Finite Group Databasebeta

← #6399898

Subgroups — #6399898

Order 512.

1619 conjugacy classes of subgroups.

The subgroup lattice has 579 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#135C211yesmaximal, characteristic
2#60253C211yesmaximal, characteristic, complemented
3Q8×#131C211yesmaximal, characteristic, complemented
4#23170C213yesmaximal, complemented
5#23080C213yesmaximal, complemented
6#22982C213yesmaximal, complemented
7#22783C213yesmaximal
8#22746C211yesmaximal, characteristic, complemented
9Q8×#104C211yesmaximal, characteristic, complemented
10Q8×#104C211yesmaximal, characteristic, complemented
11#22256C213yesmaximal, complemented
12#22130C213yesmaximal, complemented
13#21627C213yesmaximal, complemented
14#21137C213yesmaximal
15#20569C211yesmaximal, characteristic, complemented
16Q8×#41C2211yescharacteristic
17Q8×#41C2211yescharacteristic
18#3383C413yescomplemented
19Q8×Q16C411yescharacteristic, complemented
20#3381C413yescomplemented
21#3381C413yescomplemented
22#3380C413yescomplemented
23Q8×SD16C411yescharacteristic, complemented
24Q8×SD16C411yescharacteristic, complemented
25Q8×D8C411yescharacteristic, complemented
26#3160C2213yes—
27#3076C2213yes—
28#3076C2213yes—
29#2986—23no—
30#2985—23no—
31#2984—21no—
32#2982—21no—
33Q8×M16—21no—
34Q8×M16C2211yescharacteristic
35Q8×M16C2211yescharacteristic
36#2962—23no—
37#2962C2213yes—
38#2962C2213yes—
39#2960—23no—
40#2960C2213yes—
41#2960C2213yes—
42C2×C8×Q8—21no—
43C2×C8×Q8C2211yescharacteristic
44C2×C8×Q8C2211yescharacteristic
45C4×#135C2213yes—
46#2925C2213yes—
47#2922C2213yes—
48#2906C2213yes—
49C4×#131C2213yes—
50C42×Q8C2211yescharacteristic
51#1994C2213yes—
52#1994C2213yes—
53#1986C2213yes—
54#1986C2213yes—
55#1950C2213yes—
56#1950C2213yes—
57#1945C2216yes—
58#1937C2216yes—
59#1929C2213yes—
60#1929C2213yes—
61#1922C2216yes—
62#1922C2216yes—
63#1922C2213yes—
64#1922C2216yes—
65#1922C2213yes—
66#1876C2211yescharacteristic
67#1876C2213yes—
68#1866C2213yes—
69#1866C2213yes—
70#1837C2213yes—
71#1806C2213yes—
72#1806C2213yes—
73C4×#108C2213yes—
74#1772C2216yes—
75C8×C4⋊C4C2213yes—
76C8×C4⋊C4C2213yes—
77C4×C4⋊C8C2213yes—
78#1764C2216yes—
79C4×#104C2213yes—
80C4×#104C2213yes—
81#1744C2213yes—
82C2×#135C2311yescharacteristic
83C2×#131C2311yescharacteristic
84C2×#131C2313yes—
85Q82C2×C411yescharacteristic
86Q82C2×C411yescharacteristic
87#557C2×C413yes—
88#557C2×C413yes—
89#554—23no—
90#554—23no—
91#554C2×C413yes—
92#554C2×C413yes—
93D4×Q8—21no—
94D4×Q8—21no—
95D4×Q8C2×C411yescharacteristic
96D4×Q8C2×C411yescharacteristic
97#533—23no—
98#533—23no—
99#533C2313yes—
100#533C2313yes—
101C4×#41C2313yes—
102C4×#41C2313yes—
103C2×C4×Q8—21no—
104C2×C4×Q8—21no—
105C2×C4×Q8—21no—
106C2×C4×Q8C2311yescharacteristic
107#500C2×C413yes—
108#500C2×C413yes—
109#499C2×C416yes—
110#499C2×C416yes—
111#498C2×C413yes—
112#498C2×C413yes—
113#479C2×C416yes—
114#479C2×C416yes—
115#478C2×C416yes—
116#478C2×C416yes—
117#477C2×C413yes—
118#477C2×C413yes—
119#476C2×C413yes—
120#476C2×C413yes—
121#475C2×C413yes—
122#475C2×C413yes—
123#474C2×C413yes—
124#474C2×C413yes—
125#446—23no—
126#446—23no—
127#446—26no—
128#446—23no—
129#446—23no—
130#446—26no—
131#446—23no—
132#446—23no—
133#446—23no—
134#446—23no—
135C8×Q8—21no—
136C8×Q8—21no—
137C8×Q8—23no—
138C8×Q8—23no—
139C8×Q8—23no—
140C8×Q8—23no—
141C8×Q8—23no—
142C8×Q8—23no—
143C8×Q8C2×C411yescharacteristic
144C8×Q8C2×C411yescharacteristic
145C8×Q8C2×C413yes—
146C8×Q8C2×C413yes—
147C8×Q8D411yescharacteristic
148C8×Q8D413yes—
149C8×Q8D413yes—
150C8×Q8D411yescharacteristic
151C4×Q16C2×C413yes—
152C4×Q16C2×C413yes—
153C4×SD16C2×C413yes—
154C4×SD16C2×C413yes—
155C4×SD16C2×C413yes—
156C4×SD16C2×C413yes—
157C4×D8C2×C413yes—
158C4×D8C2×C413yes—
159#433—26no—
160#433—23no—
161#433—23no—
162#432—23no—
163#431—23no—
164#431—23no—
165#431—23no—
166C2×#108C2313yes—
167C2×#108C2313yes—
168#427C2316yes—
169#427C2313yes—
170#424—26no—
171#424C2313yes—
172#424C2313yes—
173#424C2316yes—
174#423—23no—
175#423C2313yes—
176#423C2313yes—
177C2×C4⋊C8—23no—
178C2×C4⋊C8C2313yes—
179C2×C4⋊C8C2313yes—
180C2×#104C2313yes—
181C2×#104C2311yescharacteristic
182C2×#104C2311yescharacteristic
183C2×#104C2313yes—
184#416C2313yes—
185#416C2313yes—
186#408C2311yescharacteristic
187#408C2313yes—
188#405—23no—
189#405—23no—
190#405C2313yes—
191#405C2313yes—
192#405C2316yes—
193C4×M16—23no—
194C4×M16C2313yes—
195C4×M16C2313yes—
196C2×C4×C8—23no—
197C2×C4×C8C2316yes—
198C2×C4×C8C2313yes—
199C2×C4×C8C2313yes—
200C4×C4⋊C4—23no—
201C4×C4⋊C4C2313yes—
202C43C2313yes—
203#341C2313yes—
204#341C2316yes—
205#341C2311yescharacteristic
206#339C2316yes—
207#339C2316yes—
208#339C2313yes—
209#339C2313yes—
210#339C2313yes—
211#339C2313yes—
212#336C2313yes—
213#336C2313yes—
214#336C2311yescharacteristic
215#336C2313yes—
216C2×#41C2411yescharacteristic
217C2×#41C2411yescharacteristic
218C22×Q8—21no—
219C22×Q8—21no—
220C22×Q8#4111yescharacteristic
221#135—22no—
222#135—22no—
223C2×Q16C22×C411yescharacteristic
224C2×SD16C22×C411yescharacteristic
225C2×SD16C22×C411yescharacteristic
226C2×D8C22×C411yescharacteristic
227#131—22no—
228#131—22no—
229#131—26no—
230#131—43no—
231C2×M16—21no—
232C2×M16—23no—
233C2×M16C2413yes—
234C2×M16C2413yes—
235C2×M16C2411yescharacteristic
236C2×M16C2411yescharacteristic
237C22×C8—21no—
238C22×C8—23no—
239C22×C8C2411yescharacteristic
240C22×C8C2413yes—
241C22×C8C2411yescharacteristic
242C22×C8C2413yes—
243C4⋊Q8—23no—
244C4⋊Q8—23no—
245C4⋊Q8—23no—
246C4⋊Q8—23no—
247C4⋊Q8C22×C413yes—
248C4⋊Q8C22×C413yes—
249C4⋊Q8C22×C413yes—
250C4⋊Q8C22×C413yes—
251#125—26no—
252#125—26no—
253#125C22×C416yes—
254#125C22×C416yes—
255C22⋊Q8—23no—
256C22⋊Q8—23no—
257C22⋊Q8—23no—
258C22⋊Q8—23no—
259C22⋊Q8—23no—
260C22⋊Q8—23no—
261C22⋊Q8—23no—
262C22⋊Q8—23no—
263C22⋊Q8—26no—
264C22⋊Q8—26no—
265C4×Q8—28no—
266C4×Q8—21no—
267C4×Q8—21no—
268C4×Q8—23no—
269C4×Q8—23no—
270C4×Q8—23no—
271C4×Q8—23no—
272C4×Q8—23no—
273C4×Q8—23no—
274C4×Q8C22×C413yes—
275C4×Q8C22×C413yes—
276C4×Q8C22×C413yes—
277C4×Q8C22×C413yes—
278C4×Q8C22×C411yescharacteristic
279C4×Q8C22×C413yes—
280C4×Q8C2×D411yescharacteristic
281C4×Q8C2×D413yes—
282C4×D4—23no—
283C4×D4—23no—
284C4×D4—23no—
285C4×D4—23no—
286C4×D4C22×C413yes—
287C4×D4C22×C413yes—
288C4×D4C22×C413yes—
289C4×D4C22×C413yes—
290#117—23no—
291#117—23no—
292#117C2413yes—
293#117C2413yes—
294C2×C4⋊C4—23no—
295C2×C4⋊C4—23no—
296C2×C4⋊C4—23no—
297C2×C4⋊C4#4113yes—
298C2×C42—23no—
299C2×C42—23no—
300C2×C42—23no—
301C2×C42—23no—
302C2×C42C2411yescharacteristic
303C2×C42C2413yes—
304C2×C42C2413yes—
305#108—26no—
306#108—43no—
307#107C22×C413yes—
308#107C22×C413yes—
309#107C22×C416yes—
310#107C22×C416yes—
311#106C22×C413yes—
312#106C22×C413yes—
313#106C22×C416yes—
314#106C22×C416yes—
315C4⋊C8—26no—
316C4⋊C8—26no—
317C4⋊C8—23no—
318C4⋊C8—23no—
319C4⋊C8—23no—
320C4⋊C8—23no—
321C4⋊C8—23no—
322C4⋊C8—23no—
323C4⋊C8—23no—
324C4⋊C8—23no—
325C4⋊C8C22×C413yes—
326C4⋊C8C22×C413yes—
327C4⋊C8C22×C416yes—
328C4⋊C8C2×D413yes—
329C4⋊C8C2×D413yes—
330C4⋊C8C2×D416yes—
331#104—26no—
332#104—22no—
333#104—22no—
334#104—43no—
335#103C22×C413yes—
336#103C22×C413yes—
337#103C22×C413yes—
338#103C22×C413yes—
339#102C22×C413yes—
340#102C22×C413yes—
341#102C22×C413yes—
342#102C22×C413yes—
343C22⋊C8—21no—
344C22⋊C8—23no—
345C22⋊C8—21no—
346C22⋊C8—23no—
347#97—23no—
348#97—23no—
349#97—26no—
350#97—23no—
351#97—23no—
352C4×C8—23no—
353C4×C8—23no—
354C4×C8—26no—
355C4×C8—23no—
356C4×C8—23no—
357C4×C8—46no—
358C4×C8C22×C413yes—
359C4×C8C22×C413yes—
360C4×C8C22×C413yes—
361C4×C8C22×C413yes—
362C4×C8C2×D413yes—
363C4×C8C2×D413yes—
364C4×C8C2×D413yes—
365C4×C8C2×D413yes—
366#95—23no—
367#41—21no—
368#41—21no—
369#41—21no—
370#41—21no—
371#41—21no—
372#41—21no—
373#412−1+412yes—
374#412−1+412yes—
375C2×Q8—24no—
376C2×Q8—23no—
377C2×Q8—23no—
378C2×Q8—23no—
379C2×Q8—23no—
380C2×Q8—44no—
381C2×Q8—44no—
382C2×Q8C4×D413yes—
383C2×Q8C4×D411yescharacteristic
384C2×Q8C23×C411yescharacteristic
385C2×Q8C23×C411yescharacteristic
386C2×D4—21no—
387C2×D4—21no—
388C2×D4C23×C411yescharacteristic
389C2×D4C23×C411yescharacteristic
390C22×C4—21no—
391C22×C4—23no—
392C22×C4—21no—
393C22×C4—23no—
394C22×C4—23no—
395C22×C4—21no—
396C22×C4—23no—
397C22×C4—23no—
398C22×C4C2×#4113yes—
399C22×C4C2×#4113yes—
400C22×C4C2511yescharacteristic
401Q16—22no—
402SD16—22no—
403SD16—22no—
404D8—22no—
405M16—24no—
406M16—21no—
407M16—23no—
408M16—21no—
409M16—23no—
410M16—23no—
411M16—23no—
412M16—21no—
413M16—46no—
414M162−1+412yes—
415C2×C8—24no—
416C2×C8—21no—
417C2×C8—23no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—23no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—23no—
424C2×C8—23no—
425C2×C8—23no—
426C2×C8—21no—
427C2×C8—23no—
428C2×C8—46no—
429C2×C8C23×C411yescharacteristic
430C2×C8C23×C411yescharacteristic
431C2×C8C23×C413yes—
432C2×C8C23×C413yes—
433C2×C8C22×D411yescharacteristic
434C2×C8C22×D413yes—
435C2×C8C22×D411yescharacteristic
436C2×C8C22×D413yes—
437C4⋊C4—212no—
438C4⋊C4—26no—
439C4⋊C4—26no—
440C4⋊C4—23no—
441C4⋊C4—23no—
442C4⋊C4—23no—
443C4⋊C4—23no—
444C4⋊C4—23no—
445C4⋊C4—23no—
446C4⋊C4—23no—
447C4⋊C4—23no—
448C4⋊C4—26no—
449C4⋊C4—26no—
450C4⋊C4C4×D413yes—
451C4⋊C4C4×D416yes—
452C4⋊C4C4×D413yes—
453C4⋊C4C23×C413yes—
454C4⋊C4C23×C413yes—
455C4⋊C4C23×C413yes—
456C4⋊C4C23×C413yes—
457C22⋊C4—23no—
458C22⋊C4—23no—
459C22⋊C4—23no—
460C22⋊C4—23no—
461C22⋊C4—23no—
462C22⋊C4—23no—
463C42—212no—
464C42—23no—
465C42—23no—
466C42—23no—
467C42—23no—
468C42—23no—
469C42—23no—
470C42—23no—
471C42—21no—
472C42—21no—
473C42—23no—
474C42—83no—
475C42C23×C413yes—
476C42C23×C413yes—
477C42C22×D413yes—
478C42C22×D413yes—
479C23—21no—
480C23—21no—
481C23C22×#4111yescharacteristic
482Q8—21no—
483Q8—26no—
484Q8—21no—
485Q8—21no—
486Q8—46no—
487Q8—46no—
488Q8—83no—
489Q8—83no—
490Q8#44314yes—
491Q8#51912yes—
492D4—22no—
493D4—22no—
494D4—21no—
495D4—21no—
496D4—21no—
497D4—41no—
498D4—41no—
499D4#51912yes—
500C2×C4—21no—
501C2×C4—26no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—26no—
506C2×C4—21no—
507C2×C4—23no—
508C2×C4—23no—
509C2×C4—23no—
510C2×C4—23no—
511C2×C4—43no—
512C2×C4—41no—
513C2×C4—43no—
514C2×C4—43no—
515C2×C4—41no—
516C2×C4—46no—
517C2×C4—46no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—83no—
521C2×C4—83no—
522C2×C4C2×C4×D413yes—
523C2×C4C2×C4×D413yes—
524C2×C4C2×C4×D413yes—
525C2×C4C2×C4×D413yes—
526C2×C4C24×C411yescharacteristic
527C2×C4C23×D411yescharacteristic
528C2×C4C2×2−1+411yescharacteristic
529C2×C4C2×2−1+411yescharacteristic
530C2×C4C2×2−1+411yescharacteristic
531C2×C4#58511yescharacteristic
532C8—21no—
533C8—21no—
534C8—23no—
535C8—23no—
536C8—21no—
537C8—22no—
538C8—21no—
539C8—23no—
540C8—23no—
541C8—43no—
542C8—43no—
543C8—41no—
544C22—21no—
545C22—21no—
546C22—22no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22C22×C4×D411yescharacteristic
553C22#351611yescharacteristic
554C22#352311yescharacteristic
555C4—23no—
556C4—23no—
557C4—23no—
558C4—22no—
559C4—46no—
560C4—42no—
561C4—43no—
562C4—43no—
563C4—43no—
564C4—43no—
565C4—41no—
566C4C2×#44316yes—
567C4C2×#51911yescharacteristic
568C4C4×2−1+411yescharacteristic
569C4#348611yescharacteristic
570C4#348911yescharacteristic
571C2—21no—
572C2—21no—
573C2—42no—
574C2—81no—
575C2#3429111yescharacteristic
576C2C22×#44311yescharacteristic
577C2#6018411yescharacteristic
subgroupG/H
centerC2×C4C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
socleC22C22×C4×D4
prime radicalC2×#135C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #60253C2semidirect
3. Q8×#131C2semidirect
4. #23170C2semidirect
5. #23080C2semidirect
6. #22982C2semidirect
8. #22746C2semidirect
9. Q8×#104C2semidirect
10. Q8×#104C2semidirect
11. #22256C2semidirect
12. #22130C2semidirect
13. #21627C2semidirect
15. #20569C2semidirect
18. #3383C4semidirect
19. Q8×Q16C4semidirect
20. #3381C4semidirect
21. #3381C4semidirect
22. #3380C4semidirect
23. Q8×SD16C4semidirect
24. Q8×SD16C4semidirect
25. Q8×D8C4semidirect

Maximal subgroups

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

subgroupindex
Q8×#1352
#602532
Q8×#1312
#231702
#230802
#229822
#227832
#227462
Q8×#1042
#222562
#221302
#216272
#211372
#205692