Finite Group Databasebeta

← Q8×#435 #6385151

Subgroups — Q8×#435 #6385151

Order 512.

2144 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×M16C211yesmaximal, characteristic, complemented
2C22×C8×Q8C211yesmaximal, characteristic, complemented
3C4×D4×Q8C211yesmaximal, characteristic
4#23150C213yesmaximal, complemented
5#22956C213yesmaximal, complemented
6#22953C213yesmaximal, complemented
7Q8×C4⋊C8C211yesmaximal, characteristic, complemented
8#22648C213yesmaximal
9Q8×C22⋊C8C211yesmaximal, characteristic, complemented
10Q8×C22⋊C8C211yesmaximal, characteristic, complemented
11#21714C213yesmaximal, complemented
12#21597C213yesmaximal, complemented
13#21596C213yesmaximal, complemented
14Q8×#97C211yesmaximal, characteristic, complemented
15C4×#435C213yesmaximal
16C2×D4×Q8C411yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19Q8×M16—28no—
20C2×#446C2213yes—
21C2×#446C2213yes—
22C2×C8×Q8—28no—
23C2×C8×Q8C2213yes—
24C2×C8×Q8C2213yes—
25C2×C8×Q8C2211yescharacteristic
26C2×C8×Q8C2211yescharacteristic
27C2×C8×Q8C2211yescharacteristic
28C2×C8×Q8C2211yescharacteristic
29C2×#435C2213yes—
30C2×#435C2211yescharacteristic
31C2×#423C2213yes—
32C22×C4⋊C8C2213yes—
33C2×C4×M16C2213yes—
34C22×C4×C8C2213yes—
35#2365C413yes—
36#2360C413yes—
37#2360C413yes—
38Q8×C4⋊C4C411yescharacteristic
39D4×C4⋊C4C413yes—
40Q8×C22⋊C4C411yescharacteristic
41Q8×C22⋊C4C411yescharacteristic
42C4×C4⋊Q8C2213yes—
43C4×C22⋊Q8C2213yes—
44C4×C22⋊Q8C2213yes—
45C42×Q8C2211yescharacteristic
46C42×D4C2213yes—
47#1984C2213yes—
48#1984C2213yes—
49#1984C2213yes—
50#1984C2213yes—
51#1940C2216yes—
52#1940C2216yes—
53#1939C2213yes—
54#1939C2213yes—
55#1917C2216yes—
56#1916C2216yes—
57#1851C2216yes—
58#1850C2216yes—
59#1842C2213yes—
60#1797C2213yes—
61#1797C2213yes—
62#1771C2213yes—
63#1771C2213yes—
64#1771C2216yes—
65#1770C2213yes—
66#1770C2213yes—
67#1767C2213yes—
68C4×C4⋊C8C2213yes—
69#1753C2213yes—
70#1753C2213yes—
71#1752C2213yes—
72#1752C2213yes—
73C4×C22⋊C8C2213yes—
74C4×C22⋊C8C2213yes—
75C4×#97C2213yes—
76C23×Q8C2×C411yescharacteristic
77C23×Q8C2×C411yescharacteristic
78C22×M16C2313yes—
79C22×M16C2311yescharacteristic
80C23×C8C2313yes—
81C23×C8C2311yescharacteristic
82D4×Q8—28no—
83C2×C4⋊Q8C2×C413yes—
84C2×C22⋊Q8C2×C413yes—
85C2×C22⋊Q8C2×C413yes—
86C2×C4×Q8—24no—
87C2×C4×Q8—24no—
88C2×C4×Q8—24no—
89C2×C4×Q8—24no—
90C2×C4×Q8C2×C411yescharacteristic
91C2×C4×Q8C2×C411yescharacteristic
92C2×C4×Q8C2×C411yescharacteristic
93C2×C4×Q8C2×C411yescharacteristic
94C2×C4×Q8C2×C413yes—
95C2×C4×Q8C2×C413yes—
96C2×C4×Q8C2313yes—
97C2×C4×Q8C2313yes—
98C2×C4×Q8C2311yescharacteristic
99C2×C4×D4C2×C413yes—
100C2×C4×D4C2×C413yes—
101C2×C4×D4C2311yescharacteristic
102C22×C4⋊C4C2×C413yes—
103C22×C4⋊C4C2×C413yes—
104C22×C42C2313yes—
105C22×C42C2313yes—
106#446—212no—
107#446—212no—
108C8×Q8—24no—
109C8×Q8—24no—
110C8×Q8—24no—
111C8×Q8—212no—
112C8×Q8—212no—
113C8×Q8D418yescomplemented
114#435—212no—
115#435Q818yesdirect product factor
116#423—212no—
117C2×C4⋊C8—212no—
118C2×C4⋊C8C2313yes—
119C2×C4⋊C8C2313yes—
120C2×C4⋊C8C2313yes—
121C2×C4⋊C8C2313yes—
122C2×C4⋊C8C2316yes—
123C2×C4⋊C8C2316yes—
124C2×C4⋊C8C2316yes—
125C2×C4⋊C8C2313yes—
126C2×C4⋊C8C2311yescharacteristic
127C2×C22⋊C8C2313yes—
128C2×C22⋊C8C2313yes—
129C2×C22⋊C8C2311yescharacteristic
130C2×C22⋊C8C2311yescharacteristic
131C4×M16—212no—
132C2×#97C2313yes—
133C2×#97C2313yes—
134C2×#97C2313yes—
135C2×#97C2313yes—
136C2×#97C2313yes—
137C2×#97C2311yescharacteristic
138C2×C4×C8—212no—
139C2×C4×C8C2313yes—
140C2×C4×C8C2313yes—
141C2×C4×C8C2313yes—
142C2×C4×C8C2313yes—
143C2×C4×C8C2313yes—
144C2×C4×C8C2313yes—
145#391C2×C413yes—
146#391C2×C413yes—
147#391C2×C413yes—
148#391C2×C413yes—
149#389C2×C416yes—
150#389C2×C416yes—
151#389C2×C413yes—
152#389C2×C413yes—
153#387C2×C416yes—
154#387C2×C416yes—
155#385C2×C416yes—
156#385C2×C416yes—
157#384C2×C413yes—
158#380C2×C413yes—
159#380C2×C413yes—
160C4×C4⋊C4C2×C413yes—
161C4×C4⋊C4C2×C413yes—
162C4×C4⋊C4C2×C413yes—
163C4×C4⋊C4C2×C413yes—
164C4×C4⋊C4C2313yes—
165C4×C4⋊C4C2316yes—
166C4×C4⋊C4C2313yes—
167C4×C22⋊C4C2×C413yes—
168C4×C22⋊C4C2×C413yes—
169C4×C22⋊C4C2×C413yes—
170C4×C22⋊C4C2×C413yes—
171C4×C22⋊C4C2313yes—
172C4×C22⋊C4C2313yes—
173C43C2313yes—
174#336C2316yes—
175#336C2316yes—
176#336C2316yes—
177#336C2313yes—
178#336C2313yes—
179#336C2313yes—
180#336C2313yes—
181#336C2313yes—
182#336C2313yes—
183C22×Q8—24no—
184C22×Q8—24no—
185C22×Q8—24no—
186C22×Q8—24no—
187C22×Q8—24no—
188C22×Q8M1618yescomplemented
189C22×Q8C22×C413yes—
190C22×Q8C22×C413yes—
191C22×Q8C22×C411yescharacteristic
192C22×D4C22×C411yescharacteristic
193C23×C4C22×C413yes—
194C23×C4C22×C413yes—
195C23×C4C22×C413yes—
196C23×C4C22×C413yes—
197C23×C4C2411yescharacteristic
198C23×C4C2411yescharacteristic
199C2×M16—26no—
200C2×M16—22no—
201C2×M16—26no—
202C2×M16—22no—
203C2×M16—43no—
204C2×M16C2×Q814yescomplemented
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—26no—
208C22×C8—26no—
209C22×C8—43no—
210C22×C8C2×Q814yescomplemented
211C22×C8C2413yes—
212C22×C8C2413yes—
213C22×C8C2413yes—
214C22×C8C2413yes—
215C22×C8C2411yescharacteristic
216C22×C8C2411yescharacteristic
217C22×C8C2411yescharacteristic
218C22×C8C2411yescharacteristic
219C4⋊Q8—212no—
220C22⋊Q8—212no—
221C22⋊Q8—212no—
222C4×Q8—24no—
223C4×Q8—24no—
224C4×Q8—24no—
225C4×Q8—24no—
226C4×Q8—212no—
227C4×Q8—212no—
228C4×Q8—26no—
229C4×Q8—26no—
230C4×Q8—26no—
231C4×Q8—26no—
232C4×Q8C2×D414yes—
233C4×Q8#4114yes—
234C4×D4—212no—
235C4×D4—46no—
236C4×D4C2×Q814yes—
237C4×D4#4114yes—
238C2×C4⋊C4—212no—
239C2×C4⋊C4—212no—
240C2×C4⋊C4C22×C413yes—
241C2×C4⋊C4C22×C413yes—
242C2×C4⋊C4C22×C416yes—
243C2×C4⋊C4C22×C416yes—
244C2×C4⋊C4C22×C413yes—
245C2×C4⋊C4C22×C413yes—
246C2×C4⋊C4C22×C413yes—
247C2×C4⋊C4C22×C416yes—
248C2×C4⋊C4C22×C413yes—
249C2×C4⋊C4C22×C416yes—
250C2×C4⋊C4C22×C413yes—
251C2×C4⋊C4C22×C411yescharacteristic
252C2×C22⋊C4C22×C413yes—
253C2×C22⋊C4C22×C413yes—
254C2×C22⋊C4C22×C413yes—
255C2×C22⋊C4C22×C413yes—
256C2×C22⋊C4C22×C411yescharacteristic
257C2×C22⋊C4C22×C411yescharacteristic
258C2×C42—26no—
259C2×C42—26no—
260C2×C42—26no—
261C2×C42—26no—
262C2×C42C22×C413yes—
263C2×C42C22×C413yes—
264C2×C42C22×C413yes—
265C2×C42C22×C413yes—
266C2×C42C22×C413yes—
267C2×C42C22×C413yes—
268C2×C42C22×C413yes—
269C2×C42C22×C413yes—
270C2×C42C2413yes—
271C2×C42C2413yes—
272C2×C42C2413yes—
273C2×C42C2413yes—
274C2×C42C2411yescharacteristic
275C4⋊C8—212no—
276C4⋊C8—212no—
277C4⋊C8—26no—
278C4⋊C8—26no—
279C4⋊C8—26no—
280C4⋊C8—26no—
281C4⋊C8—46no—
282C4⋊C8C2×D4112yes—
283C4⋊C8C2×Q814yescomplemented
284C22⋊C8—26no—
285C22⋊C8—26no—
286C22⋊C8C2×Q814yescomplemented
287C22⋊C8C2×Q814yescomplemented
288#97—26no—
289#97—26no—
290#97—26no—
291#97—26no—
292#97—43no—
293#97C2×Q814yescomplemented
294C4×C8—26no—
295C4×C8—26no—
296C4×C8—26no—
297C4×C8—26no—
298C4×C8—26no—
299C4×C8C2×D4112yes—
300#95C22×C413yes—
301#95C22×C413yes—
302#95C22×C413yes—
303#95C22×C413yes—
304#95C22×C413yes—
305#95C22×C413yes—
306#95C22×C416yes—
307#95C22×C416yes—
308#95C22×C416yes—
309C24C23×C411yescharacteristic
310C24C23×C411yescharacteristic
311C2×Q8—232no—
312C2×Q8—26no—
313C2×Q8—26no—
314C2×Q8—26no—
315C2×Q8—26no—
316C2×Q8—26no—
317C2×Q8—416no—
318C2×Q8C4×D414yes—
319C2×Q8C2×M16112yes—
320C2×Q8C2×M1614yes—
321C2×Q8#13114yes—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4C4×Q814yes—
325C22×C4—26no—
326C22×C4—26no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—26no—
336C22×C4—26no—
337C22×C4—26no—
338C22×C4—26no—
339C22×C4—26no—
340C22×C4—26no—
341C22×C4—43no—
342C22×C4—43no—
343C22×C4—43no—
344C22×C4C2×M16112yes—
345C22×C4C23×C413yes—
346C22×C4C23×C413yes—
347C22×C4C23×C413yes—
348C22×C4C23×C413yes—
349C22×C4C23×C413yes—
350C22×C4C23×C413yes—
351C22×C4C23×C411yescharacteristic
352C22×C4C23×C411yescharacteristic
353C22×C4C23×C411yescharacteristic
354C22×C4C23×C411yescharacteristic
355C22×C4C23×C413yes—
356C22×C4C23×C413yes—
357C22×C4C22×Q812yes—
358C22×C4C22×Q812yes—
359C22×C4C2×#4112yes—
360C22×C4C2×#4112yes—
361C22×C4C2511yescharacteristic
362M16—28no—
363M16—412no—
364C2×C8—21no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—23no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—23no—
372C2×C8—28no—
373C2×C8—23no—
374C2×C8—23no—
375C2×C8—23no—
376C2×C8—23no—
377C2×C8—23no—
378C2×C8—23no—
379C2×C8—23no—
380C2×C8—23no—
381C2×C8—23no—
382C2×C8—412no—
383C2×C8C22×D416yes—
384C2×C8C22×D412yes—
385C2×C8C22×Q812yes—
386C2×C8C22×Q812yes—
387C2×C8C22×Q812yes—
388C2×C8C22×Q812yes—
389C2×C82−1+412yes—
390C4⋊C4—212no—
391C4⋊C4—26no—
392C4⋊C4—26no—
393C4⋊C4—212no—
394C4⋊C4—26no—
395C4⋊C4—26no—
396C4⋊C4—212no—
397C4⋊C4—26no—
398C4⋊C4—46no—
399C4⋊C4—43no—
400C4⋊C4C4×D4112yes—
401C4⋊C4C4×Q814yes—
402C22⋊C4—26no—
403C22⋊C4—26no—
404C22⋊C4—26no—
405C22⋊C4—43no—
406C22⋊C4C4×Q814yes—
407C22⋊C4C4×Q814yes—
408C42—26no—
409C42—26no—
410C42—26no—
411C42—26no—
412C42—26no—
413C42—26no—
414C42—23no—
415C42—23no—
416C42—23no—
417C42—23no—
418C42—23no—
419C42—23no—
420C42—43no—
421C42—43no—
422C42C22×D416yes—
423C42C22×Q812yes—
424C42C2×#4116yes—
425C42C2×#4112yes—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23C2×C4×Q812yes—
437C23C2×C4×Q812yes—
438C23C22×M1612yes—
439C23#56812yes—
440C23C24×C411yescharacteristic
441Q8—224no—
442Q8—412no—
443Q8#435116yesdirect product factor
444D4—28no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—23no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—23no—
453C2×C4—21no—
454C2×C4—23no—
455C2×C4—21no—
456C2×C4—23no—
457C2×C4—224no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—23no—
467C2×C4—23no—
468C2×C4—23no—
469C2×C4—23no—
470C2×C4—23no—
471C2×C4—23no—
472C2×C4—23no—
473C2×C4—23no—
474C2×C4—23no—
475C2×C4—23no—
476C2×C4—23no—
477C2×C4—23no—
478C2×C4—23no—
479C2×C4—23no—
480C2×C4—23no—
481C2×C4—412no—
482C2×C4—412no—
483C2×C4—86no—
484C2×C4C2×C4×D416yes—
485C2×C4C2×C4×D416yes—
486C2×C4C2×C4×Q812yes—
487C2×C4C2×C4×Q812yes—
488C2×C4C2×C4×Q812yes—
489C2×C4C2×C4×Q812yes—
490C2×C4C22×M1616yes—
491C2×C4C22×M1616yes—
492C2×C4C2×#13116yes—
493C2×C4C23×D411yescharacteristic
494C2×C4C23×Q811yescharacteristic
495C2×C4C22×#4111yescharacteristic
496C2×C4C22×#4111yescharacteristic
497C2×C4C2×2−1+411yescharacteristic
498C2×C4#58511yescharacteristic
499C8—22no—
500C8—22no—
501C8—26no—
502C8—22no—
503C8—43no—
504C8—43no—
505C8—43no—
506C8D4×Q814yescomplemented
507C22—22no—
508C22—22no—
509C22—22no—
510C22—22no—
511C22—22no—
512C22—22no—
513C22—22no—
514C22—22no—
515C22—22no—
516C22—41no—
517C22—41no—
518C22—41no—
519C22—41no—
520C22Q8×M1614yescomplemented
521C22C22×C4×D411yescharacteristic
522C22C22×C4×Q811yescharacteristic
523C22C4×2−1+411yescharacteristic
524C22C23×M1611yescharacteristic
525C22C22×#13111yescharacteristic
526C22C2×#56811yescharacteristic
527C22#357311yescharacteristic
528C4—22no—
529C4—22no—
530C4—22no—
531C4—22no—
532C4—26no—
533C4—26no—
534C4—43no—
535C4—43no—
536C4—43no—
537C4—43no—
538C4—43no—
539C4C2×#435112yes—
540C4C2×D4×Q811yescharacteristic
541C4D4×#4111yescharacteristic
542C4Q8×#4111yescharacteristic
543C4#348211yescharacteristic
544C2—24no—
545C2—42no—
546C2C4×D4×Q811yescharacteristic
547C2C22×#43511yescharacteristic
548C2#6014611yescharacteristic
549C2C2×Q8×M1611yescharacteristic
550C2Q8×#13111yescharacteristic
551C2#6033311yescharacteristic
552C2#6033511yescharacteristic
subgroupG/H
center, Frattini, cosocleC22×C4C25
commutator, socleC23C24×C4
prime radicalC22×D4C22×C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×Q8×M16C2semidirect
2. C22×C8×Q8C2semidirect
4. #23150C2semidirect
5. #22956C2semidirect
6. #22953C2semidirect
7. Q8×C4⋊C8C2semidirect
9. Q8×C22⋊C8C2semidirect
10. Q8×C22⋊C8C2semidirect
11. #21714C2semidirect
12. #21597C2semidirect
13. #21596C2semidirect
14. Q8×#97C2semidirect
113. C8×Q8D4semidirect
115. #435Q8direct (N × H)
188. C22×Q8M16semidirect
204. C2×M16C2×Q8semidirect
210. C22×C8C2×Q8semidirect
283. C4⋊C8C2×Q8semidirect
286. C22⋊C8C2×Q8semidirect
287. C22⋊C8C2×Q8semidirect
293. #97C2×Q8semidirect
443. Q8#435direct (N × H)
506. C8D4×Q8semidirect
520. C22Q8×M16semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×M162
C22×C8×Q82
C4×D4×Q82
#231502
#229562
#229532
Q8×C4⋊C82
#226482
Q8×C22⋊C82
#217142
#215972
#215962
Q8×#972
C4×#4352