Finite Group Databasebeta

← #506172

Subgroups — #506172

Order 512.

471 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27840C211yesmaximal, characteristic
2#27792C211yesmaximal, characteristic
3#13530C211yesmaximal, characteristic, complemented
4#13492C211yesmaximal, characteristic
5#13452C211yesmaximal, characteristic, complemented
6#13382C211yesmaximal, characteristic
7#13282C211yesmaximal, characteristic, complemented
8#13150C211yesmaximal, characteristic, complemented
9#13148C211yesmaximal, characteristic, complemented
10#13132C211yesmaximal, characteristic, complemented
11#13109C211yesmaximal, characteristic, complemented
12#13062C211yesmaximal, characteristic
13#12836C211yesmaximal, characteristic
14#12674C211yesmaximal, characteristic, complemented
15#12612C211yesmaximal, characteristic
16C22×M32C2211yescharacteristic
17C2×#500C2211yescharacteristic
18C2×#467C2211yescharacteristic
19#2497—21no—
20#2492—21no—
21C2×#367C2211yescharacteristic
22C2×#367C2211yescharacteristic
23C2×#366C2211yescharacteristic
24C2×#366C2211yescharacteristic
25C2×#358C2211yescharacteristic
26C2×#358C2211yescharacteristic
27C2×#358C2211yescharacteristic
28C2×#358C2211yescharacteristic
29C2×C22⋊C16C2211yescharacteristic
30C2×#346C2211yescharacteristic
31C2×C4×C16C2211yescharacteristic
32#2100—21no—
33#2076—21no—
34#2036—21no—
35#2024—21no—
36#1969C2211yescharacteristic
37#1969C2211yescharacteristic
38#1942C2211yescharacteristic
39#1942C2211yescharacteristic
40#1928—21no—
41#1921—21no—
42#1845C2211yescharacteristic
43#1808C2211yescharacteristic
44#1808C2211yescharacteristic
45#1808C2211yescharacteristic
46C4×#107C2211yescharacteristic
47C4×#107C2211yescharacteristic
48C8×C22⋊C4C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×Q16C2311yescharacteristic
60C23×C8C2311yescharacteristic
61C2×C4⋊Q8—21no—
62C2×C22⋊Q8—21no—
63C2×M32—22no—
64C2×M32—22no—
65C2×M32—41no—
66C22×C16C2311yescharacteristic
67C22×C16C2311yescharacteristic
68C22×C16C2311yescharacteristic
69C22×C16C2311yescharacteristic
70#500—22no—
71#500—41no—
72#467—22no—
73#467—41no—
74C2×#107—21no—
75C2×#107—21no—
76C2×#107C2311yescharacteristic
77C2×#107C2311yescharacteristic
78C2×#107C2311yescharacteristic
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#103—21no—
84C2×#103—21no—
85C2×#103—21no—
86C2×C22⋊C8D411yescharacteristic
87C2×C4×C8C2311yescharacteristic
88C2×C4×C8C2311yescharacteristic
89#400—41no—
90#400—41no—
91#396—41no—
92#393—41no—
93#391—21no—
94#391—21no—
95#391—41no—
96#389—21no—
97#389—21no—
98#388—41no—
99#388—41no—
100#387—41no—
101#384—21no—
102#380—21no—
103#380—21no—
104#380—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C22⋊C4D411yescharacteristic
108#367—22no—
109#367—41no—
110#366—22no—
111#366—22no—
112#358—22no—
113#358—22no—
114#358—22no—
115#358—41no—
116C22⋊C16—22no—
117#346—22no—
118C4×C16—41no—
119#340—21no—
120#340—21no—
121#340—21no—
122#340—21no—
123#340—21no—
124#340—21no—
125#340—21no—
126#336D411yescharacteristic
127#336D411yescharacteristic
128C22×Q8—21no—
129C22×Q8—41no—
130C23×C4C2×D411yescharacteristic
131C2×Q16—21no—
132C2×Q16—21no—
133C2×Q16—21no—
134C2×Q16—21no—
135C2×Q16—21no—
136C2×Q16—21no—
137C22×C8—21no—
138C22×C8—21no—
139C22×C8—21no—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8C2×D411yescharacteristic
144C22×C8C2×D411yescharacteristic
145C22×C8C2×D411yescharacteristic
146C22×C8C2411yescharacteristic
147C4⋊Q8—42no—
148C4⋊Q8—81no—
149C22⋊Q8—42no—
150C22⋊Q8—81no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C22⋊C4—41no—
165C2×C22⋊C4—41no—
166C2×C22⋊C4—41no—
167C2×C22⋊C4—41no—
168C2×C22⋊C4D811yescharacteristic
169C2×C22⋊C4D811yescharacteristic
170C2×C42—41no—
171C2×C42—41no—
172C2×C42C2×D411yescharacteristic
173C2×C42C2×D411yescharacteristic
174M32—44no—
175C2×C16—21no—
176C2×C16—21no—
177C2×C16—21no—
178C2×C16—21no—
179C2×C16—21no—
180C2×C16—21no—
181C2×C16—21no—
182C2×C16—21no—
183C2×C16—21no—
184C2×C16#4112yes—
185C2×C16#4112yes—
186C2×C16#4112yes—
187#107—21no—
188#107—21no—
189#107—21no—
190#107—21no—
191#107—21no—
192#107—21no—
193#107—21no—
194#107—21no—
195#107—21no—
196#107—21no—
197#107—21no—
198#107—21no—
199#107—21no—
200#107—42no—
201#107—42no—
202#107#4112yes—
203#103—42no—
204#103—42no—
205#103—81no—
206C22⋊C8—22no—
207C4×C8—21no—
208C4×C8—21no—
209C4×C8—21no—
210C4×C8—21no—
211#95—41no—
212#95—41no—
213#95—41no—
214#95—41no—
215#95—41no—
216#95—41no—
217#95—41no—
218#95—41no—
219#95—41no—
220#95—41no—
221#95—41no—
222#95D811yescharacteristic
223#95D811yescharacteristic
224C24C2×D811yescharacteristic
225C2×Q8—41no—
226C2×Q8—41no—
227C2×Q8—41no—
228C2×Q8—41no—
229C2×Q8—41no—
230C2×Q8—41no—
231C2×Q8—82no—
232C2×Q8—82no—
233C2×Q8—161no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4C2×D811yescharacteristic
249C22×C4C2×D811yescharacteristic
250C22×C4C2×D811yescharacteristic
251C22×C4C2×D811yescharacteristic
252C22×C4C2×D811yescharacteristic
253C22×C4C22×D411yescharacteristic
254Q16—41no—
255Q16—41no—
256Q16—41no—
257Q16—41no—
258C2×C8—21no—
259C2×C8—21no—
260C2×C8—21no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8—41no—
269C2×C8—41no—
270C2×C8#13512yes—
271C2×C8C2×#4111yescharacteristic
272C2×C8C2×#4111yescharacteristic
273C2×C8C2×#4111yescharacteristic
274C2×C8C2×#4111yescharacteristic
275C2×C8H(5,𝔽2)11yescharacteristic
276C2×C82−1+411yescharacteristic
277C4⋊C4—22no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—82no—
292C4⋊C4—82no—
293C4⋊C4—82no—
294C4⋊C4—161no—
295C4⋊C4—161no—
296C4⋊C4—161no—
297C22⋊C4—22no—
298C22⋊C4—22no—
299C22⋊C4—82no—
300C22⋊C4—161no—
301C22⋊C4—161no—
302C22⋊C4—161no—
303C42—21no—
304C42—21no—
305C42—21no—
306C42—21no—
307C42—161no—
308C42—161no—
309C16—22no—
310C16—22no—
311C16—22no—
312C16—41no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23C22×D811yescharacteristic
321Q8—81no—
322Q8—81no—
323Q8—81no—
324Q8—81no—
325Q8—164no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4#51012yes—
368C2×C4#51012yes—
369C2×C4#53211yescharacteristic
370C2×C4#54611yescharacteristic
371C2×C4#54711yescharacteristic
372C2×C4C2×#13511yescharacteristic
373C2×C4#57711yescharacteristic
374C2×C4#57811yescharacteristic
375C8—22no—
376C8—41no—
377C8—41no—
378C8#52911yescharacteristic
379C8#54211yescharacteristic
380C8#55111yescharacteristic
381C8#55611yescharacteristic
382C22—41no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22#316911yescharacteristic
390C22#329411yescharacteristic
391C22#333411yescharacteristic
392C22C2×#51011yescharacteristic
393C22C2×#51011yescharacteristic
394C22#341511yescharacteristic
395C22#341611yescharacteristic
396C4—22no—
397C4—22no—
398C4—41no—
399C4—41no—
400C4—41no—
401C4—82no—
402C4—161no—
403C4—161no—
404C4—161no—
405C4—161no—
406C4—161no—
407C4—161no—
408C4—161no—
409C4#252111yescharacteristic
410C4#317811yescharacteristic
411C4#330011yescharacteristic
412C4#334011yescharacteristic
413C2—81no—
414C2#2806511yescharacteristic
415C2#3374411yescharacteristic
416C2#3374911yescharacteristic
417C2#3388111yescharacteristic
418C2#3388711yescharacteristic
419C2#3392311yescharacteristic
420C2#3392411yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #13530C2semidirect
5. #13452C2semidirect
7. #13282C2semidirect
8. #13150C2semidirect
9. #13148C2semidirect
10. #13132C2semidirect
11. #13109C2semidirect
14. #12674C2semidirect

Maximal subgroups

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

subgroupindex
#278402
#277922
#135302
#134922
#134522
#133822
#132822
#131502
#131482
#131322
#131092
#130622
#128362
#126742
#126122