Finite Group Databasebeta

← #491269

Subgroups — #491269

Order 512.

448 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33794C211yesmaximal, characteristic, complemented
2#33488C211yesmaximal, characteristic, complemented
3#19255C211yesmaximal, characteristic
4#19105C211yesmaximal, characteristic, complemented
5#12472C211yesmaximal, characteristic, complemented
6#12471C211yesmaximal, characteristic, complemented
7#12468C211yesmaximal, characteristic, complemented
8#12467C211yesmaximal, characteristic, complemented
9#12456C211yesmaximal, characteristic, complemented
10#12454C211yesmaximal, characteristic, complemented
11#12453C211yesmaximal, characteristic, complemented
12#12451C211yesmaximal, characteristic, complemented
13#12036C211yesmaximal, characteristic, complemented
14#11976C211yesmaximal, characteristic, complemented
15#11951C211yesmaximal, characteristic, complemented
16#3169C2211yescharacteristic
17#3160C2211yescharacteristic
18#3101—21no—
19#3036—21no—
20C2×C8×D4C2211yescharacteristic
21#2254C2211yescharacteristic, complemented
22#2252C2211yescharacteristic, complemented
23#2240C2211yescharacteristic, complemented
24#2240C2211yescharacteristic, complemented
25#2234—21no—
26#2234—21no—
27#2228—21no—
28#2228—21no—
29#2169C2211yescharacteristic
30#2169C2211yescharacteristic
31#2168—21no—
32#2168—21no—
33D4×C16C2211yescharacteristic
34D4×C16C2211yescharacteristic
35#2163C2211yescharacteristic
36#2162C2211yescharacteristic
37#2112—21no—
38C2×C22⋊C16—21no—
39#1681—21no—
40#1681—21no—
41#1675C2211yescharacteristic
42#1675C2211yescharacteristic
43#1673C2211yescharacteristic
44#1673C2211yescharacteristic
45#1673C2211yescharacteristic
46#1673C2211yescharacteristic
47#1671—21no—
48#1671—21no—
49#1667C2211yescharacteristic
50#1667C2211yescharacteristic
51#1567C2211yescharacteristic
52#1564C2211yescharacteristic
53#1547—21no—
54#1542—21no—
55#1372C2211yescharacteristic, complemented
56#1372C2211yescharacteristic, complemented
57#1371C2211yescharacteristic, complemented
58#1371C2211yescharacteristic, complemented
59#1365C2211yescharacteristic, complemented
60#1365C2211yescharacteristic, complemented
61#1364C2211yescharacteristic, complemented
62#1364C2211yescharacteristic, complemented
63#1364C2211yescharacteristic, complemented
64#1364C2211yescharacteristic, complemented
65#1361C2211yescharacteristic, complemented
66#1361C2211yescharacteristic, complemented
67C2×AGL(1,ℤ/8)—41no—
68C23×C8—21no—
69#533—21no—
70#532—21no—
71C2×C4×D4D411yescharacteristic
72C2×M32—21no—
73C2×M32—21no—
74C22×C16—21no—
75C22×C16—21no—
76#500C2311yescharacteristic
77#500C2311yescharacteristic
78#500C2311yescharacteristic
79#499—21no—
80#499—21no—
81#498—21no—
82#493C2311yescharacteristic
83#486—21no—
84#483—41no—
85#476—21no—
86#476—21no—
87#466—21no—
88#466—21no—
89#463—41no—
90#463—41no—
91#461—21no—
92#461—21no—
93#461—21no—
94#461—21no—
95#459—21no—
96#459—21no—
97#450—41no—
98C4×SD16—41no—
99C4×SD16—41no—
100C4×D8—21no—
101C4×D8—21no—
102C8×D4—21no—
103C8×D4—21no—
104C8×D4D411yescharacteristic
105C8×D4D411yescharacteristic
106C8×D4C2311yescharacteristic
107C8×D4C2311yescharacteristic
108#431—21no—
109#427—21no—
110#427—21no—
111#423—21no—
112C2×C4⋊C8D411yescharacteristic
113#417—41no—
114C2×C22⋊C8—21no—
115C2×C4×C8C2311yescharacteristic
116#367—21no—
117#367—21no—
118#366—21no—
119#366—21no—
120C4⋊C16D411yescharacteristic
121C4⋊C16D411yescharacteristic
122C4⋊C16C2311yescharacteristic
123C4⋊C16C2311yescharacteristic
124#357—21no—
125#357—21no—
126#357—21no—
127#357—21no—
128C22⋊C16—21no—
129C22⋊C16—21no—
130C22⋊C16—21no—
131C22⋊C16—21no—
132C22⋊C16—21no—
133C22⋊C16D411yescharacteristic
134C22⋊C16D411yescharacteristic
135#346—21no—
136C4×C16C2311yescharacteristic
137C4×C16C2311yescharacteristic
138#335—21no—
139#335C2311yescharacteristic
140#335C2311yescharacteristic
141#335C2311yescharacteristic
142#335C2311yescharacteristic
143#334—21no—
144#333—21no—
145#333—21no—
146#332—21no—
147#332—21no—
148#332—21no—
149#332—21no—
150#331—21no—
151#331—21no—
152C2×#41—41no—
153C22×D4D811yescharacteristic
154C23×C4—21no—
155AGL(1,ℤ/8)—84no—
156C2×SD16—41no—
157C2×SD16—41no—
158C2×SD16—41no—
159C2×SD16—41no—
160C2×D8—21no—
161C2×D8—21no—
162C2×D8—41no—
163C2×D8—41no—
164C2×M16—41no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—81no—
171C22×C8—81no—
172C22×C8C2×D411yescharacteristic
173C22×C8#4111yescharacteristic
174C4⋊Q8—21no—
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177#127—21no—
178#125—41no—
179#124—41no—
180C22⋊Q8—41no—
181C22⋊Q8—41no—
182#121—41no—
183#121—41no—
184C4×Q8—41no—
185C4×Q8—41no—
186C4×D4—21no—
187C4×D4—21no—
188C4×D4—41no—
189C4×D4—41no—
190C4×D4D811yescharacteristic
191C4×D4D811yescharacteristic
192C4×D4C2×D411yescharacteristic
193C4×D4C2×D411yescharacteristic
194#117—41no—
195C2×C4⋊C4D811yescharacteristic
196C2×C22⋊C4—21no—
197C2×C42C2×D411yescharacteristic
198M32—81no—
199M32—81no—
200C2×C16—21no—
201C2×C16—21no—
202C2×C16—81no—
203C2×C16—81no—
204C2×C16C2×D411yescharacteristic
205C2×C16C2×D411yescharacteristic
206C2×C16C2×D411yescharacteristic
207C2×C16C2×D411yescharacteristic
208#107—21no—
209#107—21no—
210#107—21no—
211#107—21no—
212#107—21no—
213#107—21no—
214#106—41no—
215#106—41no—
216#106—41no—
217#106—41no—
218C4⋊C8—21no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—41no—
224C4⋊C8C2×D411yescharacteristic
225C4⋊C8C2×D411yescharacteristic
226#103—41no—
227#103—41no—
228#103—41no—
229#103—41no—
230#102—41no—
231#102—41no—
232#102—41no—
233#102—41no—
234#102—41no—
235#102—41no—
236C22⋊C8—21no—
237C22⋊C8—21no—
238C22⋊C8—21no—
239C22⋊C8—21no—
240C22⋊C8—41no—
241C4×C8—41no—
242C4×C8C2×D411yescharacteristic
243C4×C8C2×D411yescharacteristic
244C4×C8#4111yescharacteristic
245C4×C8C2411yescharacteristic
246C24—21no—
247#41—82no—
248#41—82no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×D4—22no—
253C2×D4—21no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—81no—
259C2×D4C2×D811yescharacteristic
260C2×D4C2×D811yescharacteristic
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—81no—
267C22×C4—81no—
268C22×C4—81no—
269C22×C4#12111yescharacteristic
270C22×C4C2×D811yescharacteristic
271C22×C4#13511yescharacteristic
272SD16—82no—
273SD16—82no—
274SD16—82no—
275SD16—82no—
276D8—41no—
277D8—41no—
278D8—81no—
279D8—82no—
280D8—82no—
281M16—82no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—81no—
296C2×C8—81no—
297C2×C8—81no—
298C2×C8#12111yescharacteristic
299C2×C8#12111yescharacteristic
300C2×C8C22×D411yescharacteristic
301C2×C8C2×#4111yescharacteristic
302C4⋊C4—21no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—81no—
310C4⋊C4—81no—
311C4⋊C4C2×D811yescharacteristic
312C4⋊C4C2×D811yescharacteristic
313C22⋊C4—21no—
314C22⋊C4—21no—
315C22⋊C4—21no—
316C22⋊C4—21no—
317C22⋊C4—81no—
318C22⋊C4—81no—
319C22⋊C4—81no—
320C42—81no—
321C42#12111yescharacteristic
322C42C2×D811yescharacteristic
323C42#13511yescharacteristic
324C42C22×D411yescharacteristic
325C16—21no—
326C16—21no—
327C16—21no—
328C16—21no—
329C16—41no—
330C16—41no—
331C23—21no—
332C23—21no—
333C23—42no—
334C23—42no—
335C23—81no—
336C23—81no—
337C23#46611yescharacteristic
338Q8—81no—
339Q8—81no—
340Q8—81no—
341Q8—81no—
342Q8—161no—
343D4—21no—
344D4—21no—
345D4—21no—
346D4—42no—
347D4—81no—
348D4—81no—
349D4—81no—
350D4—81no—
351D4—161no—
352D4—162no—
353D4#50912yes—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—162no—
372C2×C4#46611yescharacteristic
373C2×C4#46611yescharacteristic
374C2×C4#46611yescharacteristic
375C2×C4C2×#12111yescharacteristic
376C2×C4C22×D811yescharacteristic
377C2×C4C2×#13511yescharacteristic
378C8—21no—
379C8—21no—
380C8—21no—
381C8—41no—
382C8—41no—
383C8—81no—
384C8—81no—
385C8—81no—
386C8—81no—
387C8#57711yescharacteristic
388C8#57711yescharacteristic
389C22—21no—
390C22—22no—
391C22—21no—
392C22—21no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—84no—
399C22—161no—
400C22—161no—
401C22—161no—
402C22C2×#46611yescharacteristic
403C4—21no—
404C4—21no—
405C4—21no—
406C4—21no—
407C4—41no—
408C4—41no—
409C4—161no—
410C4—161no—
411C4—161no—
412C4#304211yescharacteristic
413C4#306911yescharacteristic
414C4C2×#50911yescharacteristic
415C4#341411yescharacteristic
416C2—41no—
417C2—42no—
418C2—81no—
419C2—321no—
420C2#1939211yescharacteristic
421C2#3360511yescharacteristic
422C2#3361711yescharacteristic
subgroupG/H
center, socleC22C2×#466
commutator, Frattini, cosocleC4×C8C24
prime radical#19255C2
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C8×D4 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33794C2semidirect
2. #33488C2semidirect
4. #19105C2semidirect
5. #12472C2semidirect
6. #12471C2semidirect
7. #12468C2semidirect
8. #12467C2semidirect
9. #12456C2semidirect
10. #12454C2semidirect
11. #12453C2semidirect
12. #12451C2semidirect
13. #12036C2semidirect
14. #11976C2semidirect
15. #11951C2semidirect
21. #2254C22semidirect
22. #2252C22semidirect
23. #2240C22semidirect
24. #2240C22semidirect
55. #1372C22semidirect
56. #1372C22semidirect
57. #1371C22semidirect
58. #1371C22semidirect
59. #1365C22semidirect
60. #1365C22semidirect
61. #1364C22semidirect
62. #1364C22semidirect
63. #1364C22semidirect
64. #1364C22semidirect
65. #1361C22semidirect
66. #1361C22semidirect

Maximal subgroups

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

subgroupindex
#337942
#334882
#192552
#191052
#124722
#124712
#124682
#124672
#124562
#124542
#124532
#124512
#120362
#119762
#119512