Finite Group Databasebeta

← #491270

Subgroups — #491270

Order 512.

421 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33797C211yesmaximal, characteristic, complemented
2#33488C211yesmaximal, characteristic
3#19256C211yesmaximal, characteristic
4#19105C211yesmaximal, characteristic
5#12474C211yesmaximal, characteristic, complemented
6#12473C211yesmaximal, characteristic, complemented
7#12470C211yesmaximal, characteristic, complemented
8#12469C211yesmaximal, characteristic, complemented
9#12456C211yesmaximal, characteristic, complemented
10#12455C211yesmaximal, characteristic, complemented
11#12453C211yesmaximal, characteristic, complemented
12#12452C211yesmaximal, characteristic, complemented
13#12036C211yesmaximal, characteristic, complemented
14#11976C211yesmaximal, characteristic, complemented
15#11953C211yesmaximal, characteristic, complemented
16#3170C2211yescharacteristic
17#3160C2211yescharacteristic
18#3101—21no—
19#3077—21no—
20C2×C8×D4C2211yescharacteristic
21#2254C2211yescharacteristic, complemented
22#2252C2211yescharacteristic, complemented
23#2241C2211yescharacteristic, complemented
24#2241C2211yescharacteristic, complemented
25#2237—21no—
26#2237—21no—
27#2229—21no—
28#2229—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#1683—21no—
40#1683—21no—
41#1675C2211yescharacteristic
42#1675C2211yescharacteristic
43#1675C2211yescharacteristic
44#1675C2211yescharacteristic
45#1673C2211yescharacteristic
46#1673C2211yescharacteristic
47#1671—21no—
48#1671—21no—
49#1669C2211yescharacteristic
50#1669C2211yescharacteristic
51#1567C2211yescharacteristic
52#1564C2211yescharacteristic
53#1550—21no—
54#1542—21no—
55#1372C2211yescharacteristic, complemented
56#1372C2211yescharacteristic, complemented
57#1371C2211yescharacteristic, complemented
58#1371C2211yescharacteristic, complemented
59#1365C2211yescharacteristic, complemented
60#1365C2211yescharacteristic, complemented
61#1365C2211yescharacteristic, complemented
62#1365C2211yescharacteristic, complemented
63#1364C2211yescharacteristic, complemented
64#1364C2211yescharacteristic, complemented
65#1362C2211yescharacteristic, complemented
66#1362C2211yescharacteristic, complemented
67C23×C8—21no—
68#533—21no—
69#533—21no—
70C2×C4×D4D411yescharacteristic
71C2×M32—21no—
72C2×M32—21no—
73C22×C16—21no—
74C22×C16—21no—
75#500C2311yescharacteristic
76#500C2311yescharacteristic
77#500C2311yescharacteristic
78#499—21no—
79#499—21no—
80#498—21no—
81#494C2311yescharacteristic
82#487—21no—
83#483—41no—
84#478—41no—
85#478—41no—
86#476—21no—
87#476—21no—
88#476—21no—
89#476—21no—
90#474—21no—
91#474—21no—
92#467—21no—
93#467—21no—
94#461—21no—
95#461—21no—
96#450—41no—
97C4×Q16—21no—
98C4×Q16—21no—
99C4×SD16—41no—
100C4×SD16—41no—
101C8×D4—21no—
102C8×D4—21no—
103C8×D4D411yescharacteristic
104C8×D4D411yescharacteristic
105C8×D4C2311yescharacteristic
106C8×D4C2311yescharacteristic
107#431—21no—
108#428—41no—
109#427—21no—
110#427—21no—
111#423—21no—
112C2×C4⋊C8D411yescharacteristic
113#418—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#358—21no—
125#358—21no—
126#358—21no—
127#358—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#333—21no—
147#333—21no—
148#332—21no—
149#332—21no—
150#332—21no—
151#332—21no—
152C22×D4D811yescharacteristic
153C23×C4—21no—
154C2×Q16—21no—
155C2×Q16—21no—
156C2×SD16—41no—
157C2×SD16—41no—
158C2×M16—41no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—81no—
165C22×C8—81no—
166C22×C8C2×D411yescharacteristic
167C22×C8#4111yescharacteristic
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172#125—41no—
173#125—41no—
174C22⋊Q8—41no—
175C22⋊Q8—41no—
176C22⋊Q8—41no—
177C22⋊Q8—41no—
178C4×Q8—41no—
179C4×Q8—41no—
180C4×Q8—41no—
181C4×Q8—41no—
182C4×D4—21no—
183C4×D4—21no—
184C4×D4D811yescharacteristic
185C4×D4D811yescharacteristic
186C4×D4C2×D411yescharacteristic
187C4×D4C2×D411yescharacteristic
188#117—41no—
189#117—41no—
190C2×C4⋊C4D811yescharacteristic
191C2×C22⋊C4—21no—
192C2×C42C2×D411yescharacteristic
193M32—81no—
194M32—81no—
195C2×C16—21no—
196C2×C16—21no—
197C2×C16—81no—
198C2×C16—81no—
199C2×C16C2×D411yescharacteristic
200C2×C16C2×D411yescharacteristic
201C2×C16C2×D411yescharacteristic
202C2×C16C2×D411yescharacteristic
203#107—21no—
204#107—21no—
205#107—21no—
206#107—21no—
207#107—21no—
208#107—21no—
209#107—41no—
210#107—41no—
211#106—41no—
212#106—41no—
213#106—41no—
214#106—41no—
215#106—41no—
216#106—41no—
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—41no—
223C4⋊C8C2×D411yescharacteristic
224C4⋊C8C2×D411yescharacteristic
225#103—41no—
226#103—41no—
227#103—41no—
228#103—41no—
229#102—41no—
230#102—41no—
231#102—41no—
232#102—41no—
233#102—41no—
234#102—41no—
235C22⋊C8—21no—
236C22⋊C8—21no—
237C22⋊C8—21no—
238C22⋊C8—21no—
239C22⋊C8—41no—
240C4×C8—41no—
241C4×C8C2×D411yescharacteristic
242C4×C8C2×D411yescharacteristic
243C4×C8#4111yescharacteristic
244C4×C8C2411yescharacteristic
245C24—21no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×D4—22no—
251C2×D4—21no—
252C2×D4—41no—
253C2×D4C2×D811yescharacteristic
254C2×D4C2×D811yescharacteristic
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—81no—
261C22×C4—81no—
262C22×C4#12111yescharacteristic
263C22×C4C2×D811yescharacteristic
264C22×C4#13511yescharacteristic
265Q16—41no—
266Q16—41no—
267Q16—81no—
268SD16—82no—
269SD16—82no—
270M16—82no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—81no—
285C2×C8—81no—
286C2×C8—81no—
287C2×C8#12111yescharacteristic
288C2×C8#12111yescharacteristic
289C2×C8C22×D411yescharacteristic
290C2×C8C2×#4111yescharacteristic
291C4⋊C4—21no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—81no—
301C4⋊C4—81no—
302C4⋊C4—81no—
303C4⋊C4—81no—
304C4⋊C4C2×D811yescharacteristic
305C4⋊C4C2×D811yescharacteristic
306C22⋊C4—21no—
307C22⋊C4—21no—
308C22⋊C4—21no—
309C22⋊C4—21no—
310C22⋊C4—81no—
311C22⋊C4—81no—
312C42—81no—
313C42—81no—
314C42#12111yescharacteristic
315C42C2×D811yescharacteristic
316C42#13511yescharacteristic
317C42C22×D411yescharacteristic
318C16—21no—
319C16—21no—
320C16—21no—
321C16—21no—
322C16—41no—
323C16—41no—
324C23—21no—
325C23—21no—
326C23—42no—
327C23—42no—
328C23—81no—
329C23#46611yescharacteristic
330Q8—81no—
331Q8—81no—
332Q8—81no—
333Q8—81no—
334Q8—161no—
335Q8—161no—
336D4—21no—
337D4—21no—
338D4—21no—
339D4—42no—
340D4#51012yes—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4#46611yescharacteristic
360C2×C4#46611yescharacteristic
361C2×C4#46611yescharacteristic
362C2×C4C2×#12111yescharacteristic
363C2×C4C22×D811yescharacteristic
364C2×C4C2×#13511yescharacteristic
365C8—21no—
366C8—21no—
367C8—21no—
368C8—41no—
369C8—41no—
370C8—81no—
371C8—81no—
372C8—81no—
373C8—81no—
374C8#57711yescharacteristic
375C8#57711yescharacteristic
376C22—21no—
377C22—22no—
378C22—21no—
379C22—21no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—84no—
386C22C2×#46611yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—41no—
392C4—41no—
393C4—161no—
394C4—161no—
395C4—161no—
396C4—161no—
397C4#304211yescharacteristic
398C4#306911yescharacteristic
399C4C2×#51011yescharacteristic
400C4#341411yescharacteristic
401C2—41no—
402C2—42no—
403C2—81no—
404C2#1939211yescharacteristic
405C2#3360611yescharacteristic
406C2#3361711yescharacteristic
subgroupG/H
center, socleC22C2×#466
commutator, Frattini, cosocleC4×C8C24
prime radicalC22×D4D8
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. #33797C2semidirect
5. #12474C2semidirect
6. #12473C2semidirect
7. #12470C2semidirect
8. #12469C2semidirect
9. #12456C2semidirect
10. #12455C2semidirect
11. #12453C2semidirect
12. #12452C2semidirect
13. #12036C2semidirect
14. #11976C2semidirect
15. #11953C2semidirect
21. #2254C22semidirect
22. #2252C22semidirect
23. #2241C22semidirect
24. #2241C22semidirect
55. #1372C22semidirect
56. #1372C22semidirect
57. #1371C22semidirect
58. #1371C22semidirect
59. #1365C22semidirect
60. #1365C22semidirect
61. #1365C22semidirect
62. #1365C22semidirect
63. #1364C22semidirect
64. #1364C22semidirect
65. #1362C22semidirect
66. #1362C22semidirect

Maximal subgroups

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

subgroupindex
#337972
#334882
#192562
#191052
#124742
#124732
#124702
#124692
#124562
#124552
#124532
#124522
#120362
#119762
#119532