Finite Group Databasebeta

← #6400500

Subgroups — #6400500

Order 512.

2252 conjugacy classes of subgroups.

The subgroup lattice has 529 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3358C211yesmaximal, characteristic
2#34311C211yesmaximal, characteristic, complemented
3C4×#550C211yesmaximal, characteristic, complemented
4#23216C213yesmaximal, complemented
5#23093C213yesmaximal, complemented
6#23060C213yesmaximal, complemented
7#22807C213yesmaximal, complemented
8Q8×#107C211yesmaximal, characteristic, complemented
9#22540C211yesmaximal, characteristic, complemented
10Q8×#102C211yesmaximal, characteristic, complemented
11#22220C213yesmaximal, complemented
12#22145C213yesmaximal, complemented
13C4×#474C213yesmaximal, complemented
14#21113C213yesmaximal, complemented
15#20569C211yesmaximal, characteristic, complemented
16C2×#550C2211yescharacteristic
17C2×D4×Q8C2211yescharacteristic
18C2×#519—21no—
19C2×C4×#41—21no—
20#3358C4116yescomplemented
21C2×#486C2213yes—
22C2×#474C2213yes—
23C2×#459C2213yes—
24C2×C8×Q8C2211yescharacteristic
25C2×C8×Q8C2211yescharacteristic
26C2×C4×D8C2213yes—
27#2366C2213yes—
28#2365C2213yes—
29#2363—23no—
30#2362—23no—
31#2354C2213yes—
32#2354C2213yes—
33Q8×C4⋊C4C2211yescharacteristic
34Q8×C4⋊C4C2211yescharacteristic
35#2341—21no—
36Q8×C22⋊C4—21no—
37#2323—23no—
38C4×C4⋊Q8C2213yes—
39C4×#127C2213yes—
40C4×#121—23no—
41#2293C2213yes—
42C42×Q8C2211yescharacteristic
43C42×D4C2213yes—
44#1990C2213yes—
45#1988C2213yes—
46#1982C2213yes—
47#1970C2216yes—
48#1957C2213yes—
49#1947C2213yes—
50#1947C2213yes—
51#1934C2216yes—
52#1925C2216yes—
53#1925C2213yes—
54#1925C2216yes—
55#1925C2211yescharacteristic
56#1918C2213yes—
57#1918C2213yes—
58#1879C2213yes—
59#1879C2213yes—
60#1862C2216yes—
61#1862C2213yes—
62#1862C2213yes—
63#1845C2213yes—
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—
75C22×#41—21no—
76C23×Q8—21no—
77C22×D8C2313yes—
78#550—24no—
79#550C2×C418yes—
80D4×Q8—24no—
81D4×Q8C2×C418yescomplemented
82C2×C4⋊Q8C2313yes—
83C2×#127C2313yes—
84C2×C22⋊Q8—23no—
85C2×#121—23no—
86#519—48no—
87C4×#41—48no—
88C2×C4×Q8—21no—
89C2×C4×Q8—21no—
90C2×C4×Q8—23no—
91C2×C4×Q8—23no—
92C2×C4×Q8D411yescharacteristic
93C2×C4×Q8D411yescharacteristic
94C2×C4×Q8C2311yescharacteristic
95C2×C4×D4—23no—
96C2×C4×D4—23no—
97C2×C4×D4—23no—
98C2×C4×D4C2313yes—
99C2×C4×D4C2313yes—
100C2×C4×D4C2313yes—
101C2×C4×D4C2313yes—
102C2×#117—23no—
103C2×#117—23no—
104C2×#117—23no—
105C22×C42—23no—
106#486C2×C4124yescomplemented
107#474C2×C4124yescomplemented
108#459C2×C4124yescomplemented
109C8×Q8—24no—
110C8×Q8C2×C418yescomplemented
111C4×D8C2×C4124yescomplemented
112C2×#107C2311yescharacteristic
113C2×#107C2313yes—
114C2×#107C2316yes—
115C2×#107C2313yes—
116C2×C4⋊C8C2313yes—
117C2×C4⋊C8C2313yes—
118C2×#102C2311yescharacteristic
119C2×#102C2313yes—
120C2×#102C2316yes—
121C2×#102C2313yes—
122C2×#102C2313yes—
123C2×#102C2313yes—
124C2×#102C2316yes—
125C2×#102C2311yescharacteristic
126C2×C4×C8C2313yes—
127C2×C4×C8C2316yes—
128C2×C4×C8C2313yes—
129#391—23no—
130#391—23no—
131#390—23no—
132#390—23no—
133#390—23no—
134#390C2316yes—
135#390C2313yes—
136#390C2311yescharacteristic
137#389—23no—
138#389—23no—
139#389—23no—
140#389C2313yes—
141#389C2313yes—
142#388—23no—
143#388—23no—
144#388—26no—
145#388—26no—
146#387—26no—
147#386—26no—
148#384C2313yes—
149#384C2313yes—
150#380—23no—
151#380—23no—
152C4×C4⋊C4—23no—
153C4×C4⋊C4D413yes—
154C4×C4⋊C4D413yes—
155C4×C4⋊C4C2316yes—
156C4×C4⋊C4C2313yes—
157C4×C4⋊C4C2313yes—
158C4×C4⋊C4C2313yes—
159C4×C4⋊C4C2311yescharacteristic
160C4×C4⋊C4C2313yes—
161C4×C22⋊C4—23no—
162C4×C22⋊C4—23no—
163C4×C22⋊C4—23no—
164C4×C22⋊C4—23no—
165C43C2313yes—
166#340C2313yes—
167#340C2316yes—
168#340C2313yes—
169#340C2313yes—
170#340C2313yes—
171#340C2313yes—
172#340C2311yescharacteristic
173#336C2311yescharacteristic
174#336C2313yes—
175#336C2313yes—
176#336C2313yes—
177C2×#41—28no—
178C2×#41—44no—
179C2×#41—44no—
180C22×Q8—28no—
181C22×Q8—23no—
182C22×Q8—44no—
183C22×Q8—44no—
184C22×Q8C2×D411yescharacteristic
185C22×D4—23no—
186C22×D4—23no—
187C22×D4C2413yes—
188C22×D4C2411yescharacteristic
189C23×C4—23no—
190C23×C4—23no—
191C23×C4—23no—
192C23×C4—23no—
193C2×D8—43no—
194C2×D8—43no—
195C2×D8C22×C4112yes—
196C22×C8C2411yescharacteristic
197C22×C8C2413yes—
198C22×C8C2413yes—
199C22×C8C2411yescharacteristic
200C4⋊Q8—26no—
201C4⋊Q8C22×C4112yes—
202#127—26no—
203#127C22×C4112yes—
204C22⋊Q8—224no—
205#121—224no—
206C4×Q8—24no—
207C4×Q8—44no—
208C4×Q8—44no—
209C4×Q8—412no—
210C4×Q8—412no—
211C4×Q8D818yescomplemented
212C4×Q8C22×C414yes—
213C4×Q8#4114yes—
214C4×D4—26no—
215C4×D4—26no—
216C4×D4—26no—
217C4×D4—43no—
218C4×D4—43no—
219C4×D4—43no—
220C4×D4—412no—
221C4×D4—412no—
222C4×D4—86no—
223C4×D4C22×C4112yes—
224C4×D4C22×C4112yes—
225#117—412no—
226#117—412no—
227#117—412no—
228C2×C4⋊C4—23no—
229C2×C4⋊C4—23no—
230C2×C4⋊C4—23no—
231C2×C4⋊C4—23no—
232C2×C4⋊C4—23no—
233C2×C4⋊C4—23no—
234C2×C4⋊C4—23no—
235C2×C4⋊C4C2×D413yes—
236C2×C4⋊C4C2×D413yes—
237C2×C4⋊C4C2×D413yes—
238C2×C4⋊C4C2413yes—
239C2×C4⋊C4C2411yescharacteristic
240C2×C4⋊C4C2413yes—
241C2×C4⋊C4C2411yescharacteristic
242C2×C4⋊C4C2413yes—
243C2×C4⋊C4C2411yescharacteristic
244C2×C22⋊C4—23no—
245C2×C22⋊C4—23no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—23no—
248C2×C22⋊C4—26no—
249C2×C22⋊C4—23no—
250C2×C22⋊C4—23no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—26no—
253C2×C22⋊C4—23no—
254C2×C42—23no—
255C2×C42—23no—
256C2×C42—26no—
257C2×C42—23no—
258C2×C42—21no—
259C2×C42—412no—
260C2×C42C2×D413yes—
261C2×C42C2×D413yes—
262C2×C42C2×D413yes—
263C2×C42C2×D413yes—
264C2×C42C2411yescharacteristic
265C2×C42C2413yes—
266C2×C42C2413yes—
267#107—22no—
268#107—43no—
269#107—46no—
270#107C22×C4112yes—
271C4⋊C8—26no—
272C4⋊C8C22×C4112yes—
273#102—22no—
274#102—43no—
275#102—46no—
276#102—43no—
277#102—43no—
278#102C22×C414yes—
279#102C22×C4112yes—
280#102C22×C4124yes—
281C4×C8—26no—
282C4×C8—46no—
283C4×C8C22×C4112yes—
284#95—21no—
285#95—23no—
286#95—23no—
287#95—23no—
288#95—23no—
289#95—26no—
290#95—23no—
291#95C2×D416yes—
292C24—21no—
293C24—23no—
294#41—432no—
295C2×Q8—212no—
296C2×Q8—432no—
297C2×Q8—46no—
298C2×Q8—46no—
299C2×Q8C4×D414yes—
300C2×Q8C2×D814yes—
301C2×Q8#13514yes—
302C2×D4—212no—
303C2×D4—212no—
304C2×D4—21no—
305C2×D4—23no—
306C2×D4—21no—
307C2×D4—23no—
308C2×D4—21no—
309C2×D4—23no—
310C2×D4—23no—
311C2×D4—23no—
312C2×D4—21no—
313C2×D4—21no—
314C2×D4—46no—
315C2×D4—46no—
316C2×D4—83no—
317C2×D4—83no—
318C2×D4C23×C416yes—
319C2×D4C23×C412yes—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—23no—
323C22×C4—212no—
324C22×C4—212no—
325C22×C4—21no—
326C22×C4—23no—
327C22×C4—23no—
328C22×C4—46no—
329C22×C4—46no—
330C22×C4—46no—
331C22×C4—46no—
332C22×C4—46no—
333C22×C4—46no—
334C22×C4—46no—
335C22×C4—46no—
336C22×C4—46no—
337C22×C4—83no—
338C22×C4C22×D413yes—
339C22×C4C22×D411yescharacteristic
340C22×C4C22×D413yes—
341C22×C4C22×D413yes—
342C22×C4C22×D411yescharacteristic
343C22×C4C22×D413yes—
344C22×C4C2511yescharacteristic
345D8—412no—
346C2×C8—21no—
347C2×C8—23no—
348C2×C8—21no—
349C2×C8—23no—
350C2×C8—21no—
351C2×C8—23no—
352C2×C8—21no—
353C2×C8—23no—
354C2×C8—21no—
355C2×C8—23no—
356C2×C8C23×C412yes—
357C2×C8C23×C416yes—
358C4⋊C4—212no—
359C4⋊C4—26no—
360C4⋊C4—21no—
361C4⋊C4—23no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—23no—
366C4⋊C4—23no—
367C4⋊C4—23no—
368C4⋊C4—23no—
369C4⋊C4—46no—
370C4⋊C4—46no—
371C4⋊C4—46no—
372C4⋊C4—46no—
373C4⋊C4—83no—
374C4⋊C4—83no—
375C4⋊C4C4×D4112yes—
376C4⋊C4C2×D8112yes—
377C4⋊C4C23×C416yes—
378C4⋊C4C23×C412yes—
379C4⋊C42−1+412yes—
380C22⋊C4—212no—
381C22⋊C4—212no—
382C22⋊C4—46no—
383C22⋊C4—42no—
384C22⋊C4—46no—
385C22⋊C4—46no—
386C22⋊C4—42no—
387C22⋊C4—412no—
388C22⋊C4—83no—
389C22⋊C4—86no—
390C42—26no—
391C42—23no—
392C42—23no—
393C42—21no—
394C42—21no—
395C42—43no—
396C42—43no—
397C42—42no—
398C42—46no—
399C42—46no—
400C42—412no—
401C42—46no—
402C42C2×D8112yes—
403C42C23×C416yes—
404C42C2×#4116yes—
405C23—22no—
406C23—26no—
407C23—41no—
408C23—43no—
409C23—41no—
410C23—43no—
411C23—43no—
412C23—41no—
413C23—43no—
414C23—41no—
415C23—41no—
416C23—43no—
417C23—41no—
418C23—43no—
419C23C23×D411yescharacteristic
420Q8—424no—
421Q8C4×D8116yes—
422D4—22no—
423D4—26no—
424D4—22no—
425D4—26no—
426D4—424no—
427D4—41no—
428D4—43no—
429D4—41no—
430D4—43no—
431D4—812no—
432C2×C4—21no—
433C2×C4—23no—
434C2×C4—23no—
435C2×C4—23no—
436C2×C4—21no—
437C2×C4—23no—
438C2×C4—23no—
439C2×C4—21no—
440C2×C4—22no—
441C2×C4—26no—
442C2×C4—21no—
443C2×C4—23no—
444C2×C4—22no—
445C2×C4—21no—
446C2×C4—23no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—43no—
451C2×C4—41no—
452C2×C4—43no—
453C2×C4—424no—
454C2×C4—424no—
455C2×C4—43no—
456C2×C4—43no—
457C2×C4—41no—
458C2×C4—43no—
459C2×C4—43no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—43no—
463C2×C4—43no—
464C2×C4—812no—
465C2×C4—812no—
466C2×C4C2×C4×D416yes—
467C2×C4C2×C4×D416yes—
468C2×C4C22×D816yes—
469C2×C4C22×D812yes—
470C2×C4C22×D816yes—
471C2×C4C2×#13516yes—
472C2×C4C24×C411yescharacteristic
473C2×C4C22×#4111yescharacteristic
474C2×C4C2×2−1+411yescharacteristic
475C2×C4#58511yescharacteristic
476C2×C4#58511yescharacteristic
477C2×C4#58511yescharacteristic
478C8—26no—
479C8—22no—
480C8—41no—
481C8—43no—
482C22—46no—
483C22—42no—
484C22—42no—
485C22—46no—
486C22—42no—
487C22—46no—
488C22—83no—
489C22—81no—
490C22—81no—
491C22—83no—
492C22—83no—
493C22—81no—
494C22—81no—
495C22—83no—
496C22C22×C4×D411yescharacteristic
497C22#348711yescharacteristic
498C22#348911yescharacteristic
499C22C23×D811yescharacteristic
500C22C22×#13511yescharacteristic
501C22#358511yescharacteristic
502C22#358611yescharacteristic
503C4—26no—
504C4—26no—
505C4—22no—
506C4—42no—
507C4—43no—
508C4—41no—
509C4—42no—
510C4—46no—
511C4—81no—
512C4—83no—
513C4—83no—
514C4C2×C4×D8112yes—
515C4#342811yescharacteristic
516C4C4×2−1+411yescharacteristic
517C4#351811yescharacteristic
518C4#352411yescharacteristic
519C2—86no—
520C2—82no—
521C2#3429311yescharacteristic
522C2C22×C4×D811yescharacteristic
523C2#6017411yescharacteristic
524C2#6188011yescharacteristic
525C2#6188711yescharacteristic
526C2#6209511yescharacteristic
527C2#6210611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC2×#3358C2
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×#550C2semidirect
4. #23216C2semidirect
5. #23093C2semidirect
6. #23060C2semidirect
7. #22807C2semidirect
8. Q8×#107C2semidirect
9. #22540C2semidirect
10. Q8×#102C2semidirect
11. #22220C2semidirect
12. #22145C2semidirect
13. C4×#474C2semidirect
14. #21113C2semidirect
15. #20569C2semidirect
20. #3358C4semidirect
81. D4×Q8C2×C4semidirect
106. #486C2×C4semidirect
107. #474C2×C4semidirect
108. #459C2×C4semidirect
110. C8×Q8C2×C4semidirect
111. C4×D8C2×C4semidirect
211. C4×Q8D8semidirect

Maximal subgroups

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

subgroupindex
C2×#33582
#343112
C4×#5502
#232162
#230932
#230602
#228072
Q8×#1072
#225402
Q8×#1022
#222202
#221452
C4×#4742
#211132
#205692