Finite Group Databasebeta

← #323495

Subgroups — #323495

Order 512.

808 conjugacy classes of subgroups.

The subgroup lattice has 303 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33316C211yesmaximal, characteristic, complemented
2#29190C212yesmaximal, complemented
3#29139C212yesmaximal, complemented
4#21690C211yesmaximal, characteristic
5#10677C212yesmaximal, complemented
6#10672C212yesmaximal, complemented
7#9837C211yesmaximal, characteristic
8#9824C211yesmaximal, characteristic
9#8819C211yesmaximal, characteristic, complemented
10#8815C211yesmaximal, characteristic, complemented
11#8262C211yesmaximal, characteristic, complemented
12C2×#501—22no—
13C2×#493C2211yescharacteristic
14C2×#442—22no—
15#2868C2211yescharacteristic
16#2859C2211yescharacteristic
17C4×#127C2212yes—
18#2065—24no—
19#2023—24no—
20#1972—22no—
21#1957C2212yes—
22#1957C2212yes—
23#1957C2212yes—
24#1957C2212yes—
25#1957C2212yes—
26#1864—24no—
27#1847—22no—
28#1845C2211yescharacteristic
29#1845C2211yescharacteristic
30#1844C2212yes—
31#1844C2211yescharacteristic
32#1844C2211yescharacteristic
33#1844C2211yescharacteristic
34#1834C2212yes—
35#1834C2212yes—
36#1834C2212yes—
37#1802—24no—
38C4×C4⋊C8C2212yes—
39C4×#97C2211yescharacteristic
40C2×#322C2211yescharacteristic
41C2×C82C2211yescharacteristic
42#1273C2212yes—
43C22×D8—22no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8—22no—
46C2×#127C2312yes—
47C2×#125—21no—
48C2×C4×D4—22no—
49C2×C4×D4—22no—
50C2×C4×D4—22no—
51#501—48no—
52#493—22no—
53#493Q814yescomplemented
54#442—48no—
55C2×#107—22no—
56C2×#107—22no—
57C2×#106—22no—
58C2×#106—22no—
59C2×#106—22no—
60C2×#106—24no—
61C2×C4⋊C8D414yes—
62C2×#102—24no—
63C2×#102—24no—
64C2×#102—24no—
65C2×#102—24no—
66C2×#102—24no—
67C2×#97D412yes—
68C2×#97D412yes—
69C2×C4×C8C2312yes—
70C2×C4×C8C2312yes—
71C2×C4×C8C2311yescharacteristic
72C2×C4×C8C2311yescharacteristic
73#390—22no—
74#390—22no—
75#390—22no—
76#389—22no—
77#389—22no—
78#384—22no—
79#384—21no—
80#384C2312yes—
81#384C2311yescharacteristic
82#384C2311yescharacteristic
83#384C2311yescharacteristic
84#384C2311yescharacteristic
85#382—21no—
86C43C2311yescharacteristic
87#340—24no—
88#340—24no—
89#340—24no—
90#336D414yes—
91#322—22no—
92C82—22no—
93C22×Q8—41no—
94C22×D4—22no—
95C22×D4—22no—
96C22×D4—22no—
97C23×C4—42no—
98C2×D8—28no—
99C2×D8—44no—
100C2×D8—44no—
101C22×C8C2×D414yes—
102C22×C8C2×D414yes—
103C22×C8C2×D412yes—
104C22×C8C2×D412yes—
105C4⋊Q8—44no—
106C4⋊Q8—84no—
107#127—22no—
108#127—22no—
109#127—22no—
110#127C2×Q814yes—
111#125—44no—
112C4×D4—48no—
113C4×D4—48no—
114C4×D4—84no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—42no—
130C2×C22⋊C4—42no—
131C2×C22⋊C4—42no—
132C2×C22⋊C4—42no—
133C2×C42C2×D412yes—
134C2×C42C2×D412yes—
135C2×C42C2×D411yescharacteristic
136C2×C42C2×D411yescharacteristic
137C2×C42C2411yescharacteristic
138#107—44no—
139#107—44no—
140#106—44no—
141#106—44no—
142#106—44no—
143#106—84no—
144C4⋊C8—44no—
145#102—48no—
146#102—48no—
147#102—84no—
148#102—84no—
149#102—84no—
150#97—24no—
151#97SD1618yes—
152C4×C8—22no—
153C4×C8—22no—
154C4×C8—22no—
155C4×C8—22no—
156C4×C8—22no—
157C4×C8C2×Q812yes—
158C4×C8#4112yes—
159#95—42no—
160C24—42no—
161C2×Q8—82no—
162C2×Q8—84no—
163C2×D4—24no—
164C2×D4—24no—
165C2×D4—24no—
166C2×D4—42no—
167C2×D4—42no—
168C2×D4—42no—
169C2×D4—44no—
170C2×D4—44no—
171C2×D4—44no—
172C2×D4—44no—
173C2×D4—44no—
174C2×D4—44no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—42no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—84no—
181C22×C4—84no—
182C22×C4—84no—
183C22×C4#12711yescharacteristic
184C22×C4#12711yescharacteristic
185C22×C4#12712yes—
186C22×C4C22×D411yescharacteristic
187C22×C4C22×D412yes—
188D8—416no—
189C2×C8—22no—
190C2×C8—22no—
191C2×C8—24no—
192C2×C8—24no—
193C2×C8—24no—
194C2×C8—24no—
195C2×C8—24no—
196C2×C8—24no—
197C2×C8C22⋊Q814yes—
198C2×C8C2×SD1614yes—
199C2×C8C2×SD1614yes—
200C2×C8AGL(1,ℤ/8)14yes—
201C4⋊C4—44no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—44no—
207C4⋊C4—44no—
208C4⋊C4—44no—
209C4⋊C4—44no—
210C4⋊C4—44no—
211C4⋊C4—44no—
212C4⋊C4—82no—
213C4⋊C4—82no—
214C4⋊C4—82no—
215C4⋊C4—84no—
216C22⋊C4—84no—
217C22⋊C4—84no—
218C22⋊C4—84no—
219C42—22no—
220C42—24no—
221C42—24no—
222C42C2×SD1614yes—
223C42C22×Q811yescharacteristic
224C42C2×#4111yescharacteristic
225C422−1+412yes—
226C23—44no—
227C23—82no—
228C23—82no—
229C23—82no—
230C23—82no—
231C23—82no—
232C23—82no—
233C23C2×#12711yescharacteristic
234Q8—164no—
235D4—48no—
236D4—48no—
237D4—48no—
238D4—84no—
239D4—84no—
240D4—84no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—82no—
248C2×C4—81no—
249C2×C4—81no—
250C2×C4—81no—
251C2×C4—81no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—168no—
259C2×C4#49214yes—
260C2×C4C2×C22⋊Q811yescharacteristic
261C2×C4#53611yescharacteristic
262C2×C4#54712yes—
263C2×C4D4×Q812yes—
264C2×C4C22×SD1612yes—
265C2×C4C2×AGL(1,ℤ/8)12yes—
266C2×C4#57514yes—
267C2×C4#57614yes—
268C8—24no—
269C8—44no—
270C8—44no—
271C8#47618yes—
272C22—84no—
273C22—84no—
274C22—84no—
275C22—162no—
276C22—162no—
277C22—162no—
278C22—162no—
279C22#254211yescharacteristic
280C22C2×#49211yescharacteristic
281C22#315111yescharacteristic
282C22#315312yes—
283C22#315512yes—
284C4—24no—
285C4—41no—
286C4—41no—
287C4—161no—
288C4—161no—
289C4—161no—
290C4—161no—
291C4—162no—
292C4C2×#47612yes—
293C4#322712yes—
294C4#334714yes—
295C4#338914yes—
296C2—164no—
297C2#1962312yes—
298C2#1972711yescharacteristic
299C2#1975011yescharacteristic
300C2#2850911yescharacteristic
301C2#2854412yes—
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#493C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33316C2semidirect
2. #29190C2semidirect
3. #29139C2semidirect
5. #10677C2semidirect
6. #10672C2semidirect
9. #8819C2semidirect
10. #8815C2semidirect
11. #8262C2semidirect
53. #493Q8semidirect

Maximal subgroups

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

subgroupindex
#333162
#291902
#291392
#216902
#106772
#106722
#98372
#98242
#88192
#88152
#82622