Finite Group Databasebeta

← #484133

Subgroups — #484133

Order 512.

440 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33737C211yesmaximal, characteristic
2#18069C211yesmaximal, characteristic, complemented
3#18066C211yesmaximal, characteristic, complemented
4#12084C211yesmaximal, characteristic, complemented
5#12081C211yesmaximal, characteristic, complemented
6#12074C211yesmaximal, characteristic, complemented
7#12073C211yesmaximal, characteristic, complemented
8#12071C211yesmaximal, characteristic, complemented
9#12069C211yesmaximal, characteristic, complemented
10#12045C211yesmaximal, characteristic, complemented
11C8×SD32C211yesmaximal, characteristic, complemented
12#12009C211yesmaximal, characteristic, complemented
13#11649C211yesmaximal, characteristic, complemented
14#11648C211yesmaximal, characteristic, complemented
15#11409C211yesmaximal, characteristic, complemented
16#3169C2211yescharacteristic
17#3168C2211yescharacteristic
18#2969—21no—
19#2935—21no—
20#2255C411yescharacteristic
21#2250C411yescharacteristic
22#2244C411yescharacteristic
23#2243C411yescharacteristic
24#2220C411yescharacteristic
25#2220C411yescharacteristic
26#2218C411yescharacteristic
27#2218C411yescharacteristic
28C4×SD32C2211yescharacteristic
29C4×SD32C2211yescharacteristic
30C4×D16C2211yescharacteristic
31C4×D16C2211yescharacteristic
32#2162C2211yescharacteristic
33C4×M32C2211yescharacteristic
34#1591C2211yescharacteristic
35#1589C2211yescharacteristic
36#1589C2211yescharacteristic
37#1589C2211yescharacteristic
38#1583—21no—
39#1582—21no—
40#1581—21no—
41#1581—21no—
42C8×Q16C2211yescharacteristic
43C8×D8C2211yescharacteristic
44C8×D8C2211yescharacteristic
45C8×D8C2211yescharacteristic
46#1564C2211yescharacteristic
47C2×#335C2211yescharacteristic
48#1482—21no—
49#1480—21no—
50C8×M16C2211yescharacteristic
51#1372C2211yescharacteristic
52#1372C2211yescharacteristic
53#1371C2211yescharacteristic
54#1371C2211yescharacteristic
55#1332C2211yescharacteristic
56#1332C2211yescharacteristic
57#1331C2211yescharacteristic
58#1331C2211yescharacteristic
59#1331C2211yescharacteristic
60#1331C2211yescharacteristic
61#1331C2211yescharacteristic
62#1331C2211yescharacteristic
63#1312C2211yescharacteristic
64#1312C2211yescharacteristic
65#1311C2211yescharacteristic
66C8×C16C2211yescharacteristic
67C2×#131—41no—
68#532—21no—
69#529—21no—
70C2×SD32C2×C411yescharacteristic
71C2×SD32C2×C411yescharacteristic
72C2×D16C2×C411yescharacteristic
73C2×D16C2×C411yescharacteristic
74C2×M32C2×C411yescharacteristic
75C2×M32C2×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#436—41no—
100#435—41no—
101#435—41no—
102#435—41no—
103C8×D4—21no—
104C8×D4—21no—
105C8×D4—21no—
106#433—41no—
107#431—21no—
108#431D411yescharacteristic
109C2×C4⋊C8—21no—
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
130C4×C16C2311yescharacteristic
131C4×C16C2311yescharacteristic
132C4×C16C2311yescharacteristic
133C4×C16C2311yescharacteristic
134#335—21no—
135#335C2311yescharacteristic
136#335C2311yescharacteristic
137#335C2311yescharacteristic
138#335C2311yescharacteristic
139#334—21no—
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
150C82C2311yescharacteristic
151C82C2311yescharacteristic
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—41no—
161C2×M16—41no—
162C2×M16—41no—
163C2×M16D811yescharacteristic
164C2×M16D811yescharacteristic
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—
195M32—22no—
196M32—41no—
197C2×C16—21no—
198C2×C16—21no—
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—
217#103—21no—
218#103—21no—
219#102—21no—
220#102—21no—
221#102—21no—
222#102—21no—
223#102—21no—
224#102—21no—
225C22⋊C8—21no—
226C22⋊C8—41no—
227C22⋊C8—41no—
228C22⋊C8—41no—
229C22⋊C8—41no—
230C22⋊C8—41no—
231C22⋊C8—41no—
232C22⋊C8—41no—
233#97—41no—
234#97D811yescharacteristic
235#97D811yescharacteristic
236C4×C8—41no—
237C4×C8—41no—
238C4×C8C22×C411yescharacteristic
239C4×C8C2×D411yescharacteristic
240C4×C8C2×D411yescharacteristic
241C4×C8C2×D411yescharacteristic
242C4×C8C2×D411yescharacteristic
243C4×C8C2×D411yescharacteristic
244C4×C8#4111yescharacteristic
245C4×C8C2411yescharacteristic
246#41—82no—
247#41—82no—
248C2×Q8—21no—
249C2×Q8—41no—
250C2×D4—21no—
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4C4×D411yescharacteristic
262C22×C4C2×D811yescharacteristic
263C22×C4#13511yescharacteristic
264Q16—21no—
265Q16—21no—
266SD16—81no—
267D8—21no—
268D8—21no—
269D8—21no—
270D8—21no—
271D8—21no—
272D8—21no—
273D8—81no—
274M16—22no—
275M16—41no—
276M16—82no—
277M16—82no—
278M16—82no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—82no—
289C2×C8—82no—
290C2×C8—82no—
291C2×C8—82no—
292C2×C8C4×D411yescharacteristic
293C2×C8C4×D411yescharacteristic
294C2×C8C2×D811yescharacteristic
295C2×C8C2×D811yescharacteristic
296C2×C8C2×D811yescharacteristic
297C2×C8C2×D811yescharacteristic
298C2×C8C23×C411yescharacteristic
299C2×C8C2×#4111yescharacteristic
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C42—41no—
319C42C4×D411yescharacteristic
320C42C2×D811yescharacteristic
321C42#13511yescharacteristic
322C42C22×D411yescharacteristic
323C16—21no—
324C16—21no—
325C16—21no—
326C16—41no—
327C16—41no—
328C16#13112yescomplemented
329C23—41no—
330C23—41no—
331C23—41no—
332C23C4×D811yescharacteristic
333Q8—41no—
334Q8—41no—
335Q8—82no—
336D4—41no—
337D4—41no—
338D4—41no—
339D4—41no—
340D4—41no—
341D4—41no—
342D4—82no—
343D4—82no—
344D4—82no—
345D4—161no—
346D4—161no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4C4×D811yescharacteristic
364C2×C4C4×D811yescharacteristic
365C2×C4C4×D811yescharacteristic
366C2×C4C2×C4×D411yescharacteristic
367C2×C4C22×D811yescharacteristic
368C2×C4C2×#13511yescharacteristic
369C8—21no—
370C8—21no—
371C8—21no—
372C8—21no—
373C8—21no—
374C8—21no—
375C8—41no—
376C8—41no—
377C8—81no—
378C8—81no—
379C8—81no—
380C8—81no—
381C8#50912yes—
382C8C2×#13111yescharacteristic
383C8#56811yescharacteristic
384C22—21no—
385C22—21no—
386C22—21no—
387C22—81no—
388C22—81no—
389C22—81no—
390C22—81no—
391C22—81no—
392C22—81no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22C2×C4×D811yescharacteristic
397C4—21no—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—81no—
402C4—81no—
403C4—81no—
404C4—81no—
405C4—81no—
406C4#293511yescharacteristic
407C4#295711yescharacteristic
408C4C2×#50911yescharacteristic
409C4#341411yescharacteristic
410C2—41no—
411C2—161no—
412C2—161no—
413C2—161no—
414C2#1806611yescharacteristic
415C2C4×#50911yescharacteristic
416C2#3346411yescharacteristic
subgroupG/H
centerC2×C4C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC4×C8C24
socleC22C2×C4×D8
prime radical#33737C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C4×M16 ◁ C8×M16 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #18069C2semidirect
3. #18066C2semidirect
4. #12084C2semidirect
5. #12081C2semidirect
6. #12074C2semidirect
7. #12073C2semidirect
8. #12071C2semidirect
9. #12069C2semidirect
10. #12045C2semidirect
11. C8×SD32C2semidirect
12. #12009C2semidirect
13. #11649C2semidirect
14. #11648C2semidirect
15. #11409C2semidirect
328. C16#131semidirect

Maximal subgroups

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

subgroupindex
#337372
#180692
#180662
#120842
#120812
#120742
#120732
#120712
#120692
#120452
C8×SD322
#120092
#116492
#116482
#114092