Finite Group Databasebeta

← Q8×#468 #6625711

Subgroups — Q8×#468 #6625711

Order 512.

1997 conjugacy classes of subgroups.

The subgroup lattice has 553 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×SD16C211yesmaximal, characteristic, complemented
2C2×Q8×M16C211yesmaximal, characteristic, complemented
3Q8×C22⋊Q8C211yesmaximal, characteristic, complemented
4Q8×#121C211yesmaximal, characteristic
5#33210C213yesmaximal, complemented
6#28684C213yesmaximal, complemented
7#28272C213yesmaximal, complemented
8#25479C213yesmaximal, complemented
9#25477C213yesmaximal, complemented
10#24845C213yesmaximal, complemented
11#24843C213yesmaximal
12Q8×#107C211yesmaximal, characteristic, complemented
13Q8×#103C211yesmaximal, characteristic, complemented
14Q8×#102C211yesmaximal, characteristic, complemented
15C4×#468C213yesmaximal
16C2×Q82C2211yescharacteristic
17C2×D4×Q8—21no—
18C2×D4×Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20Q8×SD16—28no—
21C2×#498C2213yes—
22C2×#475C2213yes—
23C2×#476C2213yes—
24C2×#468C2211yescharacteristic
25Q8×M16—28no—
26C2×#446C2213yes—
27C2×#446C2213yes—
28C2×C8×Q8C2211yescharacteristic
29C2×C8×Q8C2211yescharacteristic
30C2×C4×SD16C2213yes—
31C2×#423C2213yes—
32C2×C4×M16C2213yes—
33#2857C2213yes—
34#2857C2213yes—
35#2582C2213yes—
36#2567C2213yes—
37#2546C2213yes—
38#2538C2213yes—
39#2451—23no—
40#2440—23no—
41#2429—23no—
42#2417—23no—
43Q8×C4⋊C4—21no—
44Q8×C4⋊C4C2211yescharacteristic
45Q8×C4⋊C4C2211yescharacteristic
46Q8×C22⋊C4—21no—
47Q8×C22⋊C4—21no—
48C4×C22⋊Q8C2213yes—
49C4×#121C2213yes—
50#2088C2216yes—
51#2088C2216yes—
52#2068C2216yes—
53#2067C2216yes—
54#1990C2213yes—
55#1990C2213yes—
56#1988C2213yes—
57#1987C2213yes—
58#1973C2213yes—
59#1947C2213yes—
60#1947C2213yes—
61#1943C2213yes—
62#1943C2216yes—
63#1943C2216yes—
64#1936C2213yes—
65#1935C2213yes—
66#1895C2213yes—
67#1894C2213yes—
68#1845C2213yes—
69#1809C2216yes—
70#1809C2213yes—
71#1803C2213yes—
72#1802C2213yes—
73C4×#107C2213yes—
74C4×#103C2213yes—
75C4×#102C2213yes—
76C23×Q8—21no—
77C23×Q8D411yescharacteristic
78C22×SD16C2311yescharacteristic
79C22×M16C2313yes—
80C22×M16C2311yescharacteristic
81Q82—24no—
82Q82—24no—
83D4×Q8—24no—
84D4×Q8—24no—
85D4×Q8—48no—
86C2×C4⋊Q8—23no—
87C2×C4⋊Q8—23no—
88C2×C4⋊Q8C2313yes—
89C2×C4⋊Q8C2313yes—
90C2×C22⋊Q8—23no—
91C2×C22⋊Q8—23no—
92C2×C22⋊Q8—23no—
93C2×C22⋊Q8—23no—
94C2×C22⋊Q8C2311yescharacteristic
95C2×#121C2311yescharacteristic
96C2×C4×Q8—24no—
97C2×C4×Q8—24no—
98C2×C4×Q8—21no—
99C2×C4×Q8—21no—
100C2×C4×Q8—21no—
101C2×C4×Q8D411yescharacteristic
102C2×C4×Q8D413yes—
103C2×C4×Q8C2313yes—
104C2×C4×Q8C2313yes—
105C2×C4×Q8C2311yescharacteristic
106C2×C4×D4—23no—
107C2×C4×D4C2313yes—
108C22×C4⋊C4D413yes—
109C22×C42C2313yes—
110#498—212no—
111#476—212no—
112#475—212no—
113#468Q818yesdirect product factor
114#446—212no—
115#446—212no—
116C8×Q8—24no—
117C8×Q8D418yescomplemented
118C4×SD16—212no—
119C2×#107C2313yes—
120C2×#107C2316yes—
121C2×#107C2311yescharacteristic
122C2×#106C2316yes—
123C2×#106C2313yes—
124#423—212no—
125C2×C4⋊C8C2313yes—
126C2×C4⋊C8C2316yes—
127C2×C4⋊C8C2313yes—
128C2×#103C2313yes—
129C2×#103C2313yes—
130C2×#103C2311yescharacteristic
131C2×#102C2313yes—
132C2×#102C2313yes—
133C2×#102C2311yescharacteristic
134C4×M16—212no—
135C2×#97C2313yes—
136C2×#97C2313yes—
137C2×C4×C8C2313yes—
138C2×C4×C8C2313yes—
139#398—26no—
140#398—26no—
141#396—26no—
142#395—26no—
143#391—23no—
144#391—23no—
145#391—23no—
146#391—23no—
147#391—23no—
148#391C2313yes—
149#390C2313yes—
150#389—23no—
151#389—23no—
152#389—23no—
153#389—23no—
154#389—23no—
155#389C2313yes—
156#389C2316yes—
157#389C2316yes—
158#389C2313yes—
159#388—23no—
160#387—23no—
161#386—23no—
162#385—23no—
163#384—23no—
164#384C2313yes—
165#384C2313yes—
166#380—23no—
167#380—23no—
168#380—23no—
169#380C2313yes—
170#380C2313yes—
171C4×C4⋊C4—23no—
172C4×C4⋊C4C2313yes—
173C4×C4⋊C4C2313yes—
174C4×C22⋊C4—23no—
175C4×C22⋊C4—23no—
176#340C2316yes—
177#340C2316yes—
178#340C2313yes—
179#340C2313yes—
180#340C2313yes—
181#340C2313yes—
182#340C2313yes—
183#340C2313yes—
184C22×Q8—24no—
185C22×Q8—24no—
186C22×Q8—24no—
187C22×Q8—23no—
188C22×Q8—23no—
189C22×Q8—44no—
190C22×Q8—44no—
191C22×Q8—44no—
192C22×Q8C2×D413yes—
193C22×Q8C2×D411yescharacteristic
194C22×Q8C2411yescharacteristic
195C22×D4—21no—
196C22×D4C2411yescharacteristic
197C23×C4—23no—
198C23×C4C2×D413yes—
199C23×C4C2×D413yes—
200C23×C4C2411yescharacteristic
201C2×SD16—22no—
202C2×SD16—22no—
203C2×SD16C2×Q814yescomplemented
204C2×M16—26no—
205C2×M16—22no—
206C2×M16—26no—
207C2×M16—22no—
208C2×M16—43no—
209C2×M16C2×Q814yescomplemented
210C22×C8C2413yes—
211C22×C8C2413yes—
212C22×C8C2411yescharacteristic
213C22×C8C2411yescharacteristic
214C4⋊Q8—26no—
215C4⋊Q8—26no—
216C4⋊Q8—26no—
217C4⋊Q8—26no—
218C4⋊Q8—412no—
219C4⋊Q8—412no—
220C22⋊Q8—22no—
221C22⋊Q8—412no—
222C22⋊Q8—412no—
223C22⋊Q8—412no—
224C22⋊Q8—86no—
225C22⋊Q8C2×Q814yescomplemented
226#121—22no—
227#121C2×Q814yes—
228C4×Q8—24no—
229C4×Q8—212no—
230C4×Q8—26no—
231C4×Q8—26no—
232C4×Q8—26no—
233C4×Q8—26no—
234C4×Q8—44no—
235C4×Q8—44no—
236C4×Q8—44no—
237C4×Q8C2×D414yes—
238C4×Q8#4114yes—
239C4×D4—26no—
240C4×D4—26no—
241C4×D4—412no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—212no—
244C2×C4⋊C4—23no—
245C2×C4⋊C4—23no—
246C2×C4⋊C4—23no—
247C2×C4⋊C4—23no—
248C2×C4⋊C4—23no—
249C2×C4⋊C4—26no—
250C2×C4⋊C4—23no—
251C2×C4⋊C4—26no—
252C2×C4⋊C4—23no—
253C2×C4⋊C4—23no—
254C2×C4⋊C4—23no—
255C2×C4⋊C4C2×D413yes—
256C2×C4⋊C4C2×D416yes—
257C2×C4⋊C4C2×D413yes—
258C2×C4⋊C4C2413yes—
259C2×C4⋊C4C2413yes—
260C2×C4⋊C4C2413yes—
261C2×C4⋊C4C2413yes—
262C2×C4⋊C4C2411yescharacteristic
263C2×C4⋊C4C2411yescharacteristic
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—23no—
267C2×C22⋊C4—23no—
268C2×C22⋊C4—23no—
269C2×C22⋊C4—23no—
270C2×C22⋊C4—23no—
271C2×C42—26no—
272C2×C42—26no—
273C2×C42—23no—
274C2×C42—23no—
275C2×C42—23no—
276C2×C42C2×D413yes—
277C2×C42C2×D413yes—
278C2×C42C2413yes—
279C2×C42C2413yes—
280#107—26no—
281#107—46no—
282#107C2×Q814yescomplemented
283#106—26no—
284#106—212no—
285C4⋊C8—212no—
286C4⋊C8—26no—
287C4⋊C8C2×D4112yes—
288#103—26no—
289#103—43no—
290#103C2×Q814yescomplemented
291#102—26no—
292#102—43no—
293#102C2×Q814yescomplemented
294#97—26no—
295#97—26no—
296C4×C8—26no—
297C4×C8C2×D4112yes—
298#95—23no—
299#95—23no—
300#95—23no—
301#95—23no—
302#95—23no—
303#95—23no—
304#95—23no—
305#95—23no—
306#95—26no—
307#95—26no—
308C24—21no—
309C24C22×D411yescharacteristic
310C2×Q8—26no—
311C2×Q8—26no—
312C2×Q8—26no—
313C2×Q8—21no—
314C2×Q8—21no—
315C2×Q8—21no—
316C2×Q8—21no—
317C2×Q8—21no—
318C2×Q8—416no—
319C2×Q8—46no—
320C2×Q8—46no—
321C2×Q8—46no—
322C2×Q8—46no—
323C2×Q8—83no—
324C2×Q8—83no—
325C2×Q8—816no—
326C2×Q8#12114yes—
327C2×Q8AGL(1,ℤ/8)14yes—
328C2×Q8#13714yes—
329C2×Q8C22×Q812yes—
330C2×D4—24no—
331C2×D4—21no—
332C2×D4—21no—
333C2×D4—21no—
334C2×D4—21no—
335C2×D4—21no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4C22×Q812yes—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—26no—
347C22×C4—26no—
348C22×C4—26no—
349C22×C4—26no—
350C22×C4—23no—
351C22×C4—23no—
352C22×C4—23no—
353C22×C4—23no—
354C22×C4—43no—
355C22×C4—43no—
356C22×C4—46no—
357C22×C4—46no—
358C22×C4—46no—
359C22×C4C22×D413yes—
360C22×C4C22×D413yes—
361C22×C4C22×D413yes—
362C22×C4C22×D411yescharacteristic
363C22×C4C22×D413yes—
364C22×C4C22×Q812yes—
365C22×C42−1+412yes—
366C22×C4C2511yescharacteristic
367SD16—28no—
368M16—28no—
369M16—412no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—23no—
373C2×C8—23no—
374C2×C8—23no—
375C2×C8—23no—
376C2×C8—23no—
377C2×C8C22×D416yes—
378C2×C8C22×D412yes—
379C2×C8C22×Q812yes—
380C2×C8C22×Q812yes—
381C2×C82−1+412yes—
382C4⋊C4—26no—
383C4⋊C4—212no—
384C4⋊C4—24no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—23no—
388C4⋊C4—23no—
389C4⋊C4—23no—
390C4⋊C4—23no—
391C4⋊C4—23no—
392C4⋊C4—23no—
393C4⋊C4—23no—
394C4⋊C4—23no—
395C4⋊C4—46no—
396C4⋊C4—412no—
397C4⋊C4—46no—
398C4⋊C4—46no—
399C4⋊C4—46no—
400C4⋊C4—46no—
401C4⋊C4—46no—
402C4⋊C4—83no—
403C4⋊C4—83no—
404C4⋊C4—83no—
405C4⋊C4—86no—
406C4⋊C4#121112yes—
407C4⋊C4C22×Q812yes—
408C4⋊C4C22×Q812yes—
409C22⋊C4—24no—
410C22⋊C4—24no—
411C22⋊C4—46no—
412C22⋊C4—46no—
413C22⋊C4—46no—
414C22⋊C4—46no—
415C22⋊C4—83no—
416C42—26no—
417C42—26no—
418C42—23no—
419C42—23no—
420C42—46no—
421C42—46no—
422C42—46no—
423C42C22×D416yes—
424C42C2×#4116yes—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—22no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23D4×Q812yes—
439C23C23×D411yescharacteristic
440Q8—22no—
441Q8—22no—
442Q8—41no—
443Q8—41no—
444Q8—412no—
445Q8—812no—
446Q8—166no—
447Q8#468116yesdirect product factor
448D4—22no—
449D4—22no—
450D4—48no—
451D4—41no—
452D4—41no—
453C2×C4—23no—
454C2×C4—23no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—23no—
462C2×C4—23no—
463C2×C4—23no—
464C2×C4—23no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—23no—
469C2×C4—23no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—412no—
477C2×C4—43no—
478C2×C4—43no—
479C2×C4—43no—
480C2×C4—43no—
481C2×C4—43no—
482C2×C4—43no—
483C2×C4—43no—
484C2×C4—43no—
485C2×C4—43no—
486C2×C4—43no—
487C2×C4—43no—
488C2×C4—43no—
489C2×C4—86no—
490C2×C4—812no—
491C2×C4C2×#12116yes—
492C2×C4C2×#12116yes—
493C2×C4D4×Q812yes—
494C2×C4C2×AGL(1,ℤ/8)16yes—
495C2×C4C2×#13716yes—
496C2×C4C23×D411yescharacteristic
497C2×C4C23×Q811yescharacteristic
498C2×C4C22×#4111yescharacteristic
499C2×C4C2×2−1+411yescharacteristic
500C2×C4C2×2−1+411yescharacteristic
501C2×C4#58511yescharacteristic
502C8—22no—
503C8—26no—
504C8—43no—
505C8D4×Q814yescomplemented
506C22—22no—
507C22—22no—
508C22—22no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22—42no—
514C22—42no—
515C22—42no—
516C22—81no—
517C22—81no—
518C22—81no—
519C22—81no—
520C22C22×#12111yescharacteristic
521C22C2×D4×Q811yescharacteristic
522C22#349711yescharacteristic
523C22C22×AGL(1,ℤ/8)11yescharacteristic
524C22C22×#13711yescharacteristic
525C22#358511yescharacteristic
526C22#358611yescharacteristic
527C4—26no—
528C4—22no—
529C4—43no—
530C4—43no—
531C4—42no—
532C4—42no—
533C4—42no—
534C4—83no—
535C4—83no—
536C4—83no—
537C4—83no—
538C4C2×#468112yes—
539C4C2×D4×Q811yescharacteristic
540C4Q8×#4111yescharacteristic
541C4#348911yescharacteristic
542C4#352311yescharacteristic
543C2—42no—
544C2—82no—
545C2Q8×#12111yescharacteristic
546C2C22×#46811yescharacteristic
547C2#6046011yescharacteristic
548C2Q8×AGL(1,ℤ/8)11yescharacteristic
549C2Q8×#13711yescharacteristic
550C2#6239711yescharacteristic
551C2#6239811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×Q8×SD16C2semidirect
2. C2×Q8×M16C2semidirect
3. Q8×C22⋊Q8C2semidirect
5. #33210C2semidirect
6. #28684C2semidirect
7. #28272C2semidirect
8. #25479C2semidirect
9. #25477C2semidirect
10. #24845C2semidirect
12. Q8×#107C2semidirect
13. Q8×#103C2semidirect
14. Q8×#102C2semidirect
113. #468Q8direct (N × H)
117. C8×Q8D4semidirect
203. C2×SD16C2×Q8semidirect
209. C2×M16C2×Q8semidirect
225. C22⋊Q8C2×Q8semidirect
282. #107C2×Q8semidirect
290. #103C2×Q8semidirect
293. #102C2×Q8semidirect
447. Q8#468direct (N × H)
505. C8D4×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×SD162
C2×Q8×M162
Q8×C22⋊Q82
Q8×#1212
#332102
#286842
#282722
#254792
#254772
#248452
#248432
Q8×#1072
Q8×#1032
Q8×#1022
C4×#4682