Finite Group Databasebeta

← #484138

Subgroups — #484138

Order 512.

404 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33738C211yesmaximal, characteristic
2#18087C211yesmaximal, characteristic
3#18086C211yesmaximal, characteristic, complemented
4#12085C211yesmaximal, characteristic, complemented
5#12082C211yesmaximal, characteristic, complemented
6#12076C211yesmaximal, characteristic, complemented
7#12075C211yesmaximal, characteristic, complemented
8#12072C211yesmaximal, characteristic, complemented
9#12070C211yesmaximal, characteristic, complemented
10#12043C211yesmaximal, characteristic, complemented
11C8×Q32C211yesmaximal, characteristic, complemented
12#12009C211yesmaximal, characteristic, complemented
13#11647C211yesmaximal, characteristic
14#11646C211yesmaximal, characteristic, complemented
15#11409C211yesmaximal, characteristic, complemented
16#3170C2211yescharacteristic
17#3168C2211yescharacteristic
18#2994—21no—
19#2986—21no—
20#2255C411yescharacteristic
21#2251C411yescharacteristic
22#2245C411yescharacteristic
23#2243C411yescharacteristic
24#2220C411yescharacteristic
25#2220C411yescharacteristic
26#2219C411yescharacteristic
27#2219C411yescharacteristic
28C4×Q32C2211yescharacteristic
29C4×Q32C2211yescharacteristic
30C4×SD32C2211yescharacteristic
31C4×SD32C2211yescharacteristic
32#2162C2211yescharacteristic
33C4×M32C2211yescharacteristic
34#1591C2211yescharacteristic
35#1591C2211yescharacteristic
36#1591C2211yescharacteristic
37#1589C2211yescharacteristic
38#1588—21no—
39#1587—21no—
40#1587—21no—
41#1585—21no—
42C8×Q16C2211yescharacteristic
43C8×Q16C2211yescharacteristic
44C8×Q16C2211yescharacteristic
45C8×D8C2211yescharacteristic
46#1564C2211yescharacteristic
47C2×#335C2211yescharacteristic
48#1485—21no—
49#1483—21no—
50C8×M16C2211yescharacteristic
51#1372C2211yescharacteristic
52#1372C2211yescharacteristic
53#1371C2211yescharacteristic
54#1371C2211yescharacteristic
55#1332C2211yescharacteristic
56#1332C2211yescharacteristic
57#1332C2211yescharacteristic
58#1332C2211yescharacteristic
59#1332C2211yescharacteristic
60#1332C2211yescharacteristic
61#1331C2211yescharacteristic
62#1331C2211yescharacteristic
63#1312C2211yescharacteristic
64#1312C2211yescharacteristic
65#1311C2211yescharacteristic
66C8×C16C2211yescharacteristic
67#533—21no—
68#529—21no—
69C2×Q32C2×C411yescharacteristic
70C2×Q32C2×C411yescharacteristic
71C2×SD32C2×C411yescharacteristic
72C2×SD32C2×C411yescharacteristic
73C2×M32C2×C411yescharacteristic
74C2×M32C2×C411yescharacteristic
75#500C2×C411yescharacteristic
76#499C2×C411yescharacteristic
77#495C2×C411yescharacteristic
78#494C2×C411yescharacteristic
79#488—21no—
80#487—21no—
81#467—21no—
82#467C2×C411yescharacteristic
83#467C2×C411yescharacteristic
84#467C2×C411yescharacteristic
85#466C2×C411yescharacteristic
86#465—21no—
87#446—21no—
88#446—21no—
89#446—21no—
90#446—41no—
91C8×Q8—21no—
92C8×Q8—21no—
93C8×Q8—21no—
94C8×Q8—41no—
95C4×Q16—21no—
96C4×Q16C2311yescharacteristic
97C4×Q16C2311yescharacteristic
98C4×Q16C2311yescharacteristic
99C4×SD16—21no—
100C4×D8C2311yescharacteristic
101#436—21no—
102#435—41no—
103C8×D4—21no—
104C8×D4—41no—
105#433—41no—
106#433—41no—
107#433—41no—
108#431—21no—
109#431D411yescharacteristic
110#424—41no—
111C2×C4⋊C8—21no—
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#358C2×C411yescharacteristic
122#358C2×C411yescharacteristic
123#358C2×C411yescharacteristic
124#358C2×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—
142Q8⋊C8—21no—
143Q8⋊C8—21no—
144Q8⋊C8—21no—
145Q8⋊C8—21no—
146D4⋊C8—21no—
147D4⋊C8—21no—
148#322D411yescharacteristic
149#322D411yescharacteristic
150C82C2311yescharacteristic
151C82C2311yescharacteristic
152C2×Q16—21no—
153C2×Q16C22×C411yescharacteristic
154C2×Q16C22×C411yescharacteristic
155C2×Q16C22×C411yescharacteristic
156C2×SD16—21no—
157C2×D8C22×C411yescharacteristic
158C2×M16—21no—
159C2×M16—41no—
160C22×C8—41no—
161C22×C8—41no—
162C22×C8C22×C411yescharacteristic
163C22×C8#4111yescharacteristic
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166#126—41no—
167#125—21no—
168#125—41no—
169#124—21no—
170#123—41no—
171C22⋊Q8—21no—
172C22⋊Q8—21no—
173C22⋊Q8—21no—
174C22⋊Q8—41no—
175#121—21no—
176C4×Q8—21no—
177C4×Q8—21no—
178C4×Q8—21no—
179C4×Q8—41no—
180C4×D4—21no—
181C4×D4—41no—
182#117—41no—
183#117—41no—
184C2×C42C2×D411yescharacteristic
185Q32—22no—
186Q32—41no—
187SD32—22no—
188SD32—41no—
189M32—22no—
190M32—41no—
191C2×C16—21no—
192C2×C16—21no—
193C2×C16C22×C411yescharacteristic
194C2×C16C22×C411yescharacteristic
195C2×C16C22×C411yescharacteristic
196C2×C16C22×C411yescharacteristic
197#107—21no—
198#107C22×C411yescharacteristic
199#107C22×C411yescharacteristic
200#107C22×C411yescharacteristic
201#107C22×C411yescharacteristic
202#106—21no—
203C4⋊C8—21no—
204C4⋊C8—21no—
205C4⋊C8—21no—
206C4⋊C8—21no—
207C4⋊C8—41no—
208C4⋊C8—41no—
209C4⋊C8—41no—
210C4⋊C8—41no—
211C4⋊C8—41no—
212C4⋊C8—41no—
213C4⋊C8—41no—
214C4⋊C8D811yescharacteristic
215C4⋊C8D811yescharacteristic
216#103—21no—
217#103—21no—
218#103—21no—
219#103—21no—
220#103—21no—
221#103—21no—
222#102—21no—
223#102—21no—
224C22⋊C8—41no—
225C22⋊C8—41no—
226C22⋊C8—41no—
227C22⋊C8D811yescharacteristic
228C22⋊C8D811yescharacteristic
229#97—21no—
230#97—41no—
231#97—41no—
232#97—41no—
233C4×C8—41no—
234C4×C8—41no—
235C4×C8—41no—
236C4×C8—41no—
237C4×C8C22×C411yescharacteristic
238C4×C8C2×D411yescharacteristic
239C4×C8C2×D411yescharacteristic
240C4×C8C2×D411yescharacteristic
241C4×C8C2×D411yescharacteristic
242C4×C8C2×D411yescharacteristic
243C4×C8#4111yescharacteristic
244C4×C8C2411yescharacteristic
245C2×Q8—21no—
246C2×Q8—21no—
247C2×Q8—21no—
248C2×Q8—41no—
249C2×D4—21no—
250C2×D4—41no—
251C22×C4—41no—
252C22×C4C4×D411yescharacteristic
253C22×C4C2×D811yescharacteristic
254C22×C4#13511yescharacteristic
255Q16—21no—
256Q16—21no—
257Q16—21no—
258Q16—21no—
259Q16—21no—
260Q16—21no—
261Q16—81no—
262SD16—81no—
263D8—21no—
264D8—21no—
265M16—42no—
266M16—161no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—161no—
277C2×C8—161no—
278C2×C8C4×D411yescharacteristic
279C2×C8C4×D411yescharacteristic
280C2×C8C2×D811yescharacteristic
281C2×C8C2×D811yescharacteristic
282C2×C8C2×D811yescharacteristic
283C2×C8C2×D811yescharacteristic
284C2×C8C23×C411yescharacteristic
285C2×C8C2×#4111yescharacteristic
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—41no—
291C4⋊C4—41no—
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—41no—
301C22⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—41no—
307C22⋊C4—41no—
308C42—41no—
309C42—41no—
310C42—41no—
311C42C4×D411yescharacteristic
312C42C2×D811yescharacteristic
313C42#13511yescharacteristic
314C42C22×D411yescharacteristic
315C16—21no—
316C16—21no—
317C16—21no—
318C16—41no—
319C16—41no—
320C16#13112yes—
321C23—41no—
322C23C4×D811yescharacteristic
323Q8—41no—
324Q8—41no—
325Q8—41no—
326Q8—41no—
327Q8—41no—
328Q8—41no—
329Q8—82no—
330D4—41no—
331D4—41no—
332D4—82no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4C4×D811yescharacteristic
348C2×C4C4×D811yescharacteristic
349C2×C4C4×D811yescharacteristic
350C2×C4C2×C4×D411yescharacteristic
351C2×C4C22×D811yescharacteristic
352C2×C4C2×#13511yescharacteristic
353C8—22no—
354C8—21no—
355C8—21no—
356C8—21no—
357C8—21no—
358C8—21no—
359C8—21no—
360C8—21no—
361C8—41no—
362C8—81no—
363C8—81no—
364C8—81no—
365C8—81no—
366C8C2×#13111yescharacteristic
367C8#56811yescharacteristic
368C22—21no—
369C22—21no—
370C22—21no—
371C22—81no—
372C22—81no—
373C22—81no—
374C22C2×C4×D811yescharacteristic
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—81no—
380C4—81no—
381C4—81no—
382C4—81no—
383C4—81no—
384C4—81no—
385C4—81no—
386C4#293511yescharacteristic
387C4#295711yescharacteristic
388C4#341611yescharacteristic
389C4#341711yescharacteristic
390C2—41no—
391C2—161no—
392C2#1806611yescharacteristic
393C2#3345511yescharacteristic
394C2#3346311yescharacteristic
subgroupG/H
centerC2×C4C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC4×C8C24
socleC22C2×C4×D8
prime radical#466C2×C4
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ #431 ◁ C8×M16 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #18086C2semidirect
4. #12085C2semidirect
5. #12082C2semidirect
6. #12076C2semidirect
7. #12075C2semidirect
8. #12072C2semidirect
9. #12070C2semidirect
10. #12043C2semidirect
11. C8×Q32C2semidirect
12. #12009C2semidirect
14. #11646C2semidirect
15. #11409C2semidirect

Maximal subgroups

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

subgroupindex
#337382
#180872
#180862
#120852
#120822
#120762
#120752
#120722
#120702
#120432
C8×Q322
#120092
#116472
#116462
#114092