Finite Group Databasebeta

← #510169

Subgroups — #510169

Order 512.

479 conjugacy classes of subgroups.

The subgroup lattice has 429 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26067C211yesmaximal, characteristic
2#25035C211yesmaximal, characteristic
3#13539C211yesmaximal, characteristic, complemented
4#13514C211yesmaximal, characteristic, complemented
5#13505C211yesmaximal, characteristic, complemented
6#13493C211yesmaximal, characteristic
7#13383C211yesmaximal, characteristic
8#13366C211yesmaximal, characteristic, complemented
9#13284C211yesmaximal, characteristic, complemented
10#13150C211yesmaximal, characteristic, complemented
11#13121C211yesmaximal, characteristic
12#12993C211yesmaximal, characteristic
13#12936C211yesmaximal, characteristic, complemented
14#12866C211yesmaximal, characteristic
15#12814C211yesmaximal, characteristic, complemented
16C2×#499C2211yescharacteristic
17C2×#467C2211yescharacteristic
18C2×C4×Q16C2211yescharacteristic
19C22×#107C2211yescharacteristic
20#2442—21no—
21#2423—21no—
22C2×#366C2211yescharacteristic
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#2072—21no—
34#2059—21no—
35#2036—21no—
36#1969C2211yescharacteristic
37#1942C2211yescharacteristic
38#1942C2211yescharacteristic
39#1942C2211yescharacteristic
40#1863—21no—
41#1853—21no—
42#1846C2211yescharacteristic
43#1845C2211yescharacteristic
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×#125—21no—
62C2×C22⋊Q8—21no—
63C2×C4×Q8—21no—
64C22×C4⋊C4—21no—
65C22×C16C2311yescharacteristic
66C22×C16C2311yescharacteristic
67C22×C16C2311yescharacteristic
68C22×C16C2311yescharacteristic
69#499—22no—
70#499—41no—
71#467—22no—
72#467—41no—
73C4×Q16—22no—
74C4×Q16—41no—
75C2×#107—22no—
76C2×#107—21no—
77C2×#107—21no—
78C2×#107—41no—
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
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#399—41no—
96#398—41no—
97#398—41no—
98#391—21no—
99#389—21no—
100#389—21no—
101#389—21no—
102#388—41no—
103#387—41no—
104#385—41no—
105#384—21no—
106#382—21no—
107#380—21no—
108#380—21no—
109#380—41no—
110C4×C22⋊C4D411yescharacteristic
111#366—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—
129C23×C4—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
147#125—42no—
148#125—81no—
149C22⋊Q8—42no—
150C22⋊Q8—81no—
151C4×Q8—42no—
152C4×Q8—81no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—41no—
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×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—81no—
172C2×C22⋊C4—41no—
173C2×C22⋊C4—41no—
174C2×C22⋊C4—41no—
175C2×C22⋊C4D811yescharacteristic
176C2×C22⋊C4D811yescharacteristic
177C2×C42—41no—
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
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—
190C2×C16—21no—
191C2×C16—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—21no—
204#107—21no—
205#107—42no—
206#107—81no—
207#107#4112yes—
208#106—42no—
209#106—81no—
210#103—42no—
211#103—42no—
212#103—42no—
213C22⋊C8—22no—
214C4×C8—21no—
215C4×C8—21no—
216C4×C8—21no—
217C4×C8—21no—
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—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—82no—
251C22×C4—82no—
252C22×C4—161no—
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—82no—
302C4⋊C4—161no—
303C4⋊C4—161no—
304C4⋊C4—161no—
305C4⋊C4—161no—
306C22⋊C4—22no—
307C22⋊C4—41no—
308C22⋊C4—82no—
309C22⋊C4—82no—
310C22⋊C4—161no—
311C42—21no—
312C42—21no—
313C42—21no—
314C42—21no—
315C42—82no—
316C16—41no—
317C16—41no—
318C16—41no—
319C16—41no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23#51012yes—
327C23C22×D811yescharacteristic
328Q8—42no—
329Q8—42no—
330Q8—81no—
331Q8—81no—
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—21no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—42no—
352C2×C4—42no—
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—81no—
375C2×C4—164no—
376C2×C4#53611yescharacteristic
377C2×C4#54611yescharacteristic
378C2×C4#54611yescharacteristic
379C2×C4C2×#13711yescharacteristic
380C2×C4#57711yescharacteristic
381C2×C4#57711yescharacteristic
382C8—22no—
383C8—41no—
384C8—41no—
385C8#53911yescharacteristic
386C8#54211yescharacteristic
387C8#55211yescharacteristic
388C8#56211yescharacteristic
389C22—22no—
390C22—22no—
391C22—22no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22#322611yescharacteristic
397C22#329411yescharacteristic
398C22#329411yescharacteristic
399C22C2×#51011yescharacteristic
400C22#341411yescharacteristic
401C22#341511yescharacteristic
402C22#341711yescharacteristic
403C4—41no—
404C4—41no—
405C4—41no—
406C4—41no—
407C4—41no—
408C4—82no—
409C4—82no—
410C4—161no—
411C4—161no—
412C4—161no—
413C4—161no—
414C4—161no—
415C4—161no—
416C4#268711yescharacteristic
417C4#324411yescharacteristic
418C4#332011yescharacteristic
419C4#333111yescharacteristic
420C2—42no—
421C2#3135011yescharacteristic
422C2#3380211yescharacteristic
423C2#3380911yescharacteristic
424C2#3387611yescharacteristic
425C2#3387911yescharacteristic
426C2#3388111yescharacteristic
427C2#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. #13514C2semidirect
5. #13505C2semidirect
8. #13366C2semidirect
9. #13284C2semidirect
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
#260672
#250352
#135392
#135142
#135052
#134932
#133832
#133662
#132842
#131502
#131212
#129932
#129362
#128662
#128142