Finite Group Databasebeta

← #496501

Subgroups — #496501

Order 512.

826 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60682C211yesmaximal, characteristic
2#33419C211yesmaximal, characteristic, complemented
3#20802C211yesmaximal, characteristic, complemented
4#13288C211yesmaximal, characteristic, complemented
5#13287C211yesmaximal, characteristic, complemented
6#13211C211yesmaximal, characteristic, complemented
7#13210C211yesmaximal, characteristic, complemented
8#13002C212yesmaximal, complemented
9#13001C212yesmaximal, complemented
10#12817C212yesmaximal, complemented
11#12816C212yesmaximal, complemented
12#3455—21no—
13#3177C412yescomplemented
14#3176C412yescomplemented
15#3175C414yescomplemented
16#3174—22no—
17#3173—22no—
18#3160C2211yescharacteristic
19#3147C2211yescharacteristic
20#3135—21no—
21#3054—21no—
22#3050C2212yes—
23#2943—21no—
24C4×#135C2212yes—
25C2×#428—21no—
26C2×#427C2211yescharacteristic
27C22×#106C2211yescharacteristic
28C2×#405C2211yescharacteristic
29#2317—21no—
30#2164—22no—
31#2164—22no—
32#2150C2212yes—
33C2×#360C2212yes—
34C2×#360C2212yes—
35C2×#360C2212yes—
36C2×#360C2212yes—
37#2112C2212yes—
38C4×M32C2211yescharacteristic
39C4×M32C2211yescharacteristic
40#1847—21no—
41#1847—21no—
42#1846C2211yescharacteristic
43#1846C2211yescharacteristic
44#1844C2211yescharacteristic
45#1844C2211yescharacteristic
46#1386C2212yes—
47#1386C2212yes—
48#1386C2212yes—
49#1386C2212yes—
50C2×#135—21no—
51C2×#135C2312yes—
52C22×M16D411yescharacteristic
53C23×C8C2311yescharacteristic
54#542—24no—
55#541—22no—
56#540—22no—
57#539—42no—
58#538—42no—
59#533—21no—
60#532—21no—
61#529—44no—
62#525—22no—
63C4×#41—22no—
64C2×#117—21no—
65C2×#117D411yescharacteristic
66C22×C4⋊C4—21no—
67C2×M32C2312yes—
68C2×M32C2312yes—
69#500C2×C412yescomplemented
70#499—21no—
71#499C2×C412yescomplemented
72#498—21no—
73#495—21no—
74#495C2×C412yescomplemented
75#494C2×C411yescharacteristic, complemented
76#493C2×C411yescharacteristic, complemented
77#492—21no—
78#491—22no—
79#490—22no—
80#489—24no—
81#470—22no—
82#469—22no—
83#468—24no—
84#467C2×C414yes—
85#466C2×C414yes—
86#465—24no—
87#442—22no—
88#441—22no—
89#440—24no—
90C4×Q16C2×C414yescomplemented
91C4×SD16—24no—
92C4×D8C2×C414yescomplemented
93#428—44no—
94#427—21no—
95#427—42no—
96#427C2312yes—
97C2×#107—21no—
98C2×#107—21no—
99C2×#107—21no—
100C2×#106—21no—
101C2×#106—21no—
102C2×#106—21no—
103C2×#106—42no—
104C2×#106C2311yescharacteristic
105C2×#106C2311yescharacteristic
106C2×#106C2311yescharacteristic
107C2×#106C2311yescharacteristic
108#416—22no—
109#416—22no—
110#405C2×C412yes—
111#405C2×C412yes—
112#405D412yes—
113#405D412yes—
114C2×#97D411yescharacteristic
115C2×#97D411yescharacteristic
116C2×C4×C8C2311yescharacteristic
117C2×C4×C8C2311yescharacteristic
118#389—42no—
119#385—42no—
120#384—21no—
121#384—21no—
122#382—21no—
123#382—21no—
124C4⋊C16—24no—
125#360—42no—
126#360—42no—
127#360—42no—
128#360—42no—
129C22⋊C16—24no—
130#346—21no—
131#346—21no—
132C4×C16—21no—
133C4×C16—21no—
134C2×#41—22no—
135C23×C4—41no—
136C23×C4C2×D411yescharacteristic
137#135—22no—
138#135—22no—
139#135—22no—
140#135—22no—
141#135—42no—
142#135—42no—
143C2×Q16—21no—
144C2×Q16C22×C412yes—
145C2×SD16—22no—
146C2×SD16—22no—
147C2×D8—21no—
148C2×D8C22×C412yes—
149C2×M16—22no—
150C2×M16—22no—
151C2×M16C22⋊C412yes—
152C2×M16C22⋊C412yes—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8C22×C412yes—
156C22×C8C2×D411yescharacteristic
157C22×C8C2×D411yescharacteristic
158C22×C8C2×D412yes—
159C22×C8C2×D411yescharacteristic
160C22×C8C2411yescharacteristic
161C4⋊Q8—22no—
162C4⋊Q8—21no—
163#127—21no—
164#126—44no—
165#126—44no—
166#125—22no—
167#125—44no—
168#124—22no—
169#124—44no—
170#123—42no—
171#123—44no—
172#123—44no—
173C22⋊Q8—24no—
174C22⋊Q8—42no—
175C22⋊Q8—44no—
176#121—24no—
177#121—42no—
178#121—44no—
179#120—42no—
180C4×Q8—24no—
181C4×Q8—42no—
182C4×D4—24no—
183C4×D4—44no—
184C4×D4—44no—
185C4×D4—42no—
186#117—22no—
187#117—42no—
188#117—42no—
189#117—42no—
190#117C22⋊C412yes—
191#117C22⋊C412yes—
192#117SD1614yes—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4D812yes—
203C2×C22⋊C4—41no—
204C2×C22⋊C4D812yes—
205C2×C42—42no—
206C2×C42—41no—
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209M32—22no—
210M32—22no—
211M32—22no—
212M32—22no—
213C2×C16—22no—
214C2×C16—22no—
215#107—22no—
216#107—42no—
217#107—42no—
218#107—42no—
219#107C22×C414yes—
220#106—22no—
221#106—22no—
222#106—22no—
223#106—22no—
224#106—22no—
225#106—22no—
226#106—42no—
227#106—42no—
228#106—42no—
229#103—24no—
230#103—24no—
231#102—24no—
232#102—24no—
233#97C22⋊C412yes—
234#97C22⋊C412yes—
235#97C22⋊C412yes—
236#97C22⋊C412yes—
237C4×C8C22×C412yes—
238C4×C8C22×C411yescharacteristic
239C4×C8C22×C411yescharacteristic
240C4×C8C2×D411yescharacteristic
241C4×C8C2×D411yescharacteristic
242C4×C8C2×D412yes—
243#95—42no—
244#95—42no—
245C24C2×D811yescharacteristic
246#41—42no—
247#41—42no—
248#41—42no—
249#41—42no—
250C2×Q8—22no—
251C2×Q8—42no—
252C2×D4—22no—
253C2×D4—42no—
254C2×D4—44no—
255C2×D4—42no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—42no—
261C22×C4—44no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—82no—
267C22×C4—162no—
268C22×C4C2×C22⋊C412yes—
269C22×C4C2×D811yescharacteristic
270C22×C4C2×D812yes—
271C22×C4C2×D812yes—
272C22×C4C2×SD1612yes—
273C22×C4C22×D411yescharacteristic
274Q16—22no—
275Q16—22no—
276Q16—42no—
277SD16—42no—
278SD16—42no—
279SD16—44no—
280D8—22no—
281D8—22no—
282D8—42no—
283M16—24no—
284M16—24no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—24no—
290C2×C8—24no—
291C2×C8C2×C22⋊C412yes—
292C2×C8C2×C22⋊C412yes—
293C2×C8C2×C22⋊C411yescharacteristic
294C2×C8C2×C22⋊C411yescharacteristic
295C2×C8AGL(1,ℤ/8)12yes—
296C2×C8#13712yes—
297C2×C8C23×C411yescharacteristic
298C2×C8C22×D411yescharacteristic
299C2×C8H(5,𝔽2)12yes—
300C2×C82−1+412yes—
301C4⋊C4—24no—
302C4⋊C4—44no—
303C4⋊C4—42no—
304C4⋊C4—44no—
305C4⋊C4—44no—
306C4⋊C4—42no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—82no—
316C4⋊C4—162no—
317C4⋊C4—162no—
318C4⋊C4#10214yes—
319C4⋊C4#10214yes—
320C22⋊C4—42no—
321C22⋊C4—44no—
322C22⋊C4—44no—
323C22⋊C4—44no—
324C22⋊C4—44no—
325C22⋊C4—82no—
326C22⋊C4#10214yes—
327C22⋊C4#10214yes—
328C42—42no—
329C42—44no—
330C42—82no—
331C42C2×C22⋊C412yes—
332C42C2×C22⋊C411yescharacteristic
333C42C2×C22⋊C411yescharacteristic
334C42C2×SD1612yes—
335C42C2×SD1612yes—
336C16—42no—
337C16—42no—
338C23—21no—
339C23—21no—
340C23—42no—
341C23—42no—
342C23C2×#10212yes—
343C23C22×D811yescharacteristic
344Q8—42no—
345Q8—42no—
346Q8—162no—
347D4—42no—
348D4—42no—
349D4—82no—
350D4—82no—
351D4—162no—
352D4—164no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—44no—
362C2×C4—42no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—164no—
374C2×C4—164no—
375C2×C4C2×#10214yes—
376C2×C4C2×#10212yes—
377C2×C4C2×#10212yes—
378C2×C4C2×#10212yes—
379C2×C4C22×C22⋊C411yescharacteristic
380C2×C4#53511yescharacteristic
381C2×C4#53611yescharacteristic
382C2×C4C22×SD1611yescharacteristic
383C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
384C2×C4C2×#13711yescharacteristic
385C8—22no—
386C8—22no—
387C8#41712yes—
388C8#41712yes—
389C8#52012yes—
390C8#53712yes—
391C22—22no—
392C22—22no—
393C22—24no—
394C22—42no—
395C22—42no—
396C22—82no—
397C22—84no—
398C22C22×#10211yescharacteristic
399C22#320511yescharacteristic
400C22#322611yescharacteristic
401C4—22no—
402C4—24no—
403C4—22no—
404C4—22no—
405C4—82no—
406C4—84no—
407C4—161no—
408C4—161no—
409C4—161no—
410C4—161no—
411C4#231111yescharacteristic
412C4C2×#41711yescharacteristic
413C4#320611yescharacteristic
414C4#322711yescharacteristic
415C2—21no—
416C2—21no—
417C2—42no—
418C2—162no—
419C2#2185911yescharacteristic
420C2#3338812yes—
subgroupG/H
center, socleC22C22×#102
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radical#60682C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×#117 ◁ C2×#405 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33419C2semidirect
3. #20802C2semidirect
4. #13288C2semidirect
5. #13287C2semidirect
6. #13211C2semidirect
7. #13210C2semidirect
8. #13002C2semidirect
9. #13001C2semidirect
10. #12817C2semidirect
11. #12816C2semidirect
13. #3177C4semidirect
14. #3176C4semidirect
15. #3175C4semidirect
69. #500C2×C4semidirect
71. #499C2×C4semidirect
74. #495C2×C4semidirect
75. #494C2×C4semidirect
76. #493C2×C4semidirect
90. C4×Q16C2×C4semidirect
92. C4×D8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#606822
#334192
#208022
#132882
#132872
#132112
#132102
#130022
#130012
#128172
#128162