Finite Group Databasebeta

← #314963

Subgroups — #314963

Order 512.

506 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21491C211yesmaximal, characteristic
2#20829C211yesmaximal, characteristic, complemented
3#10688C211yesmaximal, characteristic, complemented
4#10687C211yesmaximal, characteristic, complemented
5#10661C211yesmaximal, characteristic, complemented
6#10660C211yesmaximal, characteristic, complemented
7#9629C212yesmaximal, complemented
8#9036C212yesmaximal
9#8863C212yesmaximal, complemented
10#8660C212yesmaximal, complemented
11#8602C211yesmaximal, characteristic, complemented
12C2×C8×D4—21no—
13C2×C8×D4C2211yescharacteristic
14#2921C2212yes—
15C2×#405C2211yescharacteristic
16#2392C2211yescharacteristic
17#1955C2211yescharacteristic
18#1940C2211yescharacteristic
19#1924—21no—
20#1870—21no—
21#1849C2211yescharacteristic
22#1849C2211yescharacteristic
23#1837C2212yes—
24#1811—21no—
25#1810—21no—
26#1798C2211yescharacteristic
27#1778—21no—
28C4×#108—21no—
29C8×C4⋊C4C2211yescharacteristic
30C8×C22⋊C4—21no—
31#1571C2212yes—
32#1570C2212yes—
33#1564C2212yes—
34#1486C2212yes—
35#1486C2212yes—
36#1458C2212yes—
37#1303C2212yes—
38#1303C2212yes—
39#1301C2212yes—
40#1301C2212yes—
41#1275C2211yescharacteristic
42#1275C2211yescharacteristic
43C22×M16D411yescharacteristic
44C23×C8—21no—
45C23×C8D411yescharacteristic
46C2×C4×D4—21no—
47C2×C4×D4C2311yescharacteristic
48C2×#117C2311yescharacteristic
49C8×D4—22no—
50C8×D4—22no—
51C8×D4—44no—
52#433—42no—
53#433—42no—
54#431C2312yes—
55#431C2312yes—
56C2×#108—21no—
57C2×#108—21no—
58C2×#108—21no—
59C2×#108—21no—
60#423C2312yes—
61#423C2312yes—
62C2×C4⋊C8—21no—
63C2×C4⋊C8C2311yescharacteristic
64C2×C4⋊C8C2311yescharacteristic
65#407—22no—
66C2×C22⋊C8—21no—
67C2×C22⋊C8—21no—
68C2×C22⋊C8—21no—
69C2×C22⋊C8—21no—
70#405—22no—
71#405—22no—
72C4×M16—22no—
73C2×#97D411yescharacteristic
74C2×#97C2311yescharacteristic
75C2×C4×C8—21no—
76C2×C4×C8D411yescharacteristic
77C2×C4×C8C2311yescharacteristic
78#398Q811yescharacteristic
79#397Q811yescharacteristic
80#396—21no—
81#394—21no—
82#394Q811yescharacteristic
83#392Q811yescharacteristic
84C4×C4⋊C4C2311yescharacteristic
85C4×C22⋊C4—21no—
86#341—22no—
87#341—22no—
88#336—21no—
89#336—21no—
90#336—21no—
91#335—22no—
92#335—42no—
93#334—22no—
94#334—42no—
95D4⋊C8—42no—
96D4⋊C8—42no—
97#324—22no—
98#323—22no—
99C22×D4—21no—
100C22×D4C2×Q811yescharacteristic
101C23×C4—21no—
102C23×C4C2×D411yescharacteristic
103C2×M16—22no—
104C2×M16—22no—
105C2×M16—22no—
106C2×M16—22no—
107C2×M16—41no—
108C22×C8—21no—
109C22×C8—21no—
110C22×C8—22no—
111C22×C8—21no—
112C22×C8—21no—
113C22×C8—41no—
114C22×C8—42no—
115C22×C8—42no—
116C22×C8—41no—
117C22×C8—41no—
118C22×C8C2×D411yescharacteristic
119C22×C8C2×D411yescharacteristic
120C22×C8C2×D411yescharacteristic
121C22×C8C2×D411yescharacteristic
122C4×D4—22no—
123C4×D4—22no—
124C4×D4—42no—
125C4×D4—42no—
126#117—22no—
127#117—22no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4C2×Q811yescharacteristic
131C2×C4⋊C4C2×Q811yescharacteristic
132C2×C4⋊C4#4111yescharacteristic
133C2×C22⋊C4—21no—
134C2×C22⋊C4—21no—
135C2×C22⋊C4—21no—
136C2×C22⋊C4—21no—
137C2×C22⋊C4—21no—
138C2×C22⋊C4C2×Q811yescharacteristic
139C2×C22⋊C4#4111yescharacteristic
140C2×C42—21no—
141C2×C42C2×D411yescharacteristic
142C2×C42C2411yescharacteristic
143#108—42no—
144#108—42no—
145#108—42no—
146#108—42no—
147C4⋊C8—22no—
148C4⋊C8—22no—
149C4⋊C8—22no—
150C4⋊C8—22no—
151C4⋊C8—22no—
152C4⋊C8—22no—
153C4⋊C8—22no—
154C4⋊C8—22no—
155C4⋊C8—42no—
156C22⋊C8—22no—
157C22⋊C8—22no—
158C22⋊C8—42no—
159C22⋊C8—44no—
160C22⋊C8—42no—
161C22⋊C8—42no—
162C22⋊C8—42no—
163#97—22no—
164#97—22no—
165#97—42no—
166C4×C8—21no—
167C4×C8—21no—
168C4×C8—22no—
169C4×C8—22no—
170C4×C8—42no—
171C4×C8—42no—
172C4×C8#4112yes—
173C4×C8#4112yes—
174#95—21no—
175#95—21no—
176#95C2×Q811yescharacteristic
177#95C2×Q811yescharacteristic
178C24—21no—
179C24C22⋊Q811yescharacteristic
180C2×D4—22no—
181C2×D4—42no—
182C2×D4—42no—
183C2×D4—42no—
184C2×D4—42no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4C22⋊Q811yescharacteristic
201C22×C4C22⋊Q811yescharacteristic
202C22×C4C22⋊Q811yescharacteristic
203C22×C4C22×D411yescharacteristic
204C22×C4C22×Q811yescharacteristic
205C22×C4C2×#4111yescharacteristic
206M16—44no—
207M16—42no—
208M16—42no—
209M16—42no—
210M16—42no—
211C2×C8—21no—
212C2×C8—21no—
213C2×C8—21no—
214C2×C8—21no—
215C2×C8—21no—
216C2×C8—21no—
217C2×C8—21no—
218C2×C8—21no—
219C2×C8—42no—
220C2×C8—42no—
221C2×C8—41no—
222C2×C8—42no—
223C2×C8—42no—
224C2×C8—41no—
225C2×C8—42no—
226C2×C8—42no—
227C2×C8—41no—
228C2×C8—41no—
229C2×C8—41no—
230C2×C8—41no—
231C2×C8—82no—
232C2×C8—82no—
233C2×C8#12311yescharacteristic
234C2×C8#12311yescharacteristic
235C2×C8#12312yes—
236C2×C8#13512yes—
237C2×C8AGL(1,ℤ/8)12yes—
238C4⋊C4—22no—
239C4⋊C4—22no—
240C4⋊C4—22no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C22⋊C4—22no—
244C22⋊C4—22no—
245C22⋊C4—42no—
246C22⋊C4—42no—
247C22⋊C4—42no—
248C22⋊C4—42no—
249C22⋊C4—42no—
250C42—21no—
251C42—21no—
252C42—41no—
253C42—41no—
254C42C2×#4112yes—
255C42H(5,𝔽2)12yes—
256C23—22no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23C2×C22⋊Q811yescharacteristic
269D4—24no—
270D4—42no—
271D4—48no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4C2×#12311yescharacteristic
299C2×C4#53511yescharacteristic
300C2×C4#55312yes—
301C2×C4#55412yes—
302C2×C4C2×#13511yescharacteristic
303C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
304C2×C4#57511yescharacteristic
305C2×C4#57511yescharacteristic
306C8—22no—
307C8—22no—
308C8—21no—
309C8—21no—
310C8—22no—
311C8—82no—
312C8—82no—
313C8—81no—
314C8—81no—
315C8—82no—
316C8—82no—
317C22—21no—
318C22—21no—
319C22—42no—
320C22—42no—
321C22—42no—
322C22—41no—
323C22—42no—
324C22—41no—
325C22—41no—
326C22—81no—
327C22—81no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22#248711yescharacteristic
332C22#307611yescharacteristic
333C22#307711yescharacteristic
334C4—22no—
335C4—22no—
336C4—42no—
337C4—41no—
338C4—41no—
339C4—41no—
340C4—81no—
341C4—81no—
342C4#309111yescharacteristic
343C4#309411yescharacteristic
344C4#320711yescharacteristic
345C4#321911yescharacteristic
346C2—21no—
347C2—21no—
348C2—81no—
349C2—82no—
350C2#2768311yescharacteristic
351C2#2769111yescharacteristic
352C2#2769311yescharacteristic
subgroupG/H
centerC2×C4#535
commutator, Frattini, cosocleC2×C42C24
socleC22#2487
prime radical#392Q8
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20829C2semidirect
3. #10688C2semidirect
4. #10687C2semidirect
5. #10661C2semidirect
6. #10660C2semidirect
7. #9629C2semidirect
9. #8863C2semidirect
10. #8660C2semidirect
11. #8602C2semidirect

Maximal subgroups

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

subgroupindex
#214912
#208292
#106882
#106872
#106612
#106602
#96292
#90362
#88632
#86602
#86022