Finite Group Databasebeta

← #6345978

Subgroups — #6345978

Order 512.

1573 conjugacy classes of subgroups.

The subgroup lattice has 543 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#61434C211yesmaximal, characteristic
2#60345C211yesmaximal, characteristic, complemented
3#60328C211yesmaximal, characteristic, complemented
4#18089C213yesmaximal, complemented
5#18061C213yesmaximal, complemented
6#18025C213yesmaximal, complemented
7#18021C213yesmaximal, complemented
8#17960C213yesmaximal, complemented
9#17941C213yesmaximal, complemented
10#17940C213yesmaximal, complemented
11#17890C213yesmaximal, complemented
12#17837C211yesmaximal, characteristic, complemented
13#17520C211yesmaximal, characteristic, complemented
14#17518C211yesmaximal, characteristic, complemented
15#17320C211yesmaximal, characteristic, complemented
16#3483C2211yescharacteristic
17#3482C2211yescharacteristic
18#3392C411yescharacteristic
19Q8×SD16C411yescharacteristic
20#3363C411yescharacteristic
21#3358C411yescharacteristic
22#3340C413yes—
23#3304C413yes—
24#3303C413yes—
25#3284C413yes—
26#3167C2213yes—
27#3107C2213yes—
28#3106C2213yes—
29#2993—23no—
30#2991C2213yes—
31#2991C2211yescharacteristic
32#2991C2211yescharacteristic
33#2990C2213yes—
34#2988—21no—
35#2986C2213yes—
36#2982—21no—
37#2981—23no—
38#2979C2213yes—
39#2971—21no—
40#2969—23no—
41#2965C2213yes—
42#2965C2213yes—
43C8×#41C2211yescharacteristic
44C8×#41C2211yescharacteristic
45Q8×M16—21no—
46#2960C2213yes—
47C4×#135C2213yes—
48#2935—23no—
49#2931C2213yes—
50C4×#131C2211yescharacteristic
51#1594C2213yes—
52#1594C2213yes—
53#1593C2213yes—
54#1593C2213yes—
55#1592C2216yes—
56#1592C2213yes—
57#1592C2213yes—
58#1592C2213yes—
59#1592C2211yescharacteristic
60#1584C2216yes—
61#1583C2216yes—
62#1582C2216yes—
63#1581C2216yes—
64SD16⋊C8C2216yes—
65SD16⋊C8C2216yes—
66#1574C2213yes—
67#1574C2213yes—
68#1567C2213yes—
69#1564C2213yes—
70#1493C2213yes—
71#1492C2213yes—
72#1489C2213yes—
73#1489C2213yes—
74#1485C2213yes—
75#1484C2213yes—
76#1476C2213yes—
77#1476C2213yes—
78#1452C2213yes—
79#1451C2213yes—
80#1451C2213yes—
81C2×#322C2213yes—
82C2×#135C2×C413yes—
83C2×#131—21no—
84C2×#131—23no—
85Q82C2×C411yescharacteristic
86#557C2×C411yescharacteristic
87#556C2×C413yes—
88#555—23no—
89#555C2×C413yes—
90#552—23no—
91#550—21no—
92#550C2×C413yes—
93#550C2×C411yescharacteristic
94D4×Q8C2×C411yescharacteristic
95#548—21no—
96#547C2×C413yes—
97#533C2313yes—
98#532—23no—
99#532C2313yes—
100#529—23no—
101C4×#41—21no—
102C4×#41C2313yes—
103C4×#41C2313yes—
104C4×#41C2311yescharacteristic
105C2×C4×Q8—21no—
106#498C2×C413yes—
107#492C2×C413yes—
108#487C2×C413yes—
109#486C2×C413yes—
110#485C2×C416yes—
111#482C2×C416yes—
112#477C2×C413yes—
113#476C2×C413yes—
114#475C2×C413yes—
115#474C2×C413yes—
116#465C2×C416yes—
117#465C2×C416yes—
118#462C2×C413yes—
119#461C2×C413yes—
120#460C2×C413yes—
121#459C2×C413yes—
122#452C2×C416yes—
123#449C2×C416yes—
124#446—23no—
125#446—23no—
126#446C2313yes—
127#446C2313yes—
128#446C2313yes—
129#446C2311yescharacteristic
130C8×Q8—21no—
131C8×Q8—21no—
132C8×Q8—23no—
133C8×Q8—23no—
134C8×Q8C2×C411yescharacteristic
135C8×Q8C2×C411yescharacteristic
136C8×Q8D411yescharacteristic
137C8×Q8D411yescharacteristic
138C8×Q8C2316yes—
139C4×Q16C2×C413yes—
140C4×Q16C2313yes—
141C4×SD16C2×C413yes—
142C4×SD16C2×C413yes—
143C4×SD16C2316yes—
144C4×SD16C2313yes—
145C4×SD16C2311yescharacteristic
146C4×D8C2×C413yes—
147C4×D8C2313yes—
148#436—23no—
149#436—23no—
150#436—23no—
151#436—26no—
152#436—23no—
153#436C2313yes—
154#436C2313yes—
155#436C2313yes—
156#436C2311yescharacteristic
157#435—23no—
158#435—23no—
159#435—23no—
160#435—23no—
161#435—26no—
162C8×D4—23no—
163C8×D4—23no—
164C8×D4C2×C413yes—
165C8×D4C2×C413yes—
166C8×D4D413yes—
167C8×D4D413yes—
168C8×D4C2316yes—
169#433—26no—
170#432—23no—
171#431—23no—
172#431—23no—
173#431—23no—
174#431C2313yes—
175#431C2313yes—
176#431C2313yes—
177#431C2313yes—
178#427C2×C413yes—
179#424—23no—
180#423—23no—
181#423C2313yes—
182#423C2313yes—
183C2×C4⋊C8—23no—
184#416C2×C413yes—
185#416C2×C413yes—
186#408—23no—
187#408—26no—
188#405—23no—
189#405—23no—
190C4×M16—23no—
191C4×M16—23no—
192C4×M16C2313yes—
193C2×C4×C8C2313yes—
194C2×C4×C8C2313yes—
195C2×C4×C8C2313yes—
196#335C2311yescharacteristic
197#335C2313yes—
198#335C2316yes—
199#334C2313yes—
200#334C2313yes—
201Q8⋊C8C2313yes—
202Q8⋊C8C2316yes—
203Q8⋊C8C2313yes—
204Q8⋊C8C2313yes—
205Q8⋊C8C2311yescharacteristic
206D4⋊C8C2313yes—
207D4⋊C8C2316yes—
208D4⋊C8C2313yes—
209D4⋊C8C2313yes—
210D4⋊C8C2311yescharacteristic
211#322C2313yes—
212#322C2313yes—
213#322C2316yes—
214#322C2311yescharacteristic
215#322C2313yes—
216C2×#41—21no—
217C2×#41—23no—
218C2×#41C22×C413yes—
219C2×#41C22×C413yes—
220C2×#41#4111yescharacteristic
221C22×Q8—21no—
222#135—46no—
223C2×Q16C22×C413yes—
224C2×SD16C22×C411yescharacteristic
225C2×SD16C22×C413yes—
226C2×SD16C22×C416yes—
227C2×D8C22×C413yes—
228#131—44no—
229#131—86no—
230C2×M16—23no—
231C2×M16—23no—
232C2×M16—23no—
233C2×M16—23no—
234C2×M16—21no—
235C2×M16—23no—
236C2×M16—23no—
237C22×C8—23no—
238C22×C8—26no—
239C22×C8C22×C413yes—
240C22×C8C22×C413yes—
241C22×C8C2×D413yes—
242C22×C8C2×D413yes—
243C4⋊Q8—23no—
244C4⋊Q8C22×C413yes—
245C4⋊Q8C22×C413yes—
246C4⋊Q8C22×C413yes—
247#127—23no—
248#127C22×C413yes—
249#126—26no—
250#125—23no—
251#125—23no—
252#124—23no—
253#124—26no—
254#124—23no—
255#123—26no—
256C22⋊Q8—23no—
257C22⋊Q8—23no—
258C22⋊Q8C22×C416yes—
259#121—23no—
260#121—23no—
261#121—23no—
262#121—23no—
263#121—26no—
264#121C22×C416yes—
265C4×Q8—21no—
266C4×Q8—21no—
267C4×Q8—23no—
268C4×Q8—23no—
269C4×Q8C22×C413yes—
270C4×Q8C22×C413yes—
271C4×Q8C22×C411yescharacteristic
272C4×Q8C2×D411yescharacteristic
273C4×Q8C2411yescharacteristic
274C4×Q8C2413yes—
275C4×D4—23no—
276C4×D4—23no—
277C4×D4—23no—
278C4×D4—23no—
279C4×D4—23no—
280C4×D4—23no—
281C4×D4C22×C413yes—
282C4×D4C22×C413yes—
283C4×D4C22×C413yes—
284C4×D4C2×D413yes—
285C4×D4C2411yescharacteristic
286C4×D4C2413yes—
287#117—23no—
288#117—23no—
289#117—23no—
290#117—23no—
291#117C22×C413yes—
292#117C22×C413yes—
293#117#4113yes—
294C2×C4⋊C4—23no—
295C2×C42—23no—
296C2×C42—23no—
297C2×C42C2413yes—
298#107C22×C413yes—
299#107C22×C413yes—
300#106C22×C411yescharacteristic
301#106C22×C413yes—
302#106C22×C416yes—
303C4⋊C8—23no—
304C4⋊C8—26no—
305C4⋊C8—23no—
306C4⋊C8—21no—
307C4⋊C8—23no—
308C4⋊C8—23no—
309C4⋊C8—23no—
310C4⋊C8C22×C413yes—
311C4⋊C8C22×C413yes—
312C4⋊C8C2×D413yes—
313C4⋊C8C2×D413yes—
314C4⋊C8C2411yescharacteristic
315C4⋊C8C2413yes—
316C4⋊C8C2411yescharacteristic
317C4⋊C8C2413yes—
318#103C22×C411yescharacteristic
319#103C22×C413yes—
320#103C22×C413yes—
321#103C22×C413yes—
322#103C22×C416yes—
323#102C22×C411yescharacteristic
324#102C22×C413yes—
325#102C22×C413yes—
326#102C22×C413yes—
327#102C22×C416yes—
328C22⋊C8—23no—
329C22⋊C8—23no—
330C22⋊C8—23no—
331C22⋊C8—23no—
332C22⋊C8—26no—
333C22⋊C8—21no—
334C22⋊C8—23no—
335C22⋊C8C22×C416yes—
336C22⋊C8C2×D416yes—
337#97—21no—
338#97—23no—
339#97—23no—
340#97—23no—
341#97—23no—
342C4×C8—26no—
343C4×C8—23no—
344C4×C8—23no—
345C4×C8—23no—
346C4×C8C22×C413yes—
347C4×C8C22×C413yes—
348C4×C8C2×D413yes—
349C4×C8C2×D413yes—
350C4×C8C2411yescharacteristic
351C4×C8C2413yes—
352C4×C8C2413yes—
353C4×C8C2411yescharacteristic
354C4×C8C2411yescharacteristic
355C4×C8C2413yes—
356#41—24no—
357#41—43no—
358#41—43no—
359#41—43no—
360#41—43no—
361#41—44no—
362#41—83no—
363#41—83no—
364C2×Q8—23no—
365C2×Q8—23no—
366C2×Q8—44no—
367C2×Q8C4×D411yescharacteristic
368C2×Q8C23×C411yescharacteristic
369C2×Q8C23×C413yes—
370C2×D4—23no—
371C2×D4—23no—
372C2×D4—26no—
373C2×D4—23no—
374C2×D4—23no—
375C2×D4C4×D413yes—
376C2×D4C23×C411yescharacteristic
377C2×D4C23×C413yes—
378C22×C4—23no—
379C22×C4—23no—
380C22×C4—21no—
381C22×C4—23no—
382C22×C4—23no—
383C22×C4—23no—
384C22×C4C23×C413yes—
385C22×C4C22×D413yes—
386C22×C4C2×#4113yes—
387Q16—43no—
388SD16—26no—
389SD16—22no—
390SD16—46no—
391D8—43no—
392M16—42no—
393M16—46no—
394M16—46no—
395M16—83no—
396M16—83no—
397M16—83no—
398M16—83no—
399C2×C8—26no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—23no—
403C2×C8—23no—
404C2×C8—21no—
405C2×C8—23no—
406C2×C8—43no—
407C2×C8—43no—
408C2×C8—43no—
409C2×C8—46no—
410C2×C8—86no—
411C2×C8C23×C411yescharacteristic
412C2×C8C23×C413yes—
413C2×C8C23×C413yes—
414C2×C8C23×C411yescharacteristic
415C2×C8C22×D413yes—
416C2×C8C22×D411yescharacteristic
417C2×C8C22×D411yescharacteristic
418C2×C8C22×D413yes—
419C4⋊C4—23no—
420C4⋊C4—26no—
421C4⋊C4—23no—
422C4⋊C4—23no—
423C4⋊C4—23no—
424C4⋊C4—23no—
425C4⋊C4—23no—
426C4⋊C4—21no—
427C4⋊C4—23no—
428C4⋊C4C4×D413yes—
429C4⋊C4C4×D413yes—
430C4⋊C4C23×C413yes—
431C4⋊C4C23×C411yescharacteristic
432C4⋊C4C23×C413yes—
433C4⋊C4C23×C411yescharacteristic
434C22⋊C4—26no—
435C22⋊C4—23no—
436C22⋊C4—23no—
437C22⋊C4—26no—
438C22⋊C4—23no—
439C22⋊C4—23no—
440C22⋊C4—21no—
441C22⋊C4—23no—
442C22⋊C4C4×D416yes—
443C42—23no—
444C42—23no—
445C42—23no—
446C42—23no—
447C42—23no—
448C42—21no—
449C42C23×C413yes—
450C42C22×D413yes—
451C42C2×#4113yes—
452C42C2511yescharacteristic
453C23—21no—
454C23—23no—
455C23C2×C4×D413yes—
456Q8—21no—
457Q8—23no—
458Q8—23no—
459Q8—21no—
460Q8—46no—
461Q8—83no—
462Q8#44414yes—
463D4—26no—
464D4—21no—
465D4—23no—
466D4—23no—
467D4—21no—
468D4—46no—
469D4—83no—
470D4—83no—
471D4—83no—
472D4—86no—
473C2×C4—23no—
474C2×C4—23no—
475C2×C4—26no—
476C2×C4—23no—
477C2×C4—23no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—23no—
482C2×C4—23no—
483C2×C4—23no—
484C2×C4—43no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—43no—
488C2×C4—46no—
489C2×C4—46no—
490C2×C4—83no—
491C2×C4—83no—
492C2×C4C2×C4×D413yes—
493C2×C4C2×C4×D413yes—
494C2×C4C2×C4×D413yes—
495C2×C4C24×C411yescharacteristic
496C2×C4C23×D411yescharacteristic
497C2×C4C22×#4111yescharacteristic
498C8—21no—
499C8—23no—
500C8—21no—
501C8—23no—
502C8—23no—
503C8—21no—
504C8—23no—
505C8—41no—
506C8—43no—
507C8—43no—
508C8—43no—
509C8—41no—
510C8—41no—
511C8#56812yes—
512C22—23no—
513C22—23no—
514C22—23no—
515C22—43no—
516C22—41no—
517C22—41no—
518C22—43no—
519C22—41no—
520C22—43no—
521C22C22×C4×D411yescharacteristic
522C4—23no—
523C4—21no—
524C4—23no—
525C4—41no—
526C4—43no—
527C4—41no—
528C4—43no—
529C4—41no—
530C4—43no—
531C4C2×#44416yes—
532C4C2×#56811yescharacteristic
533C4#357311yescharacteristic
534C4#358511yescharacteristic
535C4#358611yescharacteristic
536C2—43no—
537C2—83no—
538C2—81no—
539C2#6014711yescharacteristic
540C2#6017411yescharacteristic
541C2C22×#44411yescharacteristic
subgroupG/H
centerC2×C4C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC42C25
socleC22C22×C4×D4
prime radical#61434C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C8×#41 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #60345C2semidirect
3. #60328C2semidirect
4. #18089C2semidirect
5. #18061C2semidirect
6. #18025C2semidirect
7. #18021C2semidirect
8. #17960C2semidirect
9. #17941C2semidirect
10. #17940C2semidirect
11. #17890C2semidirect
12. #17837C2semidirect
13. #17520C2semidirect
14. #17518C2semidirect
15. #17320C2semidirect

Maximal subgroups

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

subgroupindex
#614342
#603452
#603282
#180892
#180612
#180252
#180212
#179602
#179412
#179402
#178902
#178372
#175202
#175182
#173202