Finite Group Databasebeta

← #509005

Subgroups — #509005

Order 512.

599 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33266C211yesmaximal, characteristic, complemented
2#33241C211yesmaximal, characteristic, complemented
3#13474C212yesmaximal, complemented
4#13451C211yesmaximal, characteristic, complemented
5#13448C212yesmaximal, complemented
6#13407C211yesmaximal, characteristic, complemented
7#13363C211yesmaximal, characteristic, complemented
8#13325C211yesmaximal, characteristic
9#13235C212yesmaximal, complemented
10#13147C212yesmaximal, complemented
11C4×C4⋊C16C211yesmaximal, characteristic, complemented
12C22×D16C2212yes—
13C2×#500C2211yescharacteristic
14C2×#493C2211yescharacteristic
15C2×#486—21no—
16#2860—21no—
17#2858—21no—
18C2×#366C2212yes—
19C2×C4⋊C16C2212yes—
20C2×#357C2212yes—
21C2×#357C2212yes—
22C2×#357C2212yes—
23C2×#357C2212yes—
24C2×C4×C16C2212yes—
25#1970—21no—
26#1970—21no—
27#1970C2211yescharacteristic
28#1970C2212yes—
29#1970C2211yescharacteristic
30#1970C2212yes—
31#1970C2211yescharacteristic
32#1942C2211yescharacteristic
33#1942C2211yescharacteristic
34#1942C2211yescharacteristic
35#1934—21no—
36#1934—21no—
37#1934—21no—
38C42×C8C2211yescharacteristic
39#1380C2212yes—
40#1380C2212yes—
41#1374C2211yescharacteristic
42#1374C2211yescharacteristic
43C22×D8—21no—
44C22×D8C2312yes—
45C22×D8C2311yescharacteristic
46C2×C4⋊Q8—21no—
47C2×#127—21no—
48C2×#126—41no—
49C2×#124—41no—
50C2×#124—41no—
51C2×D16—24no—
52C2×D16—42no—
53C2×D16—42no—
54C22×C16C2312yes—
55C22×C16C2312yes—
56#500—42no—
57#493—42no—
58#486—44no—
59C2×#107—21no—
60C2×#107C2312yes—
61C2×#107C2311yescharacteristic
62C2×#107C2311yescharacteristic
63C2×#107C2311yescharacteristic
64C2×#102—22no—
65C2×#102—22no—
66C2×#102—21no—
67C2×#102—21no—
68C2×#102—21no—
69C2×C4×C8D411yescharacteristic
70C2×C4×C8D411yescharacteristic
71C2×C4×C8D411yescharacteristic
72C2×C4×C8C2311yescharacteristic
73C2×C4×C8C2311yescharacteristic
74C2×C4×C8C2311yescharacteristic
75#390—22no—
76#390—22no—
77#390—21no—
78#390—21no—
79#390—21no—
80#390—41no—
81#390—41no—
82#389—21no—
83#389—21no—
84#389—21no—
85#388—41no—
86#388—41no—
87#388—41no—
88C43D411yescharacteristic
89#366—42no—
90C4⋊C16—24no—
91#357—24no—
92#357—42no—
93#357—42no—
94#357—42no—
95C4×C16—42no—
96#340—21no—
97#340—21no—
98#340—21no—
99C22×Q8—41no—
100C22×D4—22no—
101C22×D4—21no—
102C22×D4—41no—
103C22×D4—41no—
104C2×D8—22no—
105C2×D8—22no—
106C2×D8—22no—
107C2×D8—22no—
108C2×D8—22no—
109C2×D8—22no—
110C2×D8—22no—
111C2×D8—21no—
112C2×D8—22no—
113C2×D8—21no—
114C2×D8—42no—
115C2×D8—42no—
116C2×D8—42no—
117C22×C8C2×D411yescharacteristic
118C22×C8C2×D411yescharacteristic
119C22×C8C2×D411yescharacteristic
120C22×C8C2411yescharacteristic
121C4⋊Q8—42no—
122C4⋊Q8—42no—
123#127—42no—
124#127—42no—
125#126—84no—
126#124—84no—
127#124—84no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—41no—
136C2×C22⋊C4—42no—
137C2×C22⋊C4—42no—
138C2×C22⋊C4—42no—
139C2×C22⋊C4—42no—
140C2×C22⋊C4—41no—
141C2×C22⋊C4—41no—
142C2×C22⋊C4—41no—
143C2×C22⋊C4—41no—
144C2×C22⋊C4—41no—
145C2×C22⋊C4—41no—
146C2×C42D811yescharacteristic
147C2×C42D811yescharacteristic
148C2×C42D811yescharacteristic
149C2×C42D811yescharacteristic
150C2×C42C2×D411yescharacteristic
151C2×C42C2×D411yescharacteristic
152C2×C42C2×D411yescharacteristic
153D16—48no—
154C2×C16—22no—
155C2×C16—22no—
156C2×C16—22no—
157C2×C16—22no—
158C2×C16—22no—
159C2×C16#4114yes—
160#107—22no—
161#107—21no—
162#107—22no—
163#107—22no—
164#107—22no—
165#107—21no—
166#107—42no—
167#102—42no—
168#102—44no—
169#102—42no—
170#102—42no—
171#102—44no—
172C4×C8—22no—
173C4×C8—22no—
174C4×C8—22no—
175C4×C8—21no—
176C4×C8—22no—
177C4×C8—21no—
178C4×C8—41no—
179#95—41no—
180#95—41no—
181#95—41no—
182C24—41no—
183C24—42no—
184C2×Q8—82no—
185C2×Q8—82no—
186C2×Q8—82no—
187C2×D4—42no—
188C2×D4—41no—
189C2×D4—42no—
190C2×D4—41no—
191C2×D4—42no—
192C2×D4—41no—
193C2×D4—41no—
194C2×D4—42no—
195C2×D4—42no—
196C2×D4—41no—
197C2×D4—41no—
198C2×D4—42no—
199C2×D4—82no—
200C2×D4—82no—
201C2×D4—82no—
202C2×D4—82no—
203C2×D4—82no—
204C2×D4—82no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—42no—
209C22×C4C2×D811yescharacteristic
210C22×C4C2×D811yescharacteristic
211C22×C4C2×D811yescharacteristic
212C22×C4C2×D811yescharacteristic
213C22×C4C2×D811yescharacteristic
214C22×C4C2×D811yescharacteristic
215C22×C4C22×D411yescharacteristic
216D8—42no—
217D8—42no—
218D8—42no—
219D8—42no—
220D8—42no—
221D8—42no—
222D8—84no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—21no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—21no—
229C2×C8—21no—
230C2×C8#13512yes—
231C2×C8AGL(1,ℤ/8)12yes—
232C2×C8C2×#4112yes—
233C2×C8H(5,𝔽2)11yescharacteristic
234C2×C8H(5,𝔽2)12yes—
235C2×C82−1+411yescharacteristic
236C4⋊C4—41no—
237C4⋊C4—42no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—42no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—82no—
245C4⋊C4—82no—
246C4⋊C4—82no—
247C4⋊C4—82no—
248C22⋊C4—82no—
249C22⋊C4—84no—
250C22⋊C4—84no—
251C22⋊C4—84no—
252C22⋊C4—82no—
253C22⋊C4—82no—
254C22⋊C4—82no—
255C22⋊C4—82no—
256C22⋊C4—82no—
257C22⋊C4—84no—
258C42—21no—
259C42—21no—
260C42—21no—
261C42—21no—
262C42—21no—
263C42—21no—
264C42—22no—
265C42—22no—
266C42—41no—
267C42D1614yescomplemented
268C16—24no—
269C16—42no—
270C23—81no—
271C23—82no—
272C23—81no—
273C23—82no—
274C23—82no—
275C23—81no—
276C23—81no—
277C23—82no—
278C23—81no—
279C23—82no—
280C23—82no—
281C23—81no—
282C23—81no—
283C23—82no—
284C23C22×D811yescharacteristic
285Q8—164no—
286D4—82no—
287D4—81no—
288D4—81no—
289D4—82no—
290D4—82no—
291D4—81no—
292D4—81no—
293D4—82no—
294D4—164no—
295D4—164no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—82no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—82no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—82no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4C2×D1612yes—
322C2×C4C2×D1612yes—
323C2×C4C2×D1612yes—
324C2×C4#50812yes—
325C2×C4#50912yes—
326C2×C4#53211yescharacteristic
327C2×C4#53511yescharacteristic
328C2×C4#53711yescharacteristic
329C2×C4C2×#13511yescharacteristic
330C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
331C2×C4#57511yescharacteristic
332C8—22no—
333C8—22no—
334C8—41no—
335C8#53812yes—
336C8#54212yes—
337C22—161no—
338C22—162no—
339C22—161no—
340C22—162no—
341C22—162no—
342C22—161no—
343C22—161no—
344C22—162no—
345C22—162no—
346C22—161no—
347C22—162no—
348C22—161no—
349C22—162no—
350C22—161no—
351C22#316911yescharacteristic
352C22#320511yescharacteristic
353C22#324511yescharacteristic
354C22C22×D1611yescharacteristic
355C22C2×#50811yescharacteristic
356C22C2×#50911yescharacteristic
357C22#341411yescharacteristic
358C4—22no—
359C4—22no—
360C4—22no—
361C4—22no—
362C4—22no—
363C4—41no—
364C4—161no—
365C4—161no—
366C4—161no—
367C4—162no—
368C4#266211yescharacteristic
369C4#317111yescharacteristic
370C4#321111yescharacteristic
371C4#324711yescharacteristic
372C2—322no—
373C2—321no—
374C2#3033611yescharacteristic
375C2#3373311yescharacteristic
376C2#3373711yescharacteristic
377C2#3377111yescharacteristic
378C2#3377711yescharacteristic
379C2#3381111yescharacteristic
380C2#3381311yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radical#13325C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33266C2semidirect
2. #33241C2semidirect
3. #13474C2semidirect
4. #13451C2semidirect
5. #13448C2semidirect
6. #13407C2semidirect
7. #13363C2semidirect
9. #13235C2semidirect
10. #13147C2semidirect
11. C4×C4⋊C16C2semidirect
267. C42D16semidirect

Maximal subgroups

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

subgroupindex
#332662
#332412
#134742
#134512
#134482
#134072
#133632
#133252
#132352
#131472
C4×C4⋊C162