Finite Group Databasebeta

← #481407

Subgroups — #481407

Order 512.

741 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#61430C211yesmaximal, characteristic
2#33479C211yesmaximal, characteristic, complemented
3#17828C211yesmaximal, characteristic, complemented
4#11674C212yesmaximal, complemented
5#11671C211yesmaximal, characteristic, complemented
6#11660C212yesmaximal, complemented
7#11658C211yesmaximal, characteristic, complemented
8#11648C212yesmaximal, complemented
9#11645C211yesmaximal, characteristic, complemented
10#11592C212yesmaximal, 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#2906—21no—
30#2183—21no—
31#2169—21no—
32#2168—22no—
33#2162C2211yescharacteristic
34#2162C2212yes—
35C4×M32C2212yes—
36C2×C4×C16C2211yescharacteristic
37#1571—22no—
38#1571—21no—
39#1567C2211yescharacteristic
40#1567C2212yes—
41C2×#335C2211yescharacteristic
42C2×#335C2212yes—
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
55C2×#131—41no—
56#559—21no—
57#559—41no—
58#557—21no—
59#556—21no—
60#556—41no—
61#553—22no—
62#553—42no—
63#550—21no—
64#548—22no—
65#532—21no—
66#532—41no—
67#529—22no—
68#529—42no—
69C4×#41—22no—
70C4×#41—41no—
71C4×#41D411yescharacteristic
72C2×C4×Q8—21no—
73C2×M32—22no—
74C22×C16—21no—
75#500C2×C411yescharacteristic
76#499C2×C412yes—
77#495C2×C412yes—
78#493C2×C411yescharacteristic
79#488—22no—
80#486—21no—
81#485—22no—
82#482—22no—
83#481—22no—
84#479—22no—
85#478—22no—
86#475—21no—
87#475—21no—
88#467C2×C412yes—
89#466—21no—
90#466C2×C412yes—
91#466C2×C412yes—
92#465—22no—
93#464—22no—
94#463—22no—
95#460—21no—
96#460—21no—
97#452—22no—
98#449—22no—
99#448—22no—
100C8×Q8C2×C411yescharacteristic
101C8×Q8D411yescharacteristic
102C4×Q16—21no—
103C4×Q16—21no—
104C4×Q16C2×C411yescharacteristic
105C4×Q16C2×C411yescharacteristic
106C4×Q16C2×C412yes—
107C4×Q16C2311yescharacteristic
108C4×Q16C2311yescharacteristic
109C4×SD16—22no—
110C4×SD16—22no—
111C4×SD16—22no—
112C4×D8—21no—
113C4×D8C2×C412yes—
114C4×D8C2312yes—
115C8×D4C2×C412yes—
116C8×D4C2×C411yescharacteristic
117C8×D4D412yes—
118C8×D4D411yescharacteristic
119#431D412yes—
120#431D411yescharacteristic
121#427C2×C412yes—
122C2×#107C2×C411yescharacteristic
123#424—41no—
124#424—42no—
125#423—21no—
126#423—22no—
127C2×C4⋊C8—21no—
128C2×C4⋊C8—22no—
129#416—22no—
130C2×#103—21no—
131C2×C4×C8C2311yescharacteristic
132C2×C4×C8C2312yes—
133C4⋊C16—22no—
134C4⋊C16—21no—
135C22⋊C16—21no—
136C22⋊C16—22no—
137#346—22no—
138C4×C16—21no—
139C4×C16C2311yescharacteristic
140C4×C16C2311yescharacteristic
141C4×C16C2312yes—
142#335—21no—
143#335—22no—
144#335—21no—
145#335—22no—
146#335C2311yescharacteristic
147#335C2311yescharacteristic
148#335C2312yes—
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—
164#131—84no—
165C2×M16—42no—
166C2×M16—41no—
167C22×C8—42no—
168C22×C8—41no—
169C22×C8C22×C411yescharacteristic
170C22×C8C22×C412yes—
171C22×C8C2×D411yescharacteristic
172C22×C8C2×D412yes—
173C4⋊Q8—21no—
174C4⋊Q8—41no—
175#127—21no—
176#127—41no—
177#126—42no—
178#126—42no—
179#126—42no—
180#125—22no—
181#125—42no—
182#124—22no—
183#124—41no—
184#124—41no—
185#124—42no—
186#124—41no—
187#123—42no—
188#123—42no—
189#123—42no—
190C22⋊Q8—22no—
191C22⋊Q8—42no—
192#121—22no—
193#121—22no—
194#121—42no—
195#121—42no—
196#121—41no—
197#121—42no—
198C4×Q8—21no—
199C4×Q8—21no—
200C4×Q8—22no—
201C4×Q8—21no—
202C4×Q8—21no—
203C4×Q8—41no—
204C4×Q8—41no—
205C4×Q8—41no—
206C4×Q8—41no—
207C4×Q8C22⋊C411yescharacteristic
208C4×Q8SD1611yescharacteristic
209C4×D4—22no—
210C4×D4—22no—
211C4×D4—42no—
212C4×D4—42no—
213C4×D4—42no—
214C4×D4—41no—
215C4×D4—41no—
216C4×D4—42no—
217C4×D4—42no—
218C4×D4—41no—
219C4×D4C22⋊C412yes—
220C4×D4C22⋊C411yescharacteristic
221C4×D4SD1612yes—
222C4×D4SD1611yescharacteristic
223#117—22no—
224#117—42no—
225#117—42no—
226#117D811yescharacteristic
227#117D812yes—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—41no—
230C2×C42—41no—
231C2×C42—42no—
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D412yes—
234M32—44no—
235C2×C16—21no—
236C2×C16—21no—
237C2×C16—22no—
238C2×C16—42no—
239#107—21no—
240#107—21no—
241#107C22×C411yescharacteristic
242#107C22×C411yescharacteristic
243#107C22×C412yes—
244#106—22no—
245#106—22no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—22no—
249C4⋊C8—42no—
250C4⋊C8—41no—
251C4⋊C8—42no—
252C4⋊C8—41no—
253C4⋊C8C22⋊C411yescharacteristic
254C4⋊C8C22⋊C412yes—
255C4⋊C8C22⋊C411yescharacteristic
256C4⋊C8C22⋊C412yes—
257#103—21no—
258#103—21no—
259#103—22no—
260#103—22no—
261#103—21no—
262#103—21no—
263#102—22no—
264#102—22no—
265#102—22no—
266#102—22no—
267C22⋊C8C22⋊C412yes—
268C22⋊C8C22⋊C412yes—
269C22⋊C8C22⋊C412yes—
270C4×C8C22×C412yes—
271C4×C8C22×C411yescharacteristic
272C4×C8C2×D411yescharacteristic
273C4×C8C2×D412yes—
274C4×C8C2×D411yescharacteristic
275C4×C8C2×D412yes—
276C4×C8C2411yescharacteristic
277#41—22no—
278#41—22no—
279#41—44no—
280#41—82no—
281#41—82no—
282#41—82no—
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8—42no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—81no—
289C2×Q8#10211yescharacteristic
290C2×D4—22no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4#10212yes—
297C2×D4#10211yescharacteristic
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4C2×C22⋊C411yescharacteristic
304C22×C4C2×C22⋊C412yes—
305C22×C4C2×D812yes—
306C22×C4C2×D811yescharacteristic
307C22×C4C2×SD1611yescharacteristic
308C22×C4C2×SD1612yes—
309Q16—21no—
310Q16—21no—
311Q16—21no—
312Q16—42no—
313Q16#13112yescomplemented
314SD16—44no—
315SD16—82no—
316D8—22no—
317D8—22no—
318D8—81no—
319M16—84no—
320M16—82no—
321C2×C8—21no—
322C2×C8—22no—
323C2×C8—21no—
324C2×C8—22no—
325C2×C8—41no—
326C2×C8—42no—
327C2×C8—41no—
328C2×C8—42no—
329C2×C8—41no—
330C2×C8—84no—
331C2×C8—82no—
332C2×C8C2×C22⋊C412yes—
333C2×C8C2×C22⋊C411yescharacteristic
334C2×C8C2×C22⋊C411yescharacteristic
335C2×C8C2×C22⋊C412yes—
336C2×C8C23×C411yescharacteristic
337C2×C8C22×D411yescharacteristic
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—22no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—41no—
351C4⋊C4#10211yescharacteristic
352C4⋊C4#10211yescharacteristic
353C4⋊C4#10212yes—
354C4⋊C4#10212yes—
355C22⋊C4—42no—
356C22⋊C4—42no—
357C22⋊C4—42no—
358C22⋊C4—42no—
359C22⋊C4—42no—
360C22⋊C4—42no—
361C22⋊C4—42no—
362C22⋊C4—42no—
363C22⋊C4#10212yes—
364C22⋊C4#10212yes—
365C22⋊C4#10212yes—
366C42—41no—
367C42—41no—
368C42—42no—
369C42—42no—
370C42—41no—
371C42—41no—
372C42C2×C22⋊C412yes—
373C42C2×C22⋊C411yescharacteristic
374C42C2×D812yes—
375C42C2×D811yescharacteristic
376C42C2×SD1612yes—
377C42C2×SD1611yescharacteristic
378C42C22×D411yescharacteristic
379C16—22no—
380C16—41no—
381C16—42no—
382C23—42no—
383C23C2×#10212yes—
384C23C2×#10211yescharacteristic
385Q8—22no—
386Q8—22no—
387Q8—41no—
388Q8—41no—
389Q8—41no—
390Q8—82no—
391Q8—82no—
392D4—24no—
393D4—22no—
394D4—42no—
395D4—42no—
396D4—84no—
397D4—84no—
398D4—82no—
399D4—161no—
400D4—162no—
401C2×C4—21no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—21no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—42no—
413C2×C4—41no—
414C2×C4—84no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—162no—
419C2×C4—161no—
420C2×C4C2×#10212yes—
421C2×C4C2×#10212yes—
422C2×C4C2×#10211yescharacteristic
423C2×C4C2×#10212yes—
424C2×C4C2×#10211yescharacteristic
425C2×C4C2×#10211yescharacteristic
426C2×C4#50912yes—
427C2×C4#51012yes—
428C2×C4C22×C22⋊C411yescharacteristic
429C2×C4C22×D811yescharacteristic
430C2×C4C22×SD1611yescharacteristic
431C8—22no—
432C8—21no—
433C8—21no—
434C8—21no—
435C8—22no—
436C8—81no—
437C8—82no—
438C8—81no—
439C8C2×#13111yescharacteristic
440C8#56811yescharacteristic
441C22—21no—
442C22—21no—
443C22—22no—
444C22—22no—
445C22—22no—
446C22—82no—
447C22—82no—
448C22—82no—
449C22#213912yes—
450C22C22×#10211yescharacteristic
451C4—22no—
452C4—21no—
453C4—22no—
454C4—21no—
455C4—22no—
456C4—21no—
457C4—42no—
458C4—81no—
459C4—82no—
460C4—81no—
461C4—81no—
462C4#213912yes—
463C4#288011yescharacteristic
464C4#290111yescharacteristic
465C4C2×#50911yescharacteristic
466C4C2×#51011yescharacteristic
467C2—22no—
468C2—42no—
469C2—162no—
470C2#1750711yescharacteristic
471C2C2×#213911yescharacteristic
472C2#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. #33479C2semidirect
3. #17828C2semidirect
4. #11674C2semidirect
5. #11671C2semidirect
6. #11660C2semidirect
7. #11658C2semidirect
8. #11648C2semidirect
9. #11645C2semidirect
10. #11592C2semidirect
11. C2×#1332C2semidirect
313. Q16#131semidirect

Maximal subgroups

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

subgroupindex
#614302
#334792
#178282
#116742
#116712
#116602
#116582
#116482
#116452
#115922
C2×#13322