Finite Group Databasebeta

← #482968

Subgroups — #482968

Order 512.

804 conjugacy classes of subgroups.

The subgroup lattice has 387 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62249C211yesmaximal, characteristic
2#33489C211yesmaximal, characteristic, complemented
3#22705C211yesmaximal, characteristic, complemented
4#11890C214yesmaximal, complemented
5#11857C211yesmaximal, characteristic, complemented
6#11856C211yesmaximal, characteristic, complemented
7#11743C211yesmaximal, characteristic, complemented
8#11742C211yesmaximal, characteristic, complemented
9#11704C214yesmaximal, complemented
10#3506—21no—
11#3342C412yescomplemented
12#3341C412yescomplemented
13#3304—22no—
14#3303—22no—
15#3287C412yescomplemented
16#3283C412yescomplemented
17#3246—21no—
18#3191—22no—
19#3170C2211yescharacteristic
20#3169C2211yescharacteristic
21#3050C2214yes—
22C2×C8×D4C2211yescharacteristic
23D4×C4⋊C4—21no—
24#2169—22no—
25#2169—22no—
26#2162C2211yescharacteristic
27#2162C2211yescharacteristic
28#2112C2214yes—
29#1947—21no—
30#1852—22no—
31#1834C2211yescharacteristic
32#1808C2212yes—
33C4×#107C2211yescharacteristic
34C2×#340C2212yes—
35#1348C2214yes—
36#1348C2212yes—
37#1348C2212yes—
38#1347C2214yes—
39#1347C2212yes—
40#1347C2212yes—
41C2×#135C2×C414yescomplemented
42C23×C8C2312yes—
43#559—42no—
44#556—22no—
45#555—22no—
46#553—42no—
47#550—22no—
48#547—22no—
49#538—42no—
50#535—41no—
51#533—21no—
52#532—21no—
53#525—24no—
54#520—41no—
55C2×C4×D4—41no—
56C2×C4×D4D411yescharacteristic
57C22×C4⋊C4—22no—
58C2×M32—24no—
59#500C2×C411yescharacteristic, complemented
60#500C2×C411yescharacteristic, complemented
61#494C2×C411yescharacteristic, complemented
62#493C2×C411yescharacteristic, complemented
63#487—21no—
64#486—21no—
65#485—24no—
66#482—24no—
67#476—22no—
68#475—22no—
69#467—21no—
70#467C2×C414yescomplemented
71#467C2312yes—
72#466—21no—
73#466C2×C414yescomplemented
74#466C2312yes—
75#465—24no—
76#465—24no—
77#461—22no—
78#460—22no—
79#452—24no—
80#449—24no—
81C4×Q16—21no—
82C4×Q16C2×C412yescomplemented
83C4×D8—21no—
84C4×D8C2×C412yescomplemented
85C8×D4C2×C412yes—
86C8×D4C2×C412yes—
87C8×D4D412yes—
88C8×D4D412yes—
89#427C2×C414yescomplemented
90C2×#107—21no—
91C2×#107—22no—
92C2×C4⋊C8D411yescharacteristic
93#416—24no—
94C2×C22⋊C8D412yes—
95C2×C4×C8C2311yescharacteristic
96#389—41no—
97#385—42no—
98#384—21no—
99#380—22no—
100C4×C4⋊C4—21no—
101C4⋊C16—21no—
102C4⋊C16—21no—
103C22⋊C16—24no—
104C22⋊C16C2312yes—
105C22⋊C16C2312yes—
106C4×C16—21no—
107C4×C16—21no—
108#340—21no—
109#340—22no—
110#340C2311yescharacteristic
111#340C2312yes—
112#340C2311yescharacteristic
113C2×#41—24no—
114C2×#41—41no—
115C22×D4D811yescharacteristic
116C23×C4—42no—
117C23×C4C2×D412yes—
118#135—84no—
119C2×Q16—21no—
120C2×Q16C22×C412yes—
121C2×SD16—24no—
122C2×D8—21no—
123C2×D8C22×C412yes—
124C22×C8—82no—
125C22×C8C22×C414yes—
126C22×C8C2×D412yes—
127C22×C8C2×D411yescharacteristic
128C22×C8C2×D414yes—
129C22×C8C2411yescharacteristic
130C4⋊Q8—21no—
131C4⋊Q8—21no—
132C4⋊Q8—21no—
133#127—21no—
134#127—42no—
135#126—44no—
136#125—41no—
137#124—42no—
138#124—44no—
139#124—41no—
140#123—44no—
141#123—44no—
142C22⋊Q8—24no—
143C22⋊Q8—22no—
144C22⋊Q8—41no—
145#121—24no—
146#121—22no—
147#121—42no—
148#121—42no—
149#121—41no—
150#121—44no—
151#120—44no—
152C4×Q8—22no—
153C4×Q8—42no—
154C4×Q8—41no—
155C4×D4—22no—
156C4×D4—44no—
157C4×D4—44no—
158C4×D4—42no—
159C4×D4—41no—
160C4×D4—84no—
161C4×D4C22⋊C412yes—
162C4×D4C22⋊C412yes—
163C4×D4SD1612yes—
164C4×D4SD1612yes—
165#117—24no—
166#117—41no—
167#117—42no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—82no—
177C2×C4⋊C4D811yescharacteristic
178C2×C22⋊C4—42no—
179C2×C22⋊C4D812yes—
180C2×C42—41no—
181C2×C42C2×D411yescharacteristic
182M32—84no—
183C2×C16—22no—
184C2×C16—22no—
185#107—21no—
186#107—21no—
187#107—42no—
188#107—82no—
189#107C22×C412yes—
190#107C22×C412yes—
191#106—24no—
192C4⋊C8C22⋊C412yes—
193C4⋊C8C22⋊C412yes—
194#103—24no—
195#103—22no—
196#103—22no—
197#102—24no—
198#102—22no—
199#102—22no—
200C22⋊C8C22⋊C414yes—
201C22⋊C8C22⋊C414yes—
202C4×C8C22×C411yescharacteristic
203C4×C8C22×C411yescharacteristic
204C4×C8C2×D411yescharacteristic
205C4×C8C2×D411yescharacteristic
206#95—41no—
207#95—42no—
208C24C2×D812yes—
209#41—84no—
210#41—84no—
211#41—82no—
212#41—82no—
213C2×Q8—22no—
214C2×Q8—41no—
215C2×Q8—41no—
216C2×D4—22no—
217C2×D4—22no—
218C2×D4—22no—
219C2×D4—42no—
220C2×D4—44no—
221C2×D4—44no—
222C2×D4—41no—
223C2×D4—41no—
224C2×D4#10212yes—
225C2×D4#10212yes—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—41no—
229C22×C4—42no—
230C22×C4—44no—
231C22×C4—41no—
232C22×C4—42no—
233C22×C4—84no—
234C22×C4—84no—
235C22×C4—162no—
236C22×C4C2×C22⋊C414yes—
237C22×C4C2×D811yescharacteristic
238C22×C4C2×D812yes—
239C22×C4C2×D811yescharacteristic
240C22×C4C2×SD1614yes—
241C22×C4C22×D411yescharacteristic
242Q16—42no—
243Q16—81no—
244SD16—84no—
245D8—42no—
246D8—81no—
247C2×C8—41no—
248C2×C8—42no—
249C2×C8—44no—
250C2×C8—41no—
251C2×C8—44no—
252C2×C8C2×C22⋊C411yescharacteristic
253C2×C8C2×C22⋊C411yescharacteristic
254C2×C8C2×C22⋊C414yes—
255C2×C8C23×C411yescharacteristic
256C2×C8C22×D411yescharacteristic
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—41no—
267C4⋊C4—44no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—42no—
271C4⋊C4—41no—
272C4⋊C4—84no—
273C4⋊C4—82no—
274C4⋊C4—161no—
275C4⋊C4#10212yes—
276C4⋊C4#10212yes—
277C22⋊C4—42no—
278C22⋊C4—44no—
279C22⋊C4—44no—
280C22⋊C4—42no—
281C22⋊C4—44no—
282C22⋊C4—42no—
283C22⋊C4—84no—
284C22⋊C4#10214yes—
285C22⋊C4#10214yes—
286C42—44no—
287C42—42no—
288C42—82no—
289C42C2×C22⋊C411yescharacteristic
290C42C2×C22⋊C411yescharacteristic
291C42C2×SD1611yescharacteristic
292C42C2×SD1611yescharacteristic
293C16—42no—
294C16—42no—
295C23—22no—
296C23—22no—
297C23—42no—
298C23—42no—
299C23—42no—
300C23C2×#10214yes—
301C23C22×D811yescharacteristic
302Q8—42no—
303Q8—42no—
304Q8—82no—
305Q8—161no—
306D4—24no—
307D4—24no—
308D4—42no—
309D4—42no—
310D4—82no—
311D4—82no—
312D4—82no—
313D4—161no—
314D4—164no—
315D4—164no—
316C2×C4—21no—
317C2×C4—24no—
318C2×C4—21no—
319C2×C4—24no—
320C2×C4—42no—
321C2×C4—41no—
322C2×C4—44no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—84no—
330C2×C4—82no—
331C2×C4—81no—
332C2×C4—82no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—82no—
336C2×C4—164no—
337C2×C4—168no—
338C2×C4C2×#10214yes—
339C2×C4C2×#10211yescharacteristic
340C2×C4C2×#10211yescharacteristic
341C2×C4C2×#10211yescharacteristic
342C2×C4C2×#10211yescharacteristic
343C2×C4#50912yes—
344C2×C4#51012yes—
345C2×C4C22×C22⋊C411yescharacteristic
346C2×C4C22×SD1611yescharacteristic
347C8—21no—
348C8—21no—
349C8—24no—
350C8—21no—
351C8—21no—
352C22—21no—
353C22—21no—
354C22—24no—
355C22—24no—
356C22—24no—
357C22—44no—
358C22—44no—
359C22—82no—
360C22—82no—
361C22—82no—
362C22C22×#10211yescharacteristic
363C22C2×#50911yescharacteristic
364C22C2×#51011yescharacteristic
365C4—21no—
366C4—21no—
367C4—21no—
368C4—24no—
369C4—82no—
370C4—82no—
371C4—82no—
372C4—82no—
373C4—161no—
374C4—162no—
375C4#213912yes—
376C4#213912yes—
377C4#288311yescharacteristic
378C4#290111yescharacteristic
379C2—21no—
380C2—21no—
381C2—44no—
382C2—162no—
383C2#1761311yescharacteristic
384C2C2×#213911yescharacteristic
385C2#3338811yescharacteristic
subgroupG/H
center, socleC22C22×#102
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radical#62249C2
LCSG ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33489C2semidirect
3. #22705C2semidirect
4. #11890C2semidirect
5. #11857C2semidirect
6. #11856C2semidirect
7. #11743C2semidirect
8. #11742C2semidirect
9. #11704C2semidirect
11. #3342C4semidirect
12. #3341C4semidirect
15. #3287C4semidirect
16. #3283C4semidirect
41. C2×#135C2×C4semidirect
59. #500C2×C4semidirect
60. #500C2×C4semidirect
61. #494C2×C4semidirect
62. #493C2×C4semidirect
70. #467C2×C4semidirect
73. #466C2×C4semidirect
82. C4×Q16C2×C4semidirect
84. C4×D8C2×C4semidirect
89. #427C2×C4semidirect

Maximal subgroups

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

subgroupindex
#622492
#334892
#227052
#118902
#118572
#118562
#117432
#117422
#117042