Finite Group Databasebeta

← #511231

Subgroups — #511231

Order 512.

447 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28827C211yesmaximal, characteristic, complemented
2#28014C211yesmaximal, characteristic
3#13526C211yesmaximal, characteristic
4#13526C211yesmaximal, characteristic
5#13505C211yesmaximal, characteristic, complemented
6#13505C211yesmaximal, characteristic, complemented
7#13480C211yesmaximal, characteristic
8#13474C211yesmaximal, characteristic
9#13176C211yesmaximal, characteristic, complemented
10#13147C211yesmaximal, characteristic
11#13130C211yesmaximal, characteristic
12#13109C211yesmaximal, characteristic, complemented
13#13109C211yesmaximal, characteristic, complemented
14#13104C211yesmaximal, characteristic, complemented
15#12905C211yesmaximal, characteristic, complemented
16#2552—21no—
17#2512—21no—
18C2×#366C2211yescharacteristic
19C2×#366C2211yescharacteristic
20C2×#366C2211yescharacteristic
21C2×#366C2211yescharacteristic
22C2×#366C2211yescharacteristic
23C2×#366C2211yescharacteristic
24C2×C4⋊C16C2211yescharacteristic
25C2×#357C2211yescharacteristic
26C2×#357C2211yescharacteristic
27C2×#357C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×C4×C16C2211yescharacteristic
30#2083—21no—
31#2083—21no—
32#2071—21no—
33#2067—21no—
34#1970C2211yescharacteristic
35#1970C2211yescharacteristic
36#1970C2211yescharacteristic
37#1947—21no—
38#1942C2211yescharacteristic
39#1942C2211yescharacteristic
40#1942C2211yescharacteristic
41#1925—21no—
42#1845C2211yescharacteristic
43#1845C2211yescharacteristic
44#1845C2211yescharacteristic
45#1845C2211yescharacteristic
46C4×#107C2211yescharacteristic
47C4×#107C2211yescharacteristic
48C8×C4⋊C4C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1374C2211yescharacteristic
56#1374C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×D8C2311yescharacteristic
60C2×#126—41no—
61C2×#125—41no—
62C22×C16C2311yescharacteristic
63C22×C16C2311yescharacteristic
64C22×C16C2311yescharacteristic
65C22×C16C2311yescharacteristic
66C2×#107—21no—
67C2×#107—21no—
68C2×#107—21no—
69C2×#107—21no—
70C2×#107C2311yescharacteristic
71C2×#107C2311yescharacteristic
72C2×#107C2311yescharacteristic
73C2×#107C2311yescharacteristic
74C2×#107C2311yescharacteristic
75C2×#107C2311yescharacteristic
76C2×#107C2311yescharacteristic
77C2×C4⋊C8D411yescharacteristic
78C2×#102—21no—
79C2×#102—21no—
80C2×#102—21no—
81C2×C4×C8C2311yescharacteristic
82C2×C4×C8C2311yescharacteristic
83C2×C4×C8C2311yescharacteristic
84#398—41no—
85#398—41no—
86#397—41no—
87#396—41no—
88#390—21no—
89#390—21no—
90#390—21no—
91#389—21no—
92#389—21no—
93#389—21no—
94#389—41no—
95#388—41no—
96#384—21no—
97#384—21no—
98#384—21no—
99#384—21no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4D411yescharacteristic
103#366—22no—
104#366—22no—
105#366—41no—
106#366—41no—
107#366—41no—
108#366—41no—
109C4⋊C16—22no—
110#357—22no—
111#357—22no—
112#357—41no—
113#357—41no—
114C4×C16—22no—
115#340—21no—
116#340—21no—
117#340—21no—
118#340—21no—
119#340—21no—
120#336D411yescharacteristic
121#336D411yescharacteristic
122C22×D4—21no—
123C2×D8—21no—
124C2×D8—21no—
125C2×D8—21no—
126C2×D8—21no—
127C2×D8—21no—
128C2×D8—21no—
129C22×C8C2×D411yescharacteristic
130C22×C8C2×D411yescharacteristic
131C22×C8C2×D411yescharacteristic
132C22×C8C2411yescharacteristic
133#126—84no—
134#125—84no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4D811yescharacteristic
154C2×C4⋊C4D811yescharacteristic
155C2×C22⋊C4—41no—
156C2×C22⋊C4—41no—
157C2×C22⋊C4—41no—
158C2×C22⋊C4—41no—
159C2×C22⋊C4—41no—
160C2×C22⋊C4—41no—
161C2×C42—41no—
162C2×C42—41no—
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C16—21no—
167C2×C16—21no—
168C2×C16—21no—
169C2×C16—21no—
170C2×C16—21no—
171C2×C16—21no—
172C2×C16—21no—
173C2×C16—21no—
174C2×C16—21no—
175C2×C16—21no—
176C2×C16#4112yes—
177C2×C16#4112yes—
178#107—21no—
179#107—21no—
180#107—21no—
181#107—21no—
182#107—21no—
183#107—21no—
184#107—21no—
185#107—21no—
186#107—21no—
187#107—21no—
188#107—21no—
189#107—21no—
190#107—21no—
191#107—21no—
192#107—81no—
193#107—81no—
194#107—81no—
195#107—81no—
196C4⋊C8—22no—
197#102—42no—
198#102—81no—
199#102—81no—
200C4×C8—21no—
201C4×C8—21no—
202C4×C8—21no—
203C4×C8—21no—
204C4×C8—21no—
205C4×C8—21no—
206#95—41no—
207#95—41no—
208#95—41no—
209#95—41no—
210#95—41no—
211#95D811yescharacteristic
212#95D811yescharacteristic
213C24—41no—
214C2×D4—41no—
215C2×D4—41no—
216C2×D4—41no—
217C2×D4—41no—
218C2×D4—41no—
219C2×D4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4C2×D811yescharacteristic
228C22×C4C2×D811yescharacteristic
229C22×C4C2×D811yescharacteristic
230C22×C4C2×D811yescharacteristic
231C22×C4C2×D811yescharacteristic
232C22×C4C2×D811yescharacteristic
233C22×C4C22×D411yescharacteristic
234D8—41no—
235D8—41no—
236D8—41no—
237D8—41no—
238C2×C8—21no—
239C2×C8—21no—
240C2×C8—21no—
241C2×C8—21no—
242C2×C8—21no—
243C2×C8—21no—
244C2×C8—21no—
245C2×C8—21no—
246C2×C8#13712yes—
247C2×C8C2×#4111yescharacteristic
248C2×C8C2×#4111yescharacteristic
249C2×C8H(5,𝔽2)11yescharacteristic
250C2×C82−1+411yescharacteristic
251C2×C82−1+411yescharacteristic
252C2×C82−1+411yescharacteristic
253C4⋊C4—22no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—82no—
269C4⋊C4—82no—
270C4⋊C4—82no—
271C4⋊C4—82no—
272C4⋊C4—82no—
273C4⋊C4—82no—
274C4⋊C4—82no—
275C4⋊C4—161no—
276C4⋊C4—161no—
277C4⋊C4—161no—
278C4⋊C4—161no—
279C4⋊C4D1614yes—
280C22⋊C4—82no—
281C22⋊C4—82no—
282C22⋊C4—82no—
283C22⋊C4—82no—
284C22⋊C4—161no—
285C22⋊C4—161no—
286C42—21no—
287C42—21no—
288C42—21no—
289C42—21no—
290C42—21no—
291C42—21no—
292C42—82no—
293C42—82no—
294C16—22no—
295C16—22no—
296C16—41no—
297C16—41no—
298C23—81no—
299C23—81no—
300C23—81no—
301C23—81no—
302C23—81no—
303C23—81no—
304C23—81no—
305C23C22×D811yescharacteristic
306D4—81no—
307D4—81no—
308D4—81no—
309D4—81no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4C2×D1612yes—
346C2×C4C2×D1612yes—
347C2×C4C2×D1612yes—
348C2×C4#51012yes—
349C2×C4#53611yescharacteristic
350C2×C4#54711yescharacteristic
351C2×C4#55011yescharacteristic
352C2×C4C2×#13711yescharacteristic
353C2×C4#57611yescharacteristic
354C2×C4#57811yescharacteristic
355C8—22no—
356C8—41no—
357C8—41no—
358C8#53911yescharacteristic
359C8#54111yescharacteristic
360C8#55211yescharacteristic
361C8#56311yescharacteristic
362C22—161no—
363C22—161no—
364C22—161no—
365C22—161no—
366C22—161no—
367C22—161no—
368C22—161no—
369C22#322611yescharacteristic
370C22#333411yescharacteristic
371C22#339111yescharacteristic
372C22C22×D1611yescharacteristic
373C22C2×#51011yescharacteristic
374C22#341511yescharacteristic
375C22#341711yescharacteristic
376C4—22no—
377C4—22no—
378C4—22no—
379C4—22no—
380C4—41no—
381C4—41no—
382C4—161no—
383C4—161no—
384C4—161no—
385C4—161no—
386C4—161no—
387C4—161no—
388C4—161no—
389C4#272111yescharacteristic
390C4#324311yescharacteristic
391C4#334611yescharacteristic
392C4#339811yescharacteristic
393C2—321no—
394C2#3237411yescharacteristic
395C2#3379211yescharacteristic
396C2#3380911yescharacteristic
397C2#3390111yescharacteristic
398C2#3391311yescharacteristic
399C2#3395811yescharacteristic
400C2#3396511yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D8C23
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #28827C2semidirect
5. #13505C2semidirect
6. #13505C2semidirect
9. #13176C2semidirect
12. #13109C2semidirect
13. #13109C2semidirect
14. #13104C2semidirect
15. #12905C2semidirect

Maximal subgroups

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

subgroupindex
#288272
#280142
#135262
#135052
#134802
#134742
#131762
#131472
#131302
#131092
#131042
#129052