Finite Group Databasebeta

← #510180

Subgroups — #510180

Order 512.

472 conjugacy classes of subgroups.

The subgroup lattice has 427 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26064C211yesmaximal, characteristic
2#25033C211yesmaximal, characteristic
3#13539C211yesmaximal, characteristic, complemented
4#13516C211yesmaximal, characteristic, complemented
5#13507C211yesmaximal, characteristic, complemented
6#13493C211yesmaximal, characteristic
7#13382C211yesmaximal, characteristic
8#13364C211yesmaximal, characteristic, complemented
9#13283C211yesmaximal, characteristic, complemented
10#13150C211yesmaximal, characteristic, complemented
11#13121C211yesmaximal, characteristic
12#12994C211yesmaximal, characteristic
13#12936C211yesmaximal, characteristic, complemented
14#12868C211yesmaximal, characteristic
15#12814C211yesmaximal, characteristic, complemented
16C2×#500C2211yescharacteristic
17C2×#467C2211yescharacteristic
18C2×C4×Q16C2211yescharacteristic
19C2×#427C2211yescharacteristic
20#2439—21no—
21#2421—21no—
22C2×#367C2211yescharacteristic
23C2×#366C2211yescharacteristic
24C2×C4⋊C16C2211yescharacteristic
25C2×#358C2211yescharacteristic
26C2×#358C2211yescharacteristic
27C2×#358C2211yescharacteristic
28C2×#358C2211yescharacteristic
29C2×C22⋊C16C2211yescharacteristic
30C2×C22⋊C16C2211yescharacteristic
31C2×#346C2211yescharacteristic
32#2085—21no—
33#2074—21no—
34#2058—21no—
35#2035—21no—
36#1969C2211yescharacteristic
37#1942C2211yescharacteristic
38#1942C2211yescharacteristic
39#1942C2211yescharacteristic
40#1865—21no—
41#1854—21no—
42#1846C2211yescharacteristic
43#1846C2211yescharacteristic
44#1808C2211yescharacteristic
45#1808C2211yescharacteristic
46C8×C22⋊C4C2211yescharacteristic
47#1380C2211yescharacteristic
48#1380C2211yescharacteristic
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×C4×Q8—21no—
64C2×#117—21no—
65C22×C16C2311yescharacteristic
66C22×C16C2311yescharacteristic
67C22×C16C2311yescharacteristic
68C22×C16C2311yescharacteristic
69#500—22no—
70#500—41no—
71#467—22no—
72#467—41no—
73C4×Q16—22no—
74C4×Q16—41no—
75#427—22no—
76#427—41no—
77C2×#107—21no—
78C2×#107C2311yescharacteristic
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#106—21no—
86C2×#106—21no—
87C2×#106—21no—
88C2×#103—21no—
89C2×#103—21no—
90C2×#103—21no—
91C2×C22⋊C8D411yescharacteristic
92C2×C4×C8C2311yescharacteristic
93C2×C4×C8C2311yescharacteristic
94#400—41no—
95#397—41no—
96#396—41no—
97#393—41no—
98#391—21no—
99#391—41no—
100#389—21no—
101#389—21no—
102#389—21no—
103#388—41no—
104#387—41no—
105#387—41no—
106#382—21no—
107#382—21no—
108#380—21no—
109#380—21no—
110C4×C22⋊C4D411yescharacteristic
111#367—41no—
112#366—41no—
113C4⋊C16—41no—
114#358—22no—
115#358—22no—
116#358—41no—
117#358—41no—
118C22⋊C16—41no—
119C22⋊C16—41no—
120#346—41no—
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#4112yes—
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—
151C4×Q8—42no—
152C4×Q8—81no—
153#117—42no—
154#117—81no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C22⋊C4—41no—
169C2×C22⋊C4—41no—
170C2×C22⋊C4—41no—
171C2×C22⋊C4—41no—
172C2×C22⋊C4D811yescharacteristic
173C2×C22⋊C4D811yescharacteristic
174C2×C42—41no—
175C2×C42—41no—
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C16—21no—
179C2×C16—21no—
180C2×C16—21no—
181C2×C16—21no—
182C2×C16—21no—
183C2×C16—21no—
184C2×C16—21no—
185C2×C16—21no—
186C2×C16—21no—
187C2×C16—21no—
188C2×C16—21no—
189C2×C16—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—21no—
201#107—21no—
202#107—21no—
203#107—42no—
204#107#4112yes—
205#106—42no—
206#106—81no—
207#106—81no—
208#103—42no—
209#103—42no—
210#103—42no—
211C22⋊C8—22no—
212C4×C8—21no—
213C4×C8—21no—
214C4×C8—21no—
215C4×C8—21no—
216#95—41no—
217#95—41no—
218#95—41no—
219#95—41no—
220#95—41no—
221#95—41no—
222#95—41no—
223#95—41no—
224#95—41no—
225#95—41no—
226#95—41no—
227#95D811yescharacteristic
228#95D811yescharacteristic
229C24C2×D811yescharacteristic
230C2×Q8—22no—
231C2×Q8—41no—
232C2×Q8—41no—
233C2×Q8—41no—
234C2×Q8—41no—
235C2×Q8—41no—
236C2×Q8—82no—
237C2×Q8—82no—
238C2×Q8—161no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4C2×D811yescharacteristic
254C22×C4C2×D811yescharacteristic
255C22×C4C2×D811yescharacteristic
256C22×C4C2×D811yescharacteristic
257C22×C4C2×D811yescharacteristic
258C22×C4C22×D411yescharacteristic
259Q16—22no—
260Q16—22no—
261Q16—41no—
262Q16—41no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—21no—
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#13712yes—
276C2×C8C2×#4111yescharacteristic
277C2×C8C2×#4111yescharacteristic
278C2×C8H(5,𝔽2)11yescharacteristic
279C2×C8H(5,𝔽2)11yescharacteristic
280C2×C82−1+411yescharacteristic
281C2×C82−1+411yescharacteristic
282C4⋊C4—22no—
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—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—82no—
297C4⋊C4—82no—
298C4⋊C4—82no—
299C4⋊C4—82no—
300C4⋊C4—82no—
301C4⋊C4—161no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—82no—
305C22⋊C4—82no—
306C22⋊C4—82no—
307C22⋊C4—161no—
308C42—21no—
309C42—21no—
310C42—21no—
311C42—21no—
312C42—82no—
313C42—82no—
314C16—41no—
315C16—41no—
316C16—41no—
317C16—41no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23C22×D811yescharacteristic
326Q8—42no—
327Q8—42no—
328Q8—81no—
329Q8—81no—
330Q8—164no—
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—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—42no—
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—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4#53611yescharacteristic
375C2×C4#54611yescharacteristic
376C2×C4#54611yescharacteristic
377C2×C4C2×#13711yescharacteristic
378C2×C4#57711yescharacteristic
379C2×C4#57711yescharacteristic
380C8—22no—
381C8—41no—
382C8—41no—
383C8#53911yescharacteristic
384C8#54211yescharacteristic
385C8#55211yescharacteristic
386C8#56211yescharacteristic
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22#322611yescharacteristic
395C22#329411yescharacteristic
396C22#329411yescharacteristic
397C22#341411yescharacteristic
398C22#341411yescharacteristic
399C22#341611yescharacteristic
400C22#341711yescharacteristic
401C4—41no—
402C4—41no—
403C4—41no—
404C4—41no—
405C4—41no—
406C4—82no—
407C4—82no—
408C4—161no—
409C4—161no—
410C4—161no—
411C4—161no—
412C4—161no—
413C4—161no—
414C4#268711yescharacteristic
415C4#324411yescharacteristic
416C4#332011yescharacteristic
417C4#333111yescharacteristic
418C2—81no—
419C2#3135011yescharacteristic
420C2#3380611yescharacteristic
421C2#3380811yescharacteristic
422C2#3387811yescharacteristic
423C2#3388011yescharacteristic
424C2#3388811yescharacteristic
425C2#3389411yescharacteristic
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. #13539C2semidirect
4. #13516C2semidirect
5. #13507C2semidirect
8. #13364C2semidirect
9. #13283C2semidirect
10. #13150C2semidirect
13. #12936C2semidirect
15. #12814C2semidirect

Maximal subgroups

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

subgroupindex
#260642
#250332
#135392
#135162
#135072
#134932
#133822
#133642
#132832
#131502
#131212
#129942
#129362
#128682
#128142