Finite Group Databasebeta

← #481398

Subgroups — #481398

Order 512.

729 conjugacy classes of subgroups.

The subgroup lattice has 471 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#61430C211yesmaximal, characteristic
2C16×#41C211yesmaximal, characteristic, complemented
3#17829C211yesmaximal, characteristic, complemented
4#11673C212yesmaximal, complemented
5#11671C211yesmaximal, characteristic, complemented
6#11659C212yesmaximal, complemented
7#11658C211yesmaximal, characteristic, complemented
8#11647C212yesmaximal, complemented
9#11645C211yesmaximal, characteristic, complemented
10#11591C212yesmaximal, complemented
11C2×#1332C211yesmaximal, characteristic, complemented
12#3480—21no—
13#3396C411yescharacteristic
14#3383C411yescharacteristic
15#3365—21no—
16#3342C411yescharacteristic
17#3339C412yes—
18#3304—21no—
19#3301—22no—
20#3287C411yescharacteristic
21#3285C412yes—
22#3169C2211yescharacteristic
23#3168C2212yes—
24#3103—21no—
25#3102—22no—
26C8×#41C2211yescharacteristic
27C4×#135C2212yes—
28C2×C4×Q16C2211yescharacteristic
29#2925—21no—
30Q8×C16—21no—
31D4×C16—22no—
32D4×C16—21no—
33#2162C2211yescharacteristic
34#2162C2212yes—
35C2×C4×C16C2211yescharacteristic
36C2×C4×C16C2212yes—
37#1572—22no—
38#1570—21no—
39#1568C2212yes—
40#1567C2211yescharacteristic
41#1564C2212yes—
42C2×#335C2211yescharacteristic
43#1332C2211yescharacteristic
44#1332C2211yescharacteristic
45#1332C2211yescharacteristic
46#1332C2211yescharacteristic
47#1332C2212yes—
48#1332C2212yes—
49#1331C2212yes—
50#1331C2212yes—
51#1331C2212yes—
52#1331C2212yes—
53C2×#135C2×C412yes—
54C22×Q16C2×C411yescharacteristic
55#559—21no—
56#559—41no—
57#557—21no—
58#556—21no—
59#556—41no—
60#553—22no—
61#553—42no—
62#550—21no—
63#548—22no—
64#532—21no—
65#532—41no—
66#529—22no—
67#529—42no—
68C4×#41—22no—
69C4×#41—41no—
70C4×#41D411yescharacteristic
71C2×C4×Q8—21no—
72C22×C16—21no—
73C22×C16—22no—
74#500C2×C411yescharacteristic
75#499C2×C412yes—
76#495C2×C412yes—
77#493C2×C411yescharacteristic
78#488—22no—
79#486—21no—
80#485—22no—
81#482—22no—
82#481—22no—
83#479—22no—
84#478—22no—
85#475—21no—
86#475—21no—
87#467C2×C412yes—
88#466—21no—
89#466C2×C412yes—
90#466C2×C412yes—
91#465—22no—
92#464—22no—
93#463—22no—
94#460—21no—
95#460—21no—
96#452—22no—
97#449—22no—
98#448—22no—
99C8×Q8C2×C411yescharacteristic
100C8×Q8D411yescharacteristic
101C4×Q16—21no—
102C4×Q16—21no—
103C4×Q16C2×C411yescharacteristic
104C4×Q16C2×C411yescharacteristic
105C4×Q16C2×C412yes—
106C4×Q16C2311yescharacteristic
107C4×Q16C2311yescharacteristic
108C4×SD16—22no—
109C4×SD16—22no—
110C4×SD16—22no—
111C4×D8—21no—
112C4×D8C2×C412yes—
113C4×D8C2312yes—
114C8×D4C2×C412yes—
115C8×D4C2×C411yescharacteristic
116C8×D4D412yes—
117C8×D4D411yescharacteristic
118#433—42no—
119#432—22no—
120#431—22no—
121#431D412yes—
122#431D411yescharacteristic
123#427C2×C412yes—
124C2×#107C2×C411yescharacteristic
125#423—21no—
126C2×C4⋊C8—21no—
127#416—22no—
128C2×#103—21no—
129#408—41no—
130#405—41no—
131C2×C4×C8C2311yescharacteristic
132C2×C4×C8C2312yes—
133C4⋊C16—22no—
134C4⋊C16—21no—
135C22⋊C16—21no—
136C22⋊C16—22no—
137C4×C16—22no—
138C4×C16—21no—
139C4×C16C2311yescharacteristic
140C4×C16C2312yes—
141C4×C16C2311yescharacteristic
142#335—21no—
143#335—21no—
144#335C2311yescharacteristic
145#335C2311yescharacteristic
146#335C2312yes—
147#334—22no—
148#334—22no—
149C2×#41—22no—
150C2×#41—41no—
151C2×#41D811yescharacteristic
152C22×Q8—21no—
153#135—24no—
154#135—42no—
155C2×Q16—22no—
156C2×Q16—21no—
157C2×Q16—41no—
158C2×Q16C22×C411yescharacteristic
159C2×Q16C22×C411yescharacteristic
160C2×SD16—22no—
161C2×SD16—22no—
162C2×D8—21no—
163C2×D8C22×C412yes—
164C2×M16—41no—
165C22×C8—41no—
166C22×C8C22×C411yescharacteristic
167C22×C8C22×C412yes—
168C22×C8C2×D411yescharacteristic
169C22×C8C2×D412yes—
170C4⋊Q8—21no—
171C4⋊Q8—41no—
172#127—21no—
173#127—41no—
174#126—42no—
175#126—42no—
176#126—42no—
177#125—22no—
178#125—42no—
179#124—22no—
180#124—41no—
181#124—41no—
182#124—42no—
183#124—41no—
184#123—42no—
185#123—42no—
186#123—42no—
187C22⋊Q8—22no—
188C22⋊Q8—42no—
189#121—22no—
190#121—22no—
191#121—42no—
192#121—42no—
193#121—41no—
194#121—42no—
195C4×Q8—21no—
196C4×Q8—21no—
197C4×Q8—22no—
198C4×Q8—21no—
199C4×Q8—21no—
200C4×Q8—41no—
201C4×Q8—41no—
202C4×Q8—41no—
203C4×Q8—41no—
204C4×Q8C22⋊C411yescharacteristic
205C4×Q8SD1611yescharacteristic
206C4×D4—22no—
207C4×D4—22no—
208C4×D4—42no—
209C4×D4—42no—
210C4×D4—42no—
211C4×D4—41no—
212C4×D4—41no—
213C4×D4—42no—
214C4×D4—42no—
215C4×D4—41no—
216C4×D4C22⋊C412yes—
217C4×D4C22⋊C411yescharacteristic
218C4×D4SD1612yes—
219C4×D4SD1611yescharacteristic
220#117—22no—
221#117—42no—
222#117—42no—
223#117D811yescharacteristic
224#117D812yes—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—41no—
227C2×C42—41no—
228C2×C42—42no—
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D412yes—
231C2×C16—21no—
232C2×C16—22no—
233C2×C16—21no—
234C2×C16—42no—
235C2×C16—44no—
236#107—21no—
237#107—21no—
238#107C22×C411yescharacteristic
239#107C22×C411yescharacteristic
240#107C22×C412yes—
241#106—22no—
242#106—22no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—22no—
246C4⋊C8—41no—
247C4⋊C8—41no—
248C4⋊C8C22⋊C411yescharacteristic
249C4⋊C8C22⋊C412yes—
250C4⋊C8C22⋊C411yescharacteristic
251C4⋊C8C22⋊C412yes—
252#103—21no—
253#103—21no—
254#103—22no—
255#103—22no—
256#103—21no—
257#103—21no—
258#102—22no—
259#102—22no—
260#102—22no—
261#102—22no—
262C22⋊C8—42no—
263C22⋊C8—42no—
264C22⋊C8C22⋊C412yes—
265C22⋊C8C22⋊C412yes—
266C22⋊C8C22⋊C412yes—
267#97—42no—
268C4×C8—42no—
269C4×C8C22×C412yes—
270C4×C8C22×C411yescharacteristic
271C4×C8C2×D412yes—
272C4×C8C2×D411yescharacteristic
273C4×C8C2×D412yes—
274C4×C8C2×D411yescharacteristic
275C4×C8C2411yescharacteristic
276#41—22no—
277#41—22no—
278#41—44no—
279#41—82no—
280#41—82no—
281#41—82no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8—42no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×Q8—81no—
288C2×Q8#10211yescharacteristic
289C2×D4—22no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4#10212yes—
296C2×D4#10211yescharacteristic
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4C2×C22⋊C411yescharacteristic
303C22×C4C2×C22⋊C412yes—
304C22×C4C2×D812yes—
305C22×C4C2×D811yescharacteristic
306C22×C4C2×SD1611yescharacteristic
307C22×C4C2×SD1612yes—
308Q16—21no—
309Q16—21no—
310Q16—21no—
311Q16—42no—
312Q16#13112yes—
313SD16—44no—
314SD16—82no—
315D8—22no—
316D8—22no—
317D8—81no—
318M16—82no—
319C2×C8—21no—
320C2×C8—22no—
321C2×C8—21no—
322C2×C8—22no—
323C2×C8—41no—
324C2×C8—42no—
325C2×C8—41no—
326C2×C8—42no—
327C2×C8—41no—
328C2×C8—82no—
329C2×C8C2×C22⋊C412yes—
330C2×C8C2×C22⋊C411yescharacteristic
331C2×C8C2×C22⋊C411yescharacteristic
332C2×C8C2×C22⋊C412yes—
333C2×C8C23×C411yescharacteristic
334C2×C8C22×D411yescharacteristic
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—22no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—41no—
348C4⋊C4#10211yescharacteristic
349C4⋊C4#10211yescharacteristic
350C4⋊C4#10212yes—
351C4⋊C4#10212yes—
352C22⋊C4—42no—
353C22⋊C4—42no—
354C22⋊C4—42no—
355C22⋊C4—42no—
356C22⋊C4—42no—
357C22⋊C4—42no—
358C22⋊C4—42no—
359C22⋊C4—42no—
360C22⋊C4#10212yes—
361C22⋊C4#10212yes—
362C22⋊C4#10212yes—
363C42—41no—
364C42—41no—
365C42—42no—
366C42—42no—
367C42—41no—
368C42—41no—
369C42C2×C22⋊C412yes—
370C42C2×C22⋊C411yescharacteristic
371C42C2×D812yes—
372C42C2×D811yescharacteristic
373C42C2×SD1612yes—
374C42C2×SD1611yescharacteristic
375C42C22×D411yescharacteristic
376C16—22no—
377C16—42no—
378C16—41no—
379C23—42no—
380C23C2×#10212yes—
381C23C2×#10211yescharacteristic
382Q8—22no—
383Q8—22no—
384Q8—41no—
385Q8—41no—
386Q8—41no—
387Q8—82no—
388Q8—82no—
389D4—24no—
390D4—22no—
391D4—42no—
392D4—42no—
393D4—84no—
394D4—84no—
395D4—82no—
396D4—161no—
397D4—162no—
398C2×C4—21no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—21no—
403C2×C4—22no—
404C2×C4—42no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—41no—
411C2×C4—84no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—162no—
416C2×C4—161no—
417C2×C4C2×#10212yes—
418C2×C4C2×#10212yes—
419C2×C4C2×#10211yescharacteristic
420C2×C4C2×#10212yes—
421C2×C4C2×#10211yescharacteristic
422C2×C4C2×#10211yescharacteristic
423C2×C4#50812yes—
424C2×C4#50812yes—
425C2×C4C22×C22⋊C411yescharacteristic
426C2×C4C22×D811yescharacteristic
427C2×C4C22×SD1611yescharacteristic
428C8—22no—
429C8—21no—
430C8—21no—
431C8—21no—
432C8—22no—
433C8—81no—
434C8—81no—
435C8—82no—
436C8C2×#13111yescharacteristic
437C8#56811yescharacteristic
438C22—21no—
439C22—21no—
440C22—22no—
441C22—22no—
442C22—22no—
443C22—82no—
444C22—82no—
445C22—82no—
446C22#213812yes—
447C22C22×#10211yescharacteristic
448C4—22no—
449C4—22no—
450C4—21no—
451C4—21no—
452C4—22no—
453C4—21no—
454C4—42no—
455C4—82no—
456C4—81no—
457C4—81no—
458C4—81no—
459C4#213812yes—
460C4#288011yescharacteristic
461C4#290111yescharacteristic
462C4C2×#50811yescharacteristic
463C4C2×#50811yescharacteristic
464C2—22no—
465C2—42no—
466C2—162no—
467C2#1750711yescharacteristic
468C2C2×#213811yescharacteristic
469C2#3338511yescharacteristic
subgroupG/H
centerC2×C4C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC4×C8C24
socleC22C22×#102
prime radical#61430C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C4×#41 ◁ C8×#41 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C16×#41C2semidirect
3. #17829C2semidirect
4. #11673C2semidirect
5. #11671C2semidirect
6. #11659C2semidirect
7. #11658C2semidirect
8. #11647C2semidirect
9. #11645C2semidirect
10. #11591C2semidirect
11. C2×#1332C2semidirect

Maximal subgroups

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

subgroupindex
#614302
C16×#412
#178292
#116732
#116712
#116592
#116582
#116472
#116452
#115912
C2×#13322