Finite Group Databasebeta

← #6400699

Subgroups — #6400699

Order 512.

2311 conjugacy classes of subgroups.

The subgroup lattice has 535 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3358C211yesmaximal, characteristic
2#34313C211yesmaximal, characteristic, complemented
3C4×D4×Q8C211yesmaximal, characteristic, complemented
4#23220C213yesmaximal, complemented
5#23001C213yesmaximal, complemented
6#22968C213yesmaximal, complemented
7Q8×#106C211yesmaximal, characteristic, complemented
8#22670C213yesmaximal, complemented
9#22540C211yesmaximal, characteristic, complemented
10Q8×#102C211yesmaximal, characteristic, complemented
11#22219C213yesmaximal, complemented
12#22116C213yesmaximal, complemented
13#21503C213yesmaximal, complemented
14#21118C213yesmaximal, complemented
15#20572C211yesmaximal, characteristic, complemented
16C2×#550C2211yescharacteristic
17C2×D4×Q8C2211yescharacteristic
18C2×#520—21no—
19C22×C4×Q8—21no—
20#3358C4116yescomplemented
21C2×#486C2213yes—
22C2×#474C2213yes—
23C2×#459C2213yes—
24C2×C8×Q8C2211yescharacteristic
25C2×C8×Q8C2211yescharacteristic
26C2×#442C2216yes—
27C2×C4×D8C2213yes—
28#2366C2213yes—
29#2365C2213yes—
30#2361—23no—
31#2360—23no—
32Q8×C4⋊C4C2211yescharacteristic
33Q8×C4⋊C4C2211yescharacteristic
34D4×C4⋊C4C2213yes—
35D4×C4⋊C4C2213yes—
36#2341—21no—
37Q8×C22⋊C4—21no—
38#2324—23no—
39C4×C4⋊Q8C2213yes—
40C4×#127C2213yes—
41C4×C22⋊Q8—23no—
42#2293C2213yes—
43C42×Q8C2211yescharacteristic
44C42×D4C2213yes—
45#1990C2213yes—
46#1988C2213yes—
47#1982C2213yes—
48#1957C2213yes—
49#1948C2213yes—
50#1948C2213yes—
51#1943C2216yes—
52#1936C2216yes—
53#1926C2216yes—
54#1926C2213yes—
55#1926C2213yes—
56#1921C2213yes—
57#1921C2213yes—
58#1879C2213yes—
59#1879C2213yes—
60#1864C2211yescharacteristic
61#1864C2213yes—
62#1864C2216yes—
63#1864C2216yes—
64#1844C2213yes—
65#1802C2213yes—
66#1802C2213yes—
67C4×#106C2213yes—
68#1770C2213yes—
69#1770C2213yes—
70#1767C2213yes—
71#1762C2216yes—
72C4×#102C2213yes—
73C4×#102C2213yes—
74#1744C2213yes—
75C22×#41—21no—
76C23×Q8—21no—
77C22×D8C2313yes—
78#550—24no—
79#550C2×C418yescomplemented
80D4×Q8—24no—
81D4×Q8C2×C418yescomplemented
82C2×C4⋊Q8C2313yes—
83C2×#127C2313yes—
84C2×C22⋊Q8—23no—
85C2×#121—23no—
86#520—48no—
87C2×C4×Q8—21no—
88C2×C4×Q8—21no—
89C2×C4×Q8—23no—
90C2×C4×Q8—23no—
91C2×C4×Q8—48no—
92C2×C4×Q8D411yescharacteristic
93C2×C4×Q8D411yescharacteristic
94C2×C4×Q8C2311yescharacteristic
95C2×C4×D4—23no—
96C2×C4×D4—23no—
97C2×C4×D4—23no—
98C2×C4×D4C2311yescharacteristic
99C2×C4×D4C2313yes—
100C2×C4×D4C2313yes—
101C2×C4×D4C2313yes—
102C2×C4×D4C2316yes—
103C2×#117—23no—
104C22×C4⋊C4—23no—
105C22×C4⋊C4—23no—
106C22×C42—23no—
107#486C2×C4124yescomplemented
108#474C2×C4124yescomplemented
109#459C2×C4124yescomplemented
110C8×Q8—24no—
111C8×Q8C2×C418yescomplemented
112#442—412no—
113C4×D8C2×C4124yescomplemented
114C2×#107C2313yes—
115C2×#106C2311yescharacteristic
116C2×#106C2313yes—
117C2×#106C2316yes—
118C2×C4⋊C8C2313yes—
119C2×C4⋊C8C2313yes—
120C2×#102C2313yes—
121C2×#102C2311yescharacteristic
122C2×#102C2313yes—
123C2×#102C2316yes—
124C2×#102C2313yes—
125C2×#102C2313yes—
126C2×#102C2316yes—
127C2×#102C2311yescharacteristic
128C2×#97C2316yes—
129C2×C4×C8C2313yes—
130C2×C4×C8C2313yes—
131#391—23no—
132#391—23no—
133#390—23no—
134#390—23no—
135#390—23no—
136#390C2313yes—
137#390C2313yes—
138#389—23no—
139#389—23no—
140#389—23no—
141#389C2316yes—
142#389C2313yes—
143#389C2311yescharacteristic
144#388—23no—
145#388—23no—
146#387—26no—
147#386—26no—
148#385—26no—
149#385—26no—
150#384C2313yes—
151#384C2313yes—
152#380—23no—
153#380—23no—
154C4×C4⋊C4—23no—
155C4×C4⋊C4D413yes—
156C4×C4⋊C4D413yes—
157C4×C4⋊C4C2313yes—
158C4×C4⋊C4C2313yes—
159C4×C4⋊C4C2313yes—
160C4×C4⋊C4C2313yes—
161C4×C4⋊C4C2313yes—
162C4×C22⋊C4—23no—
163C4×C22⋊C4—23no—
164C4×C22⋊C4—23no—
165C4×C22⋊C4—23no—
166C43C2313yes—
167#340C2313yes—
168#340C2316yes—
169#340C2313yes—
170#340C2311yescharacteristic
171#340C2313yes—
172#340C2313yes—
173#340C2313yes—
174#336C2313yes—
175#336C2313yes—
176#336C2311yescharacteristic
177#336C2313yes—
178C2×#41—28no—
179C2×#41—44no—
180C2×#41—44no—
181C22×Q8—28no—
182C22×Q8—23no—
183C22×Q8—44no—
184C22×Q8—44no—
185C22×Q8C2×D411yescharacteristic
186C22×D4—23no—
187C22×D4—23no—
188C22×D4C2413yes—
189C22×D4C2411yescharacteristic
190C23×C4—23no—
191C23×C4—23no—
192C23×C4—21no—
193C23×C4—23no—
194C23×C4—26no—
195C2×D8—26no—
196C2×D8—26no—
197C2×D8C22×C4112yes—
198C22×C8C2411yescharacteristic
199C22×C8C2413yes—
200C22×C8C2413yes—
201C22×C8C2411yescharacteristic
202C4⋊Q8—26no—
203C4⋊Q8C22×C4112yes—
204#127—26no—
205#127C22×C4112yes—
206C22⋊Q8—224no—
207#121—224no—
208C4×Q8—24no—
209C4×Q8—24no—
210C4×Q8—412no—
211C4×Q8—412no—
212C4×Q8—44no—
213C4×Q8—44no—
214C4×Q8C22×C414yes—
215C4×Q8#4114yes—
216C4×D4—22no—
217C4×D4—22no—
218C4×D4—26no—
219C4×D4—26no—
220C4×D4—26no—
221C4×D4—43no—
222C4×D4—412no—
223C4×D4—46no—
224C4×D4—46no—
225C4×D4—412no—
226C4×D4—86no—
227C4×D4C22×C4112yes—
228C4×D4C22×C4112yes—
229#117—412no—
230C2×C4⋊C4—23no—
231C2×C4⋊C4—23no—
232C2×C4⋊C4—23no—
233C2×C4⋊C4—23no—
234C2×C4⋊C4—26no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—23no—
237C2×C4⋊C4—23no—
238C2×C4⋊C4—412no—
239C2×C4⋊C4—412no—
240C2×C4⋊C4C2×D413yes—
241C2×C4⋊C4C2×D413yes—
242C2×C4⋊C4C2×D413yes—
243C2×C4⋊C4C2413yes—
244C2×C4⋊C4C2413yes—
245C2×C4⋊C4C2411yescharacteristic
246C2×C4⋊C4C2411yescharacteristic
247C2×C4⋊C4C2413yes—
248C2×C4⋊C4C2411yescharacteristic
249C2×C22⋊C4—23no—
250C2×C22⋊C4—23no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—23no—
253C2×C22⋊C4—23no—
254C2×C22⋊C4—23no—
255C2×C22⋊C4—23no—
256C2×C22⋊C4—26no—
257C2×C22⋊C4—23no—
258C2×C42—23no—
259C2×C42—23no—
260C2×C42—23no—
261C2×C42—23no—
262C2×C42—412no—
263C2×C42C2×D413yes—
264C2×C42C2×D413yes—
265C2×C42C2×D413yes—
266C2×C42C2×D413yes—
267C2×C42C2413yes—
268C2×C42C2411yescharacteristic
269C2×C42C2413yes—
270#107C22×C4112yes—
271#106—22no—
272#106—43no—
273#106—46no—
274C4⋊C8—26no—
275C4⋊C8C22×C4112yes—
276#102—26no—
277#102—22no—
278#102—26no—
279#102—43no—
280#102—46no—
281#102C22×C414yes—
282#102C22×C4112yes—
283#102C22×C4124yes—
284#97—46no—
285C4×C8—26no—
286C4×C8C22×C4112yes—
287#95—21no—
288#95—23no—
289#95—23no—
290#95—23no—
291#95—23no—
292#95—26no—
293#95—23no—
294#95C2×D416yes—
295C24—21no—
296C24—23no—
297#41—432no—
298C2×Q8—212no—
299C2×Q8—432no—
300C2×Q8—46no—
301C2×Q8—46no—
302C2×Q8C4×D414yes—
303C2×Q8AGL(1,ℤ/8)14yes—
304C2×Q8AGL(1,ℤ/8)14yes—
305C2×D4—212no—
306C2×D4—212no—
307C2×D4—21no—
308C2×D4—23no—
309C2×D4—23no—
310C2×D4—21no—
311C2×D4—23no—
312C2×D4—23no—
313C2×D4—21no—
314C2×D4—23no—
315C2×D4—46no—
316C2×D4—46no—
317C2×D4—83no—
318C2×D4—83no—
319C2×D4C23×C416yes—
320C2×D4C23×C412yes—
321C2×D4H(5,𝔽2)12yes—
322C2×D42−1+412yes—
323C22×C4—24no—
324C22×C4—21no—
325C22×C4—23no—
326C22×C4—21no—
327C22×C4—212no—
328C22×C4—212no—
329C22×C4—21no—
330C22×C4—23no—
331C22×C4—23no—
332C22×C4—42no—
333C22×C4—46no—
334C22×C4—46no—
335C22×C4—412no—
336C22×C4—46no—
337C22×C4—46no—
338C22×C4—46no—
339C22×C4—46no—
340C22×C4—42no—
341C22×C4—83no—
342C22×C4—86no—
343C22×C4—86no—
344C22×C4C22×D413yes—
345C22×C4C22×D411yescharacteristic
346C22×C4C22×D413yes—
347C22×C4C22×D413yes—
348C22×C4C22×D411yescharacteristic
349C22×C4C22×D413yes—
350C22×C4C2511yescharacteristic
351D8—224no—
352C2×C8—23no—
353C2×C8—21no—
354C2×C8—23no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—23no—
358C2×C8—23no—
359C2×C8—21no—
360C2×C8—23no—
361C2×C8—21no—
362C2×C8C23×C412yes—
363C2×C8C23×C416yes—
364C4⋊C4—212no—
365C4⋊C4—26no—
366C4⋊C4—26no—
367C4⋊C4—21no—
368C4⋊C4—23no—
369C4⋊C4—21no—
370C4⋊C4—23no—
371C4⋊C4—23no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—23no—
375C4⋊C4—23no—
376C4⋊C4—46no—
377C4⋊C4—412no—
378C4⋊C4—42no—
379C4⋊C4—46no—
380C4⋊C4—46no—
381C4⋊C4—83no—
382C4⋊C4—83no—
383C4⋊C4C4×D4112yes—
384C4⋊C4C23×C416yes—
385C4⋊C4C23×C412yes—
386C4⋊C42−1+412yes—
387C22⋊C4—24no—
388C22⋊C4—212no—
389C22⋊C4—212no—
390C22⋊C4—46no—
391C22⋊C4—46no—
392C22⋊C4—46no—
393C22⋊C4—46no—
394C22⋊C4—83no—
395C22⋊C4—86no—
396C42—26no—
397C42—26no—
398C42—26no—
399C42—23no—
400C42—21no—
401C42—21no—
402C42—23no—
403C42—43no—
404C42—46no—
405C42—46no—
406C42—46no—
407C42—46no—
408C42C23×C416yes—
409C42C2×#4116yes—
410C23—22no—
411C23—22no—
412C23—22no—
413C23—26no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23—43no—
419C23—43no—
420C23—43no—
421C23—43no—
422C23—43no—
423C23—43no—
424C23C23×D411yescharacteristic
425Q8—424no—
426Q8#442116yescomplemented
427D4—26no—
428D4—22no—
429D4—26no—
430D4—22no—
431D4—22no—
432D4—424no—
433D4—43no—
434D4—43no—
435D4—812no—
436D4#52014yescomplemented
437C2×C4—23no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—23no—
441C2×C4—21no—
442C2×C4—23no—
443C2×C4—22no—
444C2×C4—26no—
445C2×C4—21no—
446C2×C4—22no—
447C2×C4—23no—
448C2×C4—23no—
449C2×C4—21no—
450C2×C4—23no—
451C2×C4—23no—
452C2×C4—23no—
453C2×C4—48no—
454C2×C4—41no—
455C2×C4—43no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—43no—
459C2×C4—41no—
460C2×C4—424no—
461C2×C4—424no—
462C2×C4—41no—
463C2×C4—43no—
464C2×C4—41no—
465C2×C4—43no—
466C2×C4—43no—
467C2×C4—41no—
468C2×C4—43no—
469C2×C4—43no—
470C2×C4—43no—
471C2×C4—812no—
472C2×C4—1612no—
473C2×C4C2×C4×D416yes—
474C2×C4C2×C4×D416yes—
475C2×C4C2×AGL(1,ℤ/8)16yes—
476C2×C4C2×AGL(1,ℤ/8)16yes—
477C2×C4#57712yes—
478C2×C4C24×C411yescharacteristic
479C2×C4C22×#4111yescharacteristic
480C2×C4C2×H(5,𝔽2)11yescharacteristic
481C2×C4C2×2−1+411yescharacteristic
482C2×C4C2×2−1+411yescharacteristic
483C2×C4#58511yescharacteristic
484C8—26no—
485C8—22no—
486C8—41no—
487C8—43no—
488C22—24no—
489C22—42no—
490C22—42no—
491C22—42no—
492C22—42no—
493C22—42no—
494C22—42no—
495C22—46no—
496C22—46no—
497C22—46no—
498C22—83no—
499C22—83no—
500C22—83no—
501C22—83no—
502C22C22×C4×D411yescharacteristic
503C22#348611yescharacteristic
504C22#348811yescharacteristic
505C22C22×AGL(1,ℤ/8)11yescharacteristic
506C22C22×AGL(1,ℤ/8)11yescharacteristic
507C22C2×#57711yescharacteristic
508C22#358411yescharacteristic
509C4—26no—
510C4—22no—
511C4—43no—
512C4—43no—
513C4—41no—
514C4—42no—
515C4—42no—
516C4—46no—
517C4—83no—
518C4—83no—
519C4—81no—
520C4C2×#442112yes—
521C4C2×#52011yescharacteristic
522C4C4×2−1+411yescharacteristic
523C4#351611yescharacteristic
524C4#352211yescharacteristic
525C2—44no—
526C2—86no—
527C2#3429411yescharacteristic
528C2C22×#44211yescharacteristic
529C2C4×#57711yescharacteristic
530C2#6184011yescharacteristic
531C2#6184311yescharacteristic
532C2#6199911yescharacteristic
533C2#6200311yescharacteristic
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. #34313C2semidirect
3. C4×D4×Q8C2semidirect
4. #23220C2semidirect
5. #23001C2semidirect
6. #22968C2semidirect
7. Q8×#106C2semidirect
8. #22670C2semidirect
9. #22540C2semidirect
10. Q8×#102C2semidirect
11. #22219C2semidirect
12. #22116C2semidirect
13. #21503C2semidirect
14. #21118C2semidirect
15. #20572C2semidirect
20. #3358C4semidirect
79. #550C2×C4semidirect
81. D4×Q8C2×C4semidirect
107. #486C2×C4semidirect
108. #474C2×C4semidirect
109. #459C2×C4semidirect
111. C8×Q8C2×C4semidirect
113. C4×D8C2×C4semidirect
426. Q8#442semidirect
436. D4#520semidirect

Maximal subgroups

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

subgroupindex
C2×#33582
#343132
C4×D4×Q82
#232202
#230012
#229682
Q8×#1062
#226702
#225402
Q8×#1022
#222192
#221162
#215032
#211182
#205722