Finite Group Databasebeta

← #484140

Subgroups — #484140

Order 512.

433 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33746C211yesmaximal, characteristic
2#18093C211yesmaximal, characteristic, complemented
3#18073C211yesmaximal, 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
12C2×#1368C211yesmaximal, characteristic, complemented
13#11654C211yesmaximal, characteristic, complemented
14#11651C211yesmaximal, characteristic, complemented
15#11413C211yesmaximal, characteristic, complemented
16#3169C2211yescharacteristic
17#3168C2211yescharacteristic
18#2994—21no—
19#2935—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#1586—21no—
39#1586—21no—
40#1583—21no—
41#1582—21no—
42#1580C2211yescharacteristic
43#1580C2211yescharacteristic
44#1580C2211yescharacteristic
45#1579C2211yescharacteristic
46C2×#334C2211yescharacteristic
47C2×#334C2211yescharacteristic
48#1483—21no—
49#1480—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
67C2×#131—41no—
68#532—21no—
69#529—21no—
70C2×SD32C2×C411yescharacteristic
71C2×SD32C2×C411yescharacteristic
72C2×D16C2×C411yescharacteristic
73C2×D16C2×C411yescharacteristic
74C22×C16C2×C411yescharacteristic
75C22×C16C2×C411yescharacteristic
76#500C2×C411yescharacteristic
77#499C2×C411yescharacteristic
78#495C2×C411yescharacteristic
79#493C2×C411yescharacteristic
80#488—21no—
81#486—21no—
82#467C2×C411yescharacteristic
83#466—21no—
84#466C2×C411yescharacteristic
85#466C2×C411yescharacteristic
86#466C2×C411yescharacteristic
87#465—21no—
88#446—21no—
89C8×Q8—21no—
90C4×Q16C2311yescharacteristic
91C4×SD16—21no—
92C4×D8—21no—
93C4×D8C2311yescharacteristic
94C4×D8C2311yescharacteristic
95C4×D8C2311yescharacteristic
96#436—21no—
97#436—21no—
98#436—21no—
99#435—41no—
100#435—41no—
101#435—41no—
102C8×D4—21no—
103C8×D4—21no—
104C8×D4—21no—
105C8×D4—41no—
106#433—41no—
107#431D411yescharacteristic
108#424—41no—
109C2×C4⋊C8—21no—
110C2×C4⋊C8—21no—
111#408—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#334—21no—
135#334—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
159#131—84no—
160C2×M16—21no—
161C2×M16—41no—
162C2×M16—41no—
163C2×M16—41no—
164C22×C8—41no—
165C22×C8—41no—
166C22×C8—41no—
167C22×C8—41no—
168C22×C8—41no—
169C22×C8C22×C411yescharacteristic
170C22×C8#4111yescharacteristic
171C4⋊Q8—21no—
172#127—21no—
173#126—41no—
174#125—21no—
175#124—21no—
176#124—41no—
177#123—41no—
178C22⋊Q8—21no—
179#121—21no—
180#121—21no—
181#121—21no—
182#121—41no—
183C4×Q8—21no—
184C4×D4—21no—
185C4×D4—21no—
186C4×D4—21no—
187C4×D4—41no—
188C4×D4—41no—
189#117—41no—
190C2×C42C2×D411yescharacteristic
191SD32—22no—
192SD32—41no—
193D16—22no—
194D16—41no—
195C2×C16—22no—
196C2×C16—21no—
197C2×C16—21no—
198C2×C16—41no—
199C2×C16C22×C411yescharacteristic
200C2×C16C22×C411yescharacteristic
201C2×C16C22×C411yescharacteristic
202C2×C16C22×C411yescharacteristic
203#107—21no—
204#107C22×C411yescharacteristic
205#107C22×C411yescharacteristic
206#107C22×C411yescharacteristic
207#107C22×C411yescharacteristic
208#106—21no—
209C4⋊C8—21no—
210C4⋊C8—21no—
211C4⋊C8—21no—
212C4⋊C8—21no—
213C4⋊C8—21no—
214C4⋊C8—41no—
215C4⋊C8—41no—
216C4⋊C8—41no—
217C4⋊C8—41no—
218#103—21no—
219#103—21no—
220#102—21no—
221#102—21no—
222#102—21no—
223#102—21no—
224#102—21no—
225#102—21no—
226C22⋊C8—21no—
227C22⋊C8—41no—
228C22⋊C8—41no—
229C22⋊C8—41no—
230C22⋊C8—41no—
231C22⋊C8—41no—
232C22⋊C8—41no—
233#97—21no—
234#97—41no—
235C4×C8—41no—
236C4×C8C22×C411yescharacteristic
237C4×C8C2×D411yescharacteristic
238C4×C8C2×D411yescharacteristic
239C4×C8C2×D411yescharacteristic
240C4×C8C2×D411yescharacteristic
241C4×C8C2×D411yescharacteristic
242C4×C8#4111yescharacteristic
243C4×C8C2411yescharacteristic
244#41—82no—
245#41—82no—
246C2×Q8—21no—
247C2×Q8—41no—
248C2×D4—21no—
249C2×D4—21no—
250C2×D4—21no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4C4×D411yescharacteristic
260C22×C4AGL(1,ℤ/8)11yescharacteristic
261C22×C4AGL(1,ℤ/8)11yescharacteristic
262Q16—21no—
263Q16—21no—
264SD16—81no—
265D8—21no—
266D8—21no—
267D8—21no—
268D8—21no—
269D8—21no—
270D8—21no—
271D8—81no—
272M16—42no—
273M16—82no—
274M16—82no—
275M16—161no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—82no—
288C2×C8—82no—
289C2×C8—82no—
290C2×C8—161no—
291C2×C8—161no—
292C2×C8C4×D411yescharacteristic
293C2×C8C4×D411yescharacteristic
294C2×C8C23×C411yescharacteristic
295C2×C8C2×#4111yescharacteristic
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—41no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C42—41no—
315C42C4×D411yescharacteristic
316C42AGL(1,ℤ/8)11yescharacteristic
317C42AGL(1,ℤ/8)11yescharacteristic
318C42C22×D411yescharacteristic
319C16—21no—
320C16—21no—
321C16—21no—
322C16—41no—
323C16—41no—
324C16#13112yescomplemented
325C23—41no—
326C23—41no—
327C23—41no—
328C23#44211yescharacteristic
329Q8—41no—
330Q8—41no—
331Q8—82no—
332D4—41no—
333D4—41no—
334D4—41no—
335D4—41no—
336D4—41no—
337D4—41no—
338D4—82no—
339D4—82no—
340D4—82no—
341D4—82no—
342D4—161no—
343C2×C4—22no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—41no—
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#44211yescharacteristic
359C2×C4#44211yescharacteristic
360C2×C4#44211yescharacteristic
361C2×C4C2×C4×D411yescharacteristic
362C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
363C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
364C8—21no—
365C8—21no—
366C8—21no—
367C8—41no—
368C8—41no—
369C8—41no—
370C8—41no—
371C8—81no—
372C8—81no—
373C8—81no—
374C8—81no—
375C8C2×#13111yescharacteristic
376C8#56811yescharacteristic
377C22—21no—
378C22—21no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22—81no—
387C22—81no—
388C22AGL(1,ℤ/16)12yes—
389C22C2×#44211yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—81no—
396C4—81no—
397C4—81no—
398C4—81no—
399C4—81no—
400C4#293511yescharacteristic
401C4#295711yescharacteristic
402C2—22no—
403C2—161no—
404C2—161no—
405C2—161no—
406C2#1807311yescharacteristic
407C2C2×AGL(1,ℤ/16)11yescharacteristic
408C2#3346911yescharacteristic
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. #18093C2semidirect
3. #18073C2semidirect
4. #12089C2semidirect
5. #12087C2semidirect
6. #12080C2semidirect
7. #12078C2semidirect
8. #12078C2semidirect
9. #12077C2semidirect
10. #12067C2semidirect
11. #12065C2semidirect
12. C2×#1368C2semidirect
13. #11654C2semidirect
14. #11651C2semidirect
15. #11413C2semidirect
324. C16#131semidirect

Maximal subgroups

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

subgroupindex
#337462
#180932
#180732
#120892
#120872
#120802
#120782
#120772
#120672
#120652
C2×#13682
#116542
#116512
#114132