Finite Group Databasebeta

← #6346160

Subgroups — #6346160

Order 512.

1557 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#61435C211yesmaximal, characteristic
2#60345C211yesmaximal, characteristic, complemented
3#60331C211yesmaximal, characteristic, complemented
4#18086C213yesmaximal, complemented
5#18057C213yesmaximal, complemented
6#18019C213yesmaximal, complemented
7#18016C213yesmaximal, complemented
8#17980C213yesmaximal, complemented
9#17944C213yesmaximal, complemented
10#17913C213yesmaximal, complemented
11#17868C213yesmaximal, complemented
12#17835C211yesmaximal, characteristic, complemented
13#17518C211yesmaximal, characteristic, complemented
14#17518C211yesmaximal, characteristic, complemented
15#17316C211yesmaximal, characteristic, complemented
16#3482C2211yescharacteristic
17#3482C2211yescharacteristic
18#3395C411yescharacteristic
19Q8×D8C411yescharacteristic
20#3359C411yescharacteristic
21#3359C411yescharacteristic
22#3342C413yes—
23#3302C413yes—
24#3302C413yes—
25#3283C413yes—
26#3170C2213yes—
27#3107C2213yes—
28#3107C2213yes—
29#2994—23no—
30#2993—23no—
31#2991C2213yes—
32#2991C2211yescharacteristic
33#2991C2211yescharacteristic
34#2990C2213yes—
35#2986C2213yes—
36#2986C2213yes—
37#2984—21no—
38#2982—21no—
39#2969—23no—
40#2966—23no—
41#2965C2213yes—
42C8×#41C2213yes—
43C8×#41C2211yescharacteristic
44C8×#41C2211yescharacteristic
45Q8×M16—21no—
46#2960C2213yes—
47#2960C2213yes—
48C2×C8×Q8—21no—
49C4×#135C2213yes—
50C4×#131C2211yescharacteristic
51#1591C2216yes—
52#1590C2216yes—
53#1588C2213yes—
54#1588C2213yes—
55#1587C2216yes—
56#1587C2213yes—
57#1587C2216yes—
58#1587C2216yes—
59#1587C2213yes—
60#1586C2213yes—
61#1586C2211yescharacteristic
62#1585C2213yes—
63#1585C2213yes—
64C8×Q16C2216yes—
65C8×SD16C2216yes—
66#1574C2213yes—
67#1574C2213yes—
68#1566C2213yes—
69#1564C2213yes—
70#1493C2213yes—
71#1493C2213yes—
72#1489C2213yes—
73#1489C2213yes—
74#1485C2213yes—
75#1485C2213yes—
76#1476C2213yes—
77#1476C2213yes—
78#1450C2213yes—
79C8×M16C2213yes—
80C8×M16C2213yes—
81C2×#322C2213yes—
82C2×#135C2×C413yes—
83C2×#131—21no—
84#557C2×C411yescharacteristic
85#557C2×C411yescharacteristic
86#556C2×C413yes—
87#556C2×C413yes—
88#552—23no—
89#552—23no—
90D4×Q8C2×C411yescharacteristic
91D4×Q8C2×C411yescharacteristic
92#548—21no—
93#548—21no—
94#547C2×C413yes—
95#547C2×C413yes—
96#533C2313yes—
97#533C2313yes—
98#529—23no—
99#529—23no—
100C4×#41C2313yes—
101C4×#41C2313yes—
102C4×#41C2311yescharacteristic
103C2×C4×Q8—21no—
104C2×C4×Q8—21no—
105#500C2×C413yes—
106#494C2×C413yes—
107#487C2×C413yes—
108#487C2×C413yes—
109#485C2×C416yes—
110#485C2×C416yes—
111#475C2×C413yes—
112#475C2×C413yes—
113#474C2×C413yes—
114#474C2×C413yes—
115#467C2×C416yes—
116#467C2×C416yes—
117#462C2×C413yes—
118#462C2×C413yes—
119#461C2×C413yes—
120#461C2×C413yes—
121#452C2×C416yes—
122#452C2×C416yes—
123#446—23no—
124#446—23no—
125#446—23no—
126#446—23no—
127#446C2313yes—
128#446C2313yes—
129#446C2316yes—
130C8×Q8—21no—
131C8×Q8—21no—
132C8×Q8—23no—
133C8×Q8—23no—
134C8×Q8—23no—
135C8×Q8—23no—
136C8×Q8C2×C411yescharacteristic
137C8×Q8C2×C411yescharacteristic
138C8×Q8D411yescharacteristic
139C8×Q8D411yescharacteristic
140C8×Q8C2316yes—
141C8×Q8C2313yes—
142C8×Q8C2313yes—
143C4×Q16C2×C413yes—
144C4×Q16C2316yes—
145C4×Q16C2313yes—
146C4×SD16C2×C413yes—
147C4×SD16C2×C413yes—
148C4×SD16C2313yes—
149C4×SD16C2313yes—
150C4×D8C2×C413yes—
151C4×D8C2311yescharacteristic
152#436—23no—
153#436—23no—
154#436—26no—
155#436C2311yescharacteristic
156#436C2313yes—
157#435—23no—
158#435—23no—
159#435—26no—
160C8×D4—23no—
161C8×D4—23no—
162C8×D4C2×C413yes—
163C8×D4C2×C413yes—
164C8×D4D413yes—
165C8×D4D413yes—
166C8×D4C2313yes—
167C8×D4C2311yescharacteristic
168#433—26no—
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#424—23no—
181#423—23no—
182#423C2313yes—
183#423C2313yes—
184C2×C4⋊C8—23no—
185#416C2×C413yes—
186#416C2×C413yes—
187#405—23no—
188#405—23no—
189#405—23no—
190C4×M16—23no—
191C4×M16C2313yes—
192C2×C4×C8—23no—
193C2×C4×C8C2313yes—
194C2×C4×C8C2313yes—
195C2×C4×C8C2313yes—
196#335C2313yes—
197#335C2313yes—
198#334C2311yescharacteristic
199#334C2313yes—
200#334C2316yes—
201Q8⋊C8C2316yes—
202Q8⋊C8C2313yes—
203Q8⋊C8C2316yes—
204Q8⋊C8C2313yes—
205Q8⋊C8C2313yes—
206Q8⋊C8C2313yes—
207D4⋊C8C2313yes—
208D4⋊C8C2311yescharacteristic
209D4⋊C8C2311yescharacteristic
210D4⋊C8C2313yes—
211#322C2313yes—
212#322C2313yes—
213#322C2311yescharacteristic
214#322C2313yes—
215C82C2316yes—
216C2×#41C22×C413yes—
217C2×#41C22×C413yes—
218C2×#41#4111yescharacteristic
219C22×Q8—21no—
220C22×Q8—21no—
221#135—46no—
222C2×Q16C22×C413yes—
223C2×Q16C22×C416yes—
224C2×SD16C22×C413yes—
225C2×SD16C22×C413yes—
226C2×D8C22×C411yescharacteristic
227#131—28no—
228C2×M16—23no—
229C2×M16—23no—
230C2×M16—23no—
231C2×M16—21no—
232C2×M16—23no—
233C22×C8—23no—
234C22×C8—21no—
235C22×C8—23no—
236C22×C8C22×C413yes—
237C22×C8C22×C413yes—
238C22×C8C2×D413yes—
239C22×C8C2×D413yes—
240C4⋊Q8C22×C413yes—
241C4⋊Q8C22×C413yes—
242C4⋊Q8C22×C413yes—
243C4⋊Q8C22×C413yes—
244#126—26no—
245#126—26no—
246#125—23no—
247#125—23no—
248#125—23no—
249#125—23no—
250#124—23no—
251#124—23no—
252#123—26no—
253#123—26no—
254C22⋊Q8—23no—
255C22⋊Q8—23no—
256C22⋊Q8—23no—
257C22⋊Q8—23no—
258C22⋊Q8C22×C416yes—
259C22⋊Q8C22×C416yes—
260#121—23no—
261#121—23no—
262C4×Q8—21no—
263C4×Q8—21no—
264C4×Q8—23no—
265C4×Q8—23no—
266C4×Q8—23no—
267C4×Q8—23no—
268C4×Q8C22×C413yes—
269C4×Q8C22×C413yes—
270C4×Q8C22×C411yescharacteristic
271C4×Q8C2×D411yescharacteristic
272C4×Q8C2413yes—
273C4×Q8C2413yes—
274C4×D4—23no—
275C4×D4—23no—
276C4×D4—23no—
277C4×D4—23no—
278C4×D4C22×C413yes—
279C4×D4C22×C413yes—
280C4×D4C22×C413yes—
281C4×D4C2×D413yes—
282C4×D4C2411yescharacteristic
283C4×D4C2411yescharacteristic
284#117—23no—
285#117—23no—
286#117—23no—
287#117—23no—
288#117C22×C413yes—
289#117C22×C413yes—
290#117#4113yes—
291C2×C4⋊C4—23no—
292C2×C4⋊C4—23no—
293C2×C42—23no—
294C2×C42—23no—
295C2×C42C2413yes—
296#107C22×C411yescharacteristic
297#107C22×C413yes—
298#107C22×C416yes—
299#106C22×C413yes—
300#106C22×C413yes—
301C4⋊C8—26no—
302C4⋊C8—26no—
303C4⋊C8—23no—
304C4⋊C8—23no—
305C4⋊C8—23no—
306C4⋊C8—23no—
307C4⋊C8—23no—
308C4⋊C8—23no—
309C4⋊C8C22×C413yes—
310C4⋊C8C22×C413yes—
311C4⋊C8C2×D413yes—
312C4⋊C8C2×D413yes—
313C4⋊C8C2413yes—
314C4⋊C8C2413yes—
315C4⋊C8C2411yescharacteristic
316C4⋊C8C2411yescharacteristic
317#103C22×C413yes—
318#103C22×C413yes—
319#103C22×C413yes—
320#103C22×C413yes—
321#103C22×C416yes—
322#103C22×C416yes—
323#102C22×C411yescharacteristic
324#102C22×C411yescharacteristic
325#102C22×C413yes—
326#102C22×C413yes—
327C22⋊C8—23no—
328C22⋊C8—23no—
329C22⋊C8—21no—
330C22⋊C8—21no—
331C22⋊C8—23no—
332C22⋊C8—23no—
333C22⋊C8C22×C416yes—
334C22⋊C8C2×D416yes—
335#97—23no—
336#97—23no—
337#97—26no—
338#97—23no—
339C4×C8—23no—
340C4×C8—23no—
341C4×C8—26no—
342C4×C8—23no—
343C4×C8—23no—
344C4×C8—23no—
345C4×C8C22×C413yes—
346C4×C8C22×C413yes—
347C4×C8C2×D413yes—
348C4×C8C2×D413yes—
349C4×C8C2413yes—
350C4×C8C2411yescharacteristic
351C4×C8C2413yes—
352C4×C8C2411yescharacteristic
353C4×C8C2413yes—
354C4×C8C2411yescharacteristic
355#41—24no—
356#41—26no—
357#41—43no—
358#41—43no—
359#41—43no—
360C2×Q8—23no—
361C2×Q8—23no—
362C2×Q8—44no—
363C2×Q8—44no—
364C2×Q8C4×D411yescharacteristic
365C2×Q8C23×C413yes—
366C2×Q8C23×C413yes—
367C2×D4—23no—
368C2×D4—23no—
369C2×D4C4×D413yes—
370C2×D4C23×C411yescharacteristic
371C2×D4C23×C411yescharacteristic
372C22×C4—23no—
373C22×C4—23no—
374C22×C4—23no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—23no—
378C22×C4C23×C413yes—
379C22×C4C22×D413yes—
380C22×C4C2×#4113yes—
381Q16—46no—
382Q16—43no—
383SD16—26no—
384SD16—43no—
385D8—22no—
386M16—212no—
387M16—42no—
388M16—83no—
389M16—83no—
390M16—83no—
391C2×C8—212no—
392C2×C8—21no—
393C2×C8—23no—
394C2×C8—21no—
395C2×C8—21no—
396C2×C8—23no—
397C2×C8—23no—
398C2×C8—43no—
399C2×C8—43no—
400C2×C8—43no—
401C2×C8—43no—
402C2×C8—42no—
403C2×C8—83no—
404C2×C8C23×C411yescharacteristic
405C2×C8C23×C411yescharacteristic
406C2×C8C23×C413yes—
407C2×C8C23×C413yes—
408C2×C8C22×D411yescharacteristic
409C2×C8C22×D413yes—
410C2×C8C22×D411yescharacteristic
411C2×C8C22×D413yes—
412C4⋊C4—23no—
413C4⋊C4—23no—
414C4⋊C4—26no—
415C4⋊C4—26no—
416C4⋊C4—23no—
417C4⋊C4—23no—
418C4⋊C4—23no—
419C4⋊C4—23no—
420C4⋊C4—23no—
421C4⋊C4—23no—
422C4⋊C4—23no—
423C4⋊C4—23no—
424C4⋊C4C4×D413yes—
425C4⋊C4C4×D413yes—
426C4⋊C4C23×C411yescharacteristic
427C4⋊C4C23×C411yescharacteristic
428C4⋊C4C23×C413yes—
429C4⋊C4C23×C413yes—
430C22⋊C4—26no—
431C22⋊C4—26no—
432C22⋊C4—23no—
433C22⋊C4—23no—
434C22⋊C4—23no—
435C22⋊C4—23no—
436C22⋊C4—21no—
437C22⋊C4—21no—
438C22⋊C4C4×D416yes—
439C42—23no—
440C42—23no—
441C42—23no—
442C42—23no—
443C42—23no—
444C42—23no—
445C42C23×C413yes—
446C42C22×D413yes—
447C42C2×#4113yes—
448C42C2511yescharacteristic
449C23—21no—
450C23—21no—
451C23C2×C4×D413yes—
452Q8—23no—
453Q8—23no—
454Q8—23no—
455Q8—23no—
456Q8—46no—
457Q8—46no—
458Q8#44314yes—
459D4—26no—
460D4—21no—
461D4—21no—
462D4—21no—
463D4—46no—
464D4—83no—
465D4#56812yes—
466C2×C4—23no—
467C2×C4—23no—
468C2×C4—26no—
469C2×C4—23no—
470C2×C4—23no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—23no—
474C2×C4—23no—
475C2×C4—23no—
476C2×C4—23no—
477C2×C4—46no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—46no—
483C2×C4—46no—
484C2×C4—83no—
485C2×C4C2×C4×D413yes—
486C2×C4C2×C4×D413yes—
487C2×C4C2×C4×D413yes—
488C2×C4C24×C411yescharacteristic
489C2×C4C23×D411yescharacteristic
490C2×C4C22×#4111yescharacteristic
491C8—22no—
492C8—26no—
493C8—21no—
494C8—21no—
495C8—23no—
496C8—23no—
497C8—21no—
498C8—23no—
499C8—21no—
500C8—23no—
501C8—41no—
502C8—43no—
503C8—41no—
504C8—43no—
505C22—23no—
506C22—23no—
507C22—23no—
508C22—22no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22—41no—
514C22C22×C4×D411yescharacteristic
515C4—23no—
516C4—21no—
517C4—23no—
518C4—41no—
519C4—41no—
520C4—43no—
521C4—43no—
522C4—43no—
523C4—43no—
524C4C2×#44316yes—
525C4C2×#56811yescharacteristic
526C4#357311yescharacteristic
527C4#358611yescharacteristic
528C4#358611yescharacteristic
529C2—43no—
530C2—42no—
531C2—81no—
532C2#6014611yescharacteristic
533C2#6017511yescharacteristic
534C2C22×#44311yescharacteristic
subgroupG/H
centerC2×C4C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC42C25
socleC22C22×C4×D4
prime radical#61435C2
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. #60331C2semidirect
4. #18086C2semidirect
5. #18057C2semidirect
6. #18019C2semidirect
7. #18016C2semidirect
8. #17980C2semidirect
9. #17944C2semidirect
10. #17913C2semidirect
11. #17868C2semidirect
12. #17835C2semidirect
13. #17518C2semidirect
14. #17518C2semidirect
15. #17316C2semidirect

Maximal subgroups

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

subgroupindex
#614352
#603452
#603312
#180862
#180572
#180192
#180162
#179802
#179442
#179132
#178682
#178352
#175182
#173162