Finite Group Databasebeta

← #484141

Subgroups — #484141

Order 512.

414 conjugacy classes of subgroups.

The subgroup lattice has 407 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33746C211yesmaximal, characteristic
2#18091C211yesmaximal, characteristic, complemented
3#18075C211yesmaximal, characteristic, complemented
4#12089C211yesmaximal, characteristic, complemented
5#12087C211yesmaximal, characteristic, complemented
6#12080C211yesmaximal, characteristic, complemented
7#12078C211yesmaximal, characteristic, complemented
8#12078C211yesmaximal, characteristic, complemented
9#12077C211yesmaximal, characteristic, complemented
10#12067C211yesmaximal, characteristic, complemented
11#12065C211yesmaximal, characteristic, complemented
12#11999C211yesmaximal, characteristic, complemented
13#11656C211yesmaximal, characteristic, complemented
14#11652C211yesmaximal, characteristic, complemented
15#11413C211yesmaximal, characteristic, complemented
16#3169C2211yescharacteristic
17#3168C2211yescharacteristic
18#2979—21no—
19#2969—21no—
20#2255C411yescharacteristic
21#2250C411yescharacteristic
22#2244C411yescharacteristic
23#2243C411yescharacteristic
24#2217C411yescharacteristic
25#2217C411yescharacteristic
26#2215C411yescharacteristic
27#2215C411yescharacteristic
28#2177C2211yescharacteristic
29#2177C2211yescharacteristic
30#2175C2211yescharacteristic
31#2175C2211yescharacteristic
32#2163C2211yescharacteristic
33C2×#346C2211yescharacteristic
34#1594C2211yescharacteristic
35#1594C2211yescharacteristic
36#1594C2211yescharacteristic
37#1593C2211yescharacteristic
38#1588—21no—
39#1585—21no—
40#1581—21no—
41#1581—21no—
42#1580C2211yescharacteristic
43#1580C2211yescharacteristic
44#1580C2211yescharacteristic
45#1579C2211yescharacteristic
46#1564C2211yescharacteristic
47#1564C2211yescharacteristic
48#1484—21no—
49#1482—21no—
50#1451C2211yescharacteristic
51#1368C2211yescharacteristic
52#1368C2211yescharacteristic
53#1368C2211yescharacteristic
54#1368C2211yescharacteristic
55#1334C2211yescharacteristic
56#1334C2211yescharacteristic
57#1333C2211yescharacteristic
58#1333C2211yescharacteristic
59#1333C2211yescharacteristic
60#1333C2211yescharacteristic
61#1333C2211yescharacteristic
62#1333C2211yescharacteristic
63#1313C2211yescharacteristic
64#1313C2211yescharacteristic
65#1313C2211yescharacteristic
66#1313C2211yescharacteristic
67#532—21no—
68#529—21no—
69C2×SD32C2×C411yescharacteristic
70C2×SD32C2×C411yescharacteristic
71C2×D16C2×C411yescharacteristic
72C2×D16C2×C411yescharacteristic
73C22×C16C2×C411yescharacteristic
74C22×C16C2×C411yescharacteristic
75#500C2×C411yescharacteristic
76#499C2×C411yescharacteristic
77#495C2×C411yescharacteristic
78#493C2×C411yescharacteristic
79#488—21no—
80#486—21no—
81#467C2×C411yescharacteristic
82#466—21no—
83#466C2×C411yescharacteristic
84#466C2×C411yescharacteristic
85#466C2×C411yescharacteristic
86#465—21no—
87#446—21no—
88C8×Q8—21no—
89C4×Q16C2311yescharacteristic
90C4×SD16—21no—
91C4×D8—21no—
92C4×D8C2311yescharacteristic
93C4×D8C2311yescharacteristic
94C4×D8C2311yescharacteristic
95#436—21no—
96#436—21no—
97#436—21no—
98#436—41no—
99#436—41no—
100#435—41no—
101C8×D4—21no—
102C8×D4—21no—
103C8×D4—21no—
104C8×D4—41no—
105#433—41no—
106#431—21no—
107#431—21no—
108#431D411yescharacteristic
109#408—41no—
110#408—41no—
111#405—41no—
112C4×M16D411yescharacteristic
113C2×C4×C8C2311yescharacteristic
114#367C2×C411yescharacteristic
115#367C2×C411yescharacteristic
116#366C2×C411yescharacteristic
117#366C2×C411yescharacteristic
118C4⋊C16—21no—
119#358C2×C411yescharacteristic
120#358C2×C411yescharacteristic
121#357C2×C411yescharacteristic
122#357C2×C411yescharacteristic
123#357C2×C411yescharacteristic
124#357C2×C411yescharacteristic
125#357C2×C411yescharacteristic
126#357C2×C411yescharacteristic
127C22⋊C16—21no—
128#346C2×C411yescharacteristic
129#346C2×C411yescharacteristic
130#346C2311yescharacteristic
131#346C2311yescharacteristic
132#346C2311yescharacteristic
133#346C2311yescharacteristic
134#335—21no—
135#335—21no—
136#334C2311yescharacteristic
137#334C2311yescharacteristic
138#334C2311yescharacteristic
139#334C2311yescharacteristic
140Q8⋊C8—21no—
141Q8⋊C8—21no—
142D4⋊C8—21no—
143D4⋊C8—21no—
144D4⋊C8—21no—
145D4⋊C8—21no—
146D4⋊C8—21no—
147D4⋊C8—21no—
148#322D411yescharacteristic
149#322D411yescharacteristic
150#322C2311yescharacteristic
151#322C2311yescharacteristic
152C2×#41—41no—
153C2×Q16C22×C411yescharacteristic
154C2×SD16—21no—
155C2×D8—21no—
156C2×D8C22×C411yescharacteristic
157C2×D8C22×C411yescharacteristic
158C2×D8C22×C411yescharacteristic
159C2×M16—21no—
160C2×M16—41no—
161C2×M16—41no—
162C2×M16—41no—
163C22×C8—41no—
164C22×C8—41no—
165C22×C8—41no—
166C22×C8C22×C411yescharacteristic
167C22×C8#4111yescharacteristic
168C4⋊Q8—21no—
169#127—21no—
170#126—41no—
171#125—21no—
172#124—21no—
173#124—41no—
174#123—41no—
175C22⋊Q8—21no—
176#121—21no—
177#121—21no—
178#121—21no—
179#121—41no—
180C4×Q8—21no—
181C4×D4—21no—
182C4×D4—21no—
183C4×D4—21no—
184C4×D4—41no—
185C4×D4—41no—
186#117—41no—
187C2×C42C2×D411yescharacteristic
188SD32—22no—
189SD32—41no—
190D16—22no—
191D16—41no—
192C2×C16—22no—
193C2×C16—21no—
194C2×C16—21no—
195C2×C16—41no—
196C2×C16C22×C411yescharacteristic
197C2×C16C22×C411yescharacteristic
198C2×C16C22×C411yescharacteristic
199C2×C16C22×C411yescharacteristic
200#107—21no—
201#107C22×C411yescharacteristic
202#107C22×C411yescharacteristic
203#107C22×C411yescharacteristic
204#107C22×C411yescharacteristic
205#106—21no—
206C4⋊C8—21no—
207C4⋊C8—21no—
208C4⋊C8—21no—
209C4⋊C8—21no—
210C4⋊C8—21no—
211C4⋊C8—41no—
212C4⋊C8—41no—
213#103—21no—
214#103—21no—
215#102—21no—
216#102—21no—
217#102—21no—
218#102—21no—
219#102—21no—
220#102—21no—
221C22⋊C8—21no—
222C22⋊C8—41no—
223C22⋊C8—41no—
224C22⋊C8—41no—
225C22⋊C8—41no—
226C22⋊C8—41no—
227C22⋊C8—41no—
228C22⋊C8—41no—
229C22⋊C8—41no—
230#97—21no—
231#97—41no—
232C4×C8—41no—
233C4×C8—41no—
234C4×C8—41no—
235C4×C8C22×C411yescharacteristic
236C4×C8C2×D411yescharacteristic
237C4×C8C2×D411yescharacteristic
238C4×C8C2×D411yescharacteristic
239C4×C8C2×D411yescharacteristic
240C4×C8C2×D411yescharacteristic
241C4×C8#4111yescharacteristic
242C4×C8C2411yescharacteristic
243#41—161no—
244#41—161no—
245C2×Q8—21no—
246C2×Q8—41no—
247C2×D4—21no—
248C2×D4—21no—
249C2×D4—21no—
250C2×D4—41no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4C4×D411yescharacteristic
259C22×C4AGL(1,ℤ/8)11yescharacteristic
260C22×C4AGL(1,ℤ/8)11yescharacteristic
261Q16—21no—
262Q16—21no—
263SD16—81no—
264D8—21no—
265D8—21no—
266D8—21no—
267D8—21no—
268D8—21no—
269D8—21no—
270D8—81no—
271M16—81no—
272M16—82no—
273M16—161no—
274M16—161no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—82no—
287C2×C8—161no—
288C2×C8—161no—
289C2×C8C4×D411yescharacteristic
290C2×C8C4×D411yescharacteristic
291C2×C8C23×C411yescharacteristic
292C2×C8C2×#4111yescharacteristic
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—41no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C42—41no—
312C42C4×D411yescharacteristic
313C42AGL(1,ℤ/8)11yescharacteristic
314C42AGL(1,ℤ/8)11yescharacteristic
315C42C22×D411yescharacteristic
316C16—21no—
317C16—21no—
318C16—21no—
319C16—41no—
320C16—41no—
321C16#13112yes—
322C23—41no—
323C23—41no—
324C23—41no—
325C23#44211yescharacteristic
326Q8—41no—
327Q8—41no—
328Q8—161no—
329D4—41no—
330D4—41no—
331D4—41no—
332D4—41no—
333D4—41no—
334D4—41no—
335D4—82no—
336D4—161no—
337D4—161no—
338D4—161no—
339D4—161no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4#44211yescharacteristic
356C2×C4#44211yescharacteristic
357C2×C4#44211yescharacteristic
358C2×C4C2×C4×D411yescharacteristic
359C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
360C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
361C8—21no—
362C8—21no—
363C8—21no—
364C8—41no—
365C8—41no—
366C8—41no—
367C8—41no—
368C8—81no—
369C8—81no—
370C8—81no—
371C8—81no—
372C8C2×#13111yescharacteristic
373C8#56811yescharacteristic
374C22—21no—
375C22—21no—
376C22—21no—
377C22—81no—
378C22—81no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22C2×#44211yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—81no—
393C4—81no—
394C4—81no—
395C4—81no—
396C4—81no—
397C4#293511yescharacteristic
398C4#295711yescharacteristic
399C2—41no—
400C2—161no—
401C2—161no—
402C2—161no—
403C2#1807311yescharacteristic
404C2#3346811yescharacteristic
405C2#3347111yescharacteristic
subgroupG/H
center, socleC22C2×#442
commutatorC2×C8C23×C4
Frattini, cosocleC4×C8C24
prime radical#33746C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1451 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #18091C2semidirect
3. #18075C2semidirect
4. #12089C2semidirect
5. #12087C2semidirect
6. #12080C2semidirect
7. #12078C2semidirect
8. #12078C2semidirect
9. #12077C2semidirect
10. #12067C2semidirect
11. #12065C2semidirect
12. #11999C2semidirect
13. #11656C2semidirect
14. #11652C2semidirect
15. #11413C2semidirect

Maximal subgroups

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

subgroupindex
#337462
#180912
#180752
#120892
#120872
#120802
#120782
#120772
#120672
#120652
#119992
#116562
#116522
#114132