4—83no—
505C4—81no—
506C4—83no—
507C4C2×C4×D8112yes—
508C4C2×#51911yescharacteristic
509C4C4×2−1+411yescharacteristic
510C4#351611yescharacteristic
511C4#352311yescharacteristic
512C2—44no—
513C2—82no—
514C2#3429111yescharacteristic
515C2C22×C4×D811yescharacteristic
516C2#6017511yescharacteristic
517C2#6181711yescharacteristic
518C2#6182411yescharacteristic
519C2#6209511yescharacteristic
520C2#6210511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC22×D8C23
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. #34311C2semidirect
3. C4×D4×Q8C2semidirect
4. #23157C2semidirect
5. #23060C2semidirect
6. #22962C2semidirect
8. #22740C2semidirect
9. Q8×#102C2semidirect
10. Q8×#102C2semidirect
11. #22247C2semidirect
12. #22110C2semidirect
13. C4×#474C2semidirect
15. #20569C2semidirect
20. Q8×D8C4semidirect
81. D4×Q8C2×C4semidirect
105. #500C2×C4semidirect
107. #474C2×C4semidirect
109. C8×Q8C2×C4semidirect
212. C4×Q8D8semidirect
424. D4#519semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×D82
#343112
C4×D4×Q82
#231572
#230602
#229622
#227662
#227402
Q8×#1022
#222472
#221102
C4×#4742
#211132
#205692