Finite Group Databasebeta

← #483949

Subgroups — #483949

Order 512.

433 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33737C211yesmaximal, characteristic
2#17973C211yesmaximal, characteristic, complemented
3#17970C211yesmaximal, characteristic, complemented
4#12086C211yesmaximal, characteristic, complemented
5#12084C211yesmaximal, characteristic, complemented
6#12082C211yesmaximal, characteristic, complemented
7#12081C211yesmaximal, characteristic, complemented
8#12045C211yesmaximal, characteristic, complemented
9#12043C211yesmaximal, characteristic, complemented
10C8×SD32C211yesmaximal, characteristic, complemented
11C8×D16C211yesmaximal, characteristic, complemented
12#12025C211yesmaximal, characteristic, complemented
13#11673C211yesmaximal, characteristic, complemented
14#11671C211yesmaximal, characteristic, complemented
15#11399C211yesmaximal, characteristic, complemented
16#3169C2211yescharacteristic
17#3168C2211yescharacteristic
18#2993—21no—
19#2981—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
32C4×M32C2211yescharacteristic
33C2×C4×C16C2211yescharacteristic
34#1594—21no—
35#1592—21no—
36#1591C2211yescharacteristic
37#1589C2211yescharacteristic
38#1589C2211yescharacteristic
39#1589C2211yescharacteristic
40#1580—21no—
41SD16⋊C8—21no—
42C8×Q16C2211yescharacteristic
43C8×D8C2211yescharacteristic
44C8×D8C2211yescharacteristic
45C8×D8C2211yescharacteristic
46#1568C2211yescharacteristic
47#1567C2211yescharacteristic
48#1496—21no—
49#1495—21no—
50#1450C2211yescharacteristic
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#1311C2211yescharacteristic
64#1311C2211yescharacteristic
65C8×C16C2211yescharacteristic
66C8×C16C2211yescharacteristic
67#532—21no—
68#529—21no—
69C2×SD32C2×C411yescharacteristic
70C2×SD32C2×C411yescharacteristic
71C2×D16C2×C411yescharacteristic
72C2×D16C2×C411yescharacteristic
73C2×M32C2×C411yescharacteristic
74C2×M32C2×C411yescharacteristic
75C22×C16—21no—
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#432—21no—
108#431D411yescharacteristic
109#431D411yescharacteristic
110#424—41no—
111#423—21no—
112#408—41no—
113#408—41no—
114C2×C4×C8C2311yescharacteristic
115#367C2×C411yescharacteristic
116#367C2×C411yescharacteristic
117#366C2×C411yescharacteristic
118#366C2×C411yescharacteristic
119#358C2×C411yescharacteristic
120#358C2×C411yescharacteristic
121#357C2×C411yescharacteristic
122#357C2×C411yescharacteristic
123#357C2×C411yescharacteristic
124#357C2×C411yescharacteristic
125#357C2×C411yescharacteristic
126#357C2×C411yescharacteristic
127#346C2×C411yescharacteristic
128#346C2×C411yescharacteristic
129C4×C16—21no—
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
159C2×M16—41no—
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—22no—
190D16—22no—
191D16—22no—
192M32—22no—
193M32—22no—
194C2×C16—24no—
195C2×C16—21no—
196C2×C16—21no—
197C2×C16C22×C411yescharacteristic
198C2×C16C22×C411yescharacteristic
199C2×C16C22×C411yescharacteristic
200C2×C16C22×C411yescharacteristic
201#107—21no—
202#107C22×C411yescharacteristic
203#107C22×C411yescharacteristic
204#107C22×C411yescharacteristic
205#107C22×C411yescharacteristic
206#106—21no—
207C4⋊C8—21no—
208C4⋊C8—21no—
209C4⋊C8—21no—
210C4⋊C8—21no—
211C4⋊C8—21no—
212C4⋊C8—41no—
213C4⋊C8—41no—
214C4⋊C8—41no—
215C4⋊C8D811yescharacteristic
216C4⋊C8D811yescharacteristic
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—
233C22⋊C8D811yescharacteristic
234C22⋊C8D811yescharacteristic
235#97—41no—
236#97—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—161no—
247#41—161no—
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—42no—
267D8—21no—
268D8—21no—
269D8—21no—
270D8—21no—
271D8—21no—
272D8—21no—
273D8—42no—
274M16—161no—
275M16—161no—
276M16—161no—
277M16—161no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—22no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—161no—
288C2×C8—161no—
289C2×C8—161no—
290C2×C8C4×D411yescharacteristic
291C2×C8C4×D411yescharacteristic
292C2×C8C2×D811yescharacteristic
293C2×C8C2×D811yescharacteristic
294C2×C8C2×D811yescharacteristic
295C2×C8C2×D811yescharacteristic
296C2×C8C23×C411yescharacteristic
297C2×C8C2×#4111yescharacteristic
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C42—41no—
317C42C4×D411yescharacteristic
318C42C2×D811yescharacteristic
319C42#13511yescharacteristic
320C42C22×D411yescharacteristic
321C16—22no—
322C16—22no—
323C16—21no—
324C16—21no—
325C16#13112yes—
326C16#13112yes—
327C23—41no—
328C23—41no—
329C23—41no—
330C23C4×D811yescharacteristic
331Q8—41no—
332Q8—41no—
333Q8—161no—
334D4—41no—
335D4—41no—
336D4—41no—
337D4—41no—
338D4—41no—
339D4—41no—
340D4—161no—
341D4—161no—
342D4—161no—
343D4—161no—
344D4—161no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4C4×D811yescharacteristic
362C2×C4C4×D811yescharacteristic
363C2×C4C4×D811yescharacteristic
364C2×C4C2×C4×D411yescharacteristic
365C2×C4C22×D811yescharacteristic
366C2×C4C2×#13511yescharacteristic
367C8—21no—
368C8—21no—
369C8—21no—
370C8—21no—
371C8—21no—
372C8—21no—
373C8—41no—
374C8—41no—
375C8—81no—
376C8—81no—
377C8—81no—
378C8—81no—
379C8#44312yes—
380C8C2×#13111yescharacteristic
381C8C2×#13111yescharacteristic
382C22—21no—
383C22—21no—
384C22—21no—
385C22—81no—
386C22—81no—
387C22—81no—
388C22—81no—
389C22—81no—
390C22—81no—
391C22—81no—
392C22—81no—
393C22—81no—
394C22C2×C4×D811yescharacteristic
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—81no—
400C4—81no—
401C4—81no—
402C4—81no—
403C4—81no—
404C4#293111yescharacteristic
405C4C2×#44311yescharacteristic
406C4#341511yescharacteristic
407C4#341611yescharacteristic
408C2—41no—
409C2—161no—
410C2—161no—
411C2—161no—
412C2#1795611yescharacteristic
413C2#3345411yescharacteristic
414C2#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 ◁ #431 ◁ #1450 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17973C2semidirect
3. #17970C2semidirect
4. #12086C2semidirect
5. #12084C2semidirect
6. #12082C2semidirect
7. #12081C2semidirect
8. #12045C2semidirect
9. #12043C2semidirect
10. C8×SD32C2semidirect
11. C8×D16C2semidirect
12. #12025C2semidirect
13. #11673C2semidirect
14. #11671C2semidirect
15. #11399C2semidirect

Maximal subgroups

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

subgroupindex
#337372
#179732
#179702
#120862
#120842
#120822
#120812
#120452
#120432
C8×SD322
C8×D162
#120252
#116732
#116712
#113992