Finite Group Databasebeta

← #483077

Subgroups — #483077

Order 512.

723 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60713C211yesmaximal, characteristic
2#33423C211yesmaximal, characteristic, complemented
3#17788C211yesmaximal, characteristic, complemented
4#11946C212yesmaximal, complemented
5#11945C212yesmaximal, complemented
6#11944C212yesmaximal, complemented
7#11943C212yesmaximal, complemented
8#11936C211yesmaximal, characteristic, complemented
9#11934C211yesmaximal, characteristic, complemented
10#11926C211yesmaximal, characteristic, complemented
11C2×#1360C211yesmaximal, characteristic, complemented
12#3510—21no—
13#3183—22no—
14#3175C412yes—
15#3175C412yes—
16#3173—22no—
17#3168C2212yes—
18#3168C2212yes—
19#3160C2211yescharacteristic
20C2×#495C2211yescharacteristic
21#3140—21no—
22#3059C411yescharacteristic
23#3058C411yescharacteristic
24#3057—21no—
25#2988—21no—
26#2950C411yescharacteristic
27#2949—21no—
28#2948C411yescharacteristic
29C4×#131C2211yescharacteristic
30#2164—22no—
31#2163C2212yescomplemented
32#2162C2212yescomplemented
33#2150C2211yescharacteristic
34C2×C4⋊C16C2211yescharacteristic
35#2113—21no—
36#2108—21no—
37#1565—22no—
38#1565—21no—
39#1564C2211yescharacteristic
40#1564C2212yescomplemented
41C2×#334C2211yescharacteristic
42C2×#334C2212yescomplemented
43#1363C2211yescharacteristic, complemented
44#1363C2211yescharacteristic, complemented
45#1363C2211yescharacteristic, complemented
46#1363C2211yescharacteristic, complemented
47#1363C2212yescomplemented
48#1363C2212yescomplemented
49#1360C2212yes—
50#1360C2212yes—
51#1360C2212yes—
52#1360C2212yes—
53C22×Q16C2×C411yescharacteristic
54C22×SD16—21no—
55C22×D8C2×C411yescharacteristic
56C2×#131C2×C411yescharacteristic
57C2×#131D411yescharacteristic
58#559—41no—
59#559—41no—
60#556—41no—
61#556—41no—
62#551—42no—
63#551—42no—
64#542—22no—
65#542—22no—
66#536—21no—
67#535—21no—
68#533—21no—
69#529—22no—
70#529—22no—
71C2×#124—21no—
72#519—21no—
73#518—21no—
74C4×#41D411yescharacteristic
75C2×M32—21no—
76C22×C16—21no—
77#500C2×C411yescharacteristic
78#500C2×C411yescharacteristic
79#499—21no—
80#499C2311yescharacteristic
81#499C2311yescharacteristic
82#498—21no—
83#495—21no—
84#495—21no—
85#495C2×C412yes—
86#495C2312yes—
87#489—22no—
88#489—22no—
89#488—22no—
90#488—22no—
91#470—22no—
92#469—22no—
93#468—22no—
94#468—22no—
95#467C2×C412yes—
96#467C2×C412yes—
97#466C2×C412yes—
98#466C2×C412yes—
99#465—22no—
100#465—22no—
101C8×Q8—41no—
102#442—22no—
103#441—22no—
104#440—22no—
105#440—22no—
106C4×Q16C2×C412yes—
107C4×Q16C2×C412yes—
108C4×SD16—22no—
109C4×SD16—22no—
110C4×D8C2×C412yes—
111C4×D8C2×C412yes—
112C8×D4—42no—
113C8×D4—41no—
114#431—21no—
115#431—22no—
116#427—21no—
117#427C2×C411yescharacteristic
118#427C2×C411yescharacteristic
119C2×C4⋊C8—21no—
120C2×C4⋊C8—22no—
121#419—21no—
122#419—21no—
123#418—21no—
124#418—21no—
125#405C2×C412yes—
126#405C2×C411yescharacteristic
127#405D411yescharacteristic
128#405D412yes—
129C4×M16D411yescharacteristic
130C4×M16D412yes—
131C2×C4×C8C2311yescharacteristic
132C2×C4×C8C2312yes—
133C4⋊C16—21no—
134C4⋊C16—21no—
135C4⋊C16C2311yescharacteristic
136C4⋊C16C2311yescharacteristic
137C4⋊C16C2312yes—
138C22⋊C16—22no—
139C22⋊C16—22no—
140#346—22no—
141C4×C16—22no—
142#335—21no—
143#335—22no—
144#334—21no—
145#334—22no—
146#334C2312yes—
147#334C2311yescharacteristic
148#334C2311yescharacteristic
149C2×#41C22⋊C411yescharacteristic
150C22×Q8—21no—
151C22×D4—21no—
152C2×Q16—21no—
153C2×Q16—42no—
154C2×Q16C22×C412yes—
155C2×SD16—22no—
156C2×SD16—22no—
157C2×SD16—42no—
158C2×SD16—42no—
159C2×D8—21no—
160C2×D8—42no—
161C2×D8C22×C412yes—
162#131—22no—
163#131—22no—
164#131—42no—
165C2×M16C22⋊C411yescharacteristic
166C2×M16C22⋊C411yescharacteristic
167C2×M16C22⋊C412yes—
168C2×M16C22⋊C412yes—
169C22×C8—41no—
170C22×C8—42no—
171C22×C8C22×C411yescharacteristic
172C22×C8C22×C412yes—
173C22×C8C2×D412yes—
174C22×C8C2×D411yescharacteristic
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177C4⋊Q8—41no—
178#127—41no—
179#126—42no—
180#126—42no—
181#126—42no—
182#126—42no—
183#125—21no—
184#125—21no—
185#124—22no—
186#124—22no—
187#124—41no—
188#124—41no—
189#124—42no—
190#124—42no—
191#124—41no—
192#124—41no—
193#123—42no—
194#123—42no—
195#123—42no—
196#123—42no—
197#123—42no—
198C22⋊Q8—22no—
199C22⋊Q8—22no—
200C22⋊Q8—41no—
201C22⋊Q8—41no—
202C22⋊Q8—42no—
203C22⋊Q8—42no—
204#121—22no—
205#121—22no—
206#120—42no—
207C4×Q8—22no—
208C4×Q8—22no—
209C4×Q8—41no—
210C4×Q8—41no—
211C4×Q8—41no—
212C4×Q8—41no—
213C4×Q8D811yescharacteristic
214C4×Q8SD1611yescharacteristic
215C4×D4—22no—
216C4×D4—22no—
217C4×D4—42no—
218C4×D4—42no—
219C4×D4—41no—
220C4×D4—41no—
221C4×D4D812yes—
222C4×D4D811yescharacteristic
223C4×D4SD1612yes—
224C4×D4SD1611yescharacteristic
225#117—21no—
226#117—21no—
227#117—42no—
228#117—42no—
229#117—42no—
230#117C22⋊C411yescharacteristic
231#117C22⋊C412yes—
232C2×C22⋊C4—41no—
233C2×C22⋊C4—41no—
234C2×C22⋊C4—42no—
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D412yes—
237M32—42no—
238C2×C16—21no—
239C2×C16—22no—
240C2×C16—21no—
241C2×C16—42no—
242#107—21no—
243#107—21no—
244#107C22×C411yescharacteristic
245#107C22×C411yescharacteristic
246#107C22×C411yescharacteristic
247#107C22×C411yescharacteristic
248#106—21no—
249#106—21no—
250#106—21no—
251#106—21no—
252C4⋊C8—22no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—42no—
256C4⋊C8—41no—
257#103—22no—
258#103—22no—
259#103—22no—
260#103—22no—
261#102—22no—
262#102—22no—
263#102—22no—
264#102—22no—
265C22⋊C8—41no—
266C22⋊C8—42no—
267#97C22⋊C412yes—
268#97C22⋊C412yes—
269#97C22⋊C412yes—
270C4×C8—42no—
271C4×C8—41no—
272C4×C8C22×C412yes—
273C4×C8C22×C411yescharacteristic
274C4×C8C2×D412yes—
275C4×C8C2×D411yescharacteristic
276C4×C8C2×D412yes—
277C4×C8C2×D411yescharacteristic
278C4×C8C2411yescharacteristic
279C24—41no—
280#41—22no—
281#41—22no—
282C2×Q8—22no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×Q8—41no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8#10211yescharacteristic
289C2×D4—22no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—41no—
294C2×D4—42no—
295C2×D4#10212yes—
296C2×D4#10211yescharacteristic
297C22×C4—41no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4C2×C22⋊C412yes—
301C22×C4C2×C22⋊C411yescharacteristic
302C22×C4C2×D811yescharacteristic
303C22×C4C2×D812yes—
304C22×C4C2×SD1611yescharacteristic
305C22×C4C2×SD1612yes—
306Q16—42no—
307Q16—42no—
308SD16—42no—
309SD16—42no—
310SD16—44no—
311D8—42no—
312D8—42no—
313M16—22no—
314M16—22no—
315M16—24no—
316M16—42no—
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—42no—
321C2×C8—41no—
322C2×C8—42no—
323C2×C8—41no—
324C2×C8—162no—
325C2×C8—161no—
326C2×C8C2×C22⋊C412yes—
327C2×C8C2×C22⋊C412yes—
328C2×C8C2×C22⋊C411yescharacteristic
329C2×C8C2×C22⋊C411yescharacteristic
330C2×C8#13512yes—
331C2×C8#13512yes—
332C2×C8C23×C411yescharacteristic
333C2×C8C22×D411yescharacteristic
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4#10211yescharacteristic
349C4⋊C4#10211yescharacteristic
350C4⋊C4#10212yes—
351C4⋊C4#10212yes—
352C22⋊C4—41no—
353C22⋊C4—41no—
354C22⋊C4—42no—
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—42no—
364C22⋊C4#10212yes—
365C22⋊C4#10212yes—
366C22⋊C4#10212yes—
367C42—41no—
368C42—41no—
369C42—42no—
370C42—42no—
371C42C2×C22⋊C412yes—
372C42C2×C22⋊C411yescharacteristic
373C42C2×D812yes—
374C42C2×D811yescharacteristic
375C42C2×SD1612yes—
376C42C2×SD1611yescharacteristic
377C42C22×D411yescharacteristic
378C16—41no—
379C16—41no—
380C16—42no—
381C23—42no—
382C23—82no—
383C23—161no—
384C23—161no—
385C23C2×#10212yes—
386C23C2×#10211yescharacteristic
387Q8—22no—
388Q8—42no—
389Q8—42no—
390Q8—82no—
391Q8—161no—
392Q8—161no—
393D4—24no—
394D4—22no—
395D4—42no—
396D4—42no—
397D4—82no—
398D4—162no—
399D4—162no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—42no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—82no—
412C2×C4—162no—
413C2×C4—162no—
414C2×C4#41312yes—
415C2×C4C2×#10212yes—
416C2×C4C2×#10212yes—
417C2×C4C2×#10211yescharacteristic
418C2×C4C2×#10211yescharacteristic
419C2×C4C2×#10212yes—
420C2×C4C2×#10211yescharacteristic
421C2×C4#41612yes—
422C2×C4C22×C22⋊C411yescharacteristic
423C2×C4C22×D811yescharacteristic
424C2×C4C22×SD1611yescharacteristic
425C8—21no—
426C8—21no—
427C8—22no—
428C8—22no—
429C8—81no—
430C8—82no—
431C8—81no—
432C8#41612yes—
433C8C2×#13511yescharacteristic
434C8C2×#13511yescharacteristic
435C22—21no—
436C22—21no—
437C22—22no—
438C22—22no—
439C22—22no—
440C22—82no—
441C22—82no—
442C22—82no—
443C22—322no—
444C22#154012yes—
445C22C22×#10211yescharacteristic
446C4—22no—
447C4—21no—
448C4—22no—
449C4—21no—
450C4—21no—
451C4—22no—
452C4—21no—
453C4—82no—
454C4—81no—
455C4—81no—
456C4—81no—
457C4—81no—
458C4#154012yes—
459C4C2×#41311yescharacteristic
460C4C2×#41611yescharacteristic
461C2—22no—
462C2—42no—
463C2—162no—
464C2C2×#154011yescharacteristic
465C2#3338511yescharacteristic
466C2#3338811yescharacteristic
subgroupG/H
center, socleC22C22×#102
commutatorC2×C8C23×C4
Frattini, cosocleC4×C8C24
prime radical#3058C4
LCSG ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C4×#41 ◁ C4×#131 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33423C2semidirect
3. #17788C2semidirect
4. #11946C2semidirect
5. #11945C2semidirect
6. #11944C2semidirect
7. #11943C2semidirect
8. #11936C2semidirect
9. #11934C2semidirect
10. #11926C2semidirect
11. C2×#1360C2semidirect
31. #2163C22semidirect
32. #2162C22semidirect
40. #1564C22semidirect
42. C2×#334C22semidirect
43. #1363C22semidirect
44. #1363C22semidirect
45. #1363C22semidirect
46. #1363C22semidirect
47. #1363C22semidirect
48. #1363C22semidirect

Maximal subgroups

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

subgroupindex
#607132
#334232
#177882
#119462
#119452
#119442
#119432
#119362
#119342
#119262
C2×#13602