Finite Group Databasebeta

← #315875

Subgroups — #315875

Order 512.

844 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 844 conjugacy classes of subgroups — too many to draw. Only its 178 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33175C211yesmaximal, characteristic
2#27906C212yesmaximal, complemented
3#27891C212yesmaximal, complemented
4#20764C211yesmaximal, characteristic, complemented
5#10677C212yesmaximal, complemented
6#10672C212yesmaximal, complemented
7#9832C212yesmaximal, complemented
8#8819C212yesmaximal, complemented
9#8147C211yesmaximal, characteristic, complemented
10C2×#493C2211yescharacteristic
11C2×#481—24no—
12C2×#469—22no—
13C2×C4×M16C2211yescharacteristic
14#2856C2212yes—
15#2285C2211yescharacteristic
16C2×#384C2211yescharacteristic
17#2094—24no—
18#2033—24no—
19#1972—22no—
20#1957C2212yes—
21#1957C2212yes—
22#1957C2212yes—
23#1957C2212yes—
24#1957C2212yes—
25#1925—24no—
26#1845C2212yes—
27#1844C2212yes—
28#1844C2212yes—
29#1844C2211yescharacteristic
30#1836—22no—
31#1834C2212yes—
32#1834C2212yes—
33#1834C2212yes—
34C4×C22⋊C8C2212yes—
35#1736—22no—
36C2×#322C2212yes—
37#1273C2211yescharacteristic
38#1273C2211yescharacteristic
39C22×D8—22no—
40C22×M16D412yes—
41C2×#127C2312yes—
42C2×#121—22no—
43C2×#121—22no—
44C2×#121—22no—
45C2×#117—21no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—21no—
48C22×C42C2311yescharacteristic
49#493—24no—
50#481—416no—
51#469—48no—
52C2×#107—24no—
53C2×#106—24no—
54C2×#106—24no—
55C2×#106—22no—
56C2×#102—24no—
57C2×#102—24no—
58C2×#102—24no—
59C2×#102—24no—
60C2×#102—24no—
61C2×C22⋊C8D414yes—
62C4×M16—24no—
63C2×#97D412yes—
64C2×C4×C8C2311yescharacteristic
65C2×C4×C8C2312yes—
66C2×C4×C8C2312yes—
67C2×C4×C8C2311yescharacteristic
68#390—22no—
69#390—22no—
70#390—22no—
71#384—21no—
72#384—22no—
73#384C2312yes—
74#384C2312yes—
75#384C2312yes—
76#382—21no—
77#380—22no—
78C4×C4⋊C4—22no—
79#340—24no—
80#340—24no—
81#340—24no—
82#336D414yes—
83#322—24no—
84C22×D4—22no—
85C22×D4—22no—
86C22×D4—22no—
87C22×D4—42no—
88C23×C4—41no—
89C23×C4C2×D412yes—
90C23×C4C2×D411yescharacteristic
91C2×D8—44no—
92C2×D8—44no—
93C2×D8—44no—
94C2×M16—24no—
95C2×M16—24no—
96C2×M16—42no—
97C22×C8C2×D412yes—
98C22×C8C2×D414yes—
99C22×C8C2×D412yes—
100C22×C8C2×D414yes—
101#127—22no—
102#127—22no—
103#127—22no—
104#127—22no—
105#121—48no—
106#121—48no—
107#121—84no—
108#117—44no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—44no—
119C2×C4⋊C4—48no—
120C2×C4⋊C4—42no—
121C2×C4⋊C4—41no—
122C2×C22⋊C4—41no—
123C2×C22⋊C4—42no—
124C2×C22⋊C4—42no—
125C2×C22⋊C4—42no—
126C2×C42—22no—
127C2×C42—21no—
128C2×C42—21no—
129C2×C42—41no—
130C2×C42C2×D412yes—
131C2×C42C2×D411yescharacteristic
132C2×C42C2411yescharacteristic
133#107—84no—
134#106—44no—
135#106—48no—
136#106—48no—
137#102—48no—
138#102—48no—
139#102—48no—
140#102—48no—
141#102—84no—
142C22⋊C8—28no—
143#97—24no—
144C4×C8—22no—
145C4×C8—22no—
146C4×C8—22no—
147C4×C8—22no—
148C4×C8—22no—
149C4×C8#4114yes—
150#95—42no—
151C24—42no—
152C24#12711yescharacteristic
153C2×D4—42no—
154C2×D4—42no—
155C2×D4—42no—
156C2×D4—42no—
157C2×D4—42no—
158C2×D4—42no—
159C2×D4—44no—
160C2×D4—44no—
161C2×D4—44no—
162C2×D4—44no—
163C2×D4—44no—
164C2×D4—44no—
165C2×D4—84no—
166C2×D4—84no—
167C2×D4—84no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—24no—
174C22×C4—24no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—84no—
179C22×C4—82no—
180C22×C4#12712yes—
181C22×C4#12711yescharacteristic
182C22×C4AGL(1,ℤ/8)14yes—
183C22×C4C22×D411yescharacteristic
184C22×C4C22×D412yes—
185D8—88no—
186M16—48no—
187C2×C8—24no—
188C2×C8—22no—
189C2×C8—22no—
190C2×C8—24no—
191C2×C8—22no—
192C2×C8—24no—
193C2×C8—24no—
194C2×C8—22no—
195C2×C8—22no—
196C2×C8—24no—
197C2×C8#12314yes—
198C2×C8AGL(1,ℤ/8)18yes—
199C4⋊C4—44no—
200C4⋊C4—44no—
201C4⋊C4—44no—
202C4⋊C4—44no—
203C4⋊C4—44no—
204C4⋊C4—44no—
205C4⋊C4—44no—
206C4⋊C4—44no—
207C4⋊C4—44no—
208C4⋊C4—82no—
209C4⋊C4—84no—
210C22⋊C4—84no—
211C22⋊C4—84no—
212C22⋊C4—82no—
213C22⋊C4—84no—
214C42—21no—
215C42—21no—
216C42—22no—
217C42—22no—
218C42—82no—
219C42C2×#4112yes—
220C422−1+411yescharacteristic
221C422−1+411yescharacteristic
222C23—22no—
223C23—21no—
224C23—22no—
225C23—21no—
226C23—82no—
227C23—82no—
228C23—82no—
229C23—82no—
230C23—82no—
231C23—82no—
232C23—82no—
233C23#49612yes—
234C23C2×#12711yescharacteristic
235D4—84no—
236D4—84no—
237D4—84no—
238D4—84no—
239D4—84no—
240D4—84no—
241D4—168no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—24no—
245C2×C4—24no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—44no—
250C2×C4—44no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—164no—
261C2×C4#49612yes—
262C2×C4C2×#12311yescharacteristic
263C2×C4#53611yescharacteristic
264C2×C4#54712yes—
265C2×C4#54712yes—
266C2×C4C2×AGL(1,ℤ/8)14yes—
267C2×C4C2×AGL(1,ℤ/8)12yes—
268C2×C4#57512yes—
269C2×C4#57714yes—
270C8—28no—
271C8—24no—
272C8—42no—
273C8—44no—
274C22—22no—
275C22—24no—
276C22—41no—
277C22—41no—
278C22—42no—
279C22—162no—
280C22—162no—
281C22—162no—
282C22—162no—
283C22—162no—
284C22—162no—
285C22—162no—
286C22#250111yescharacteristic
287C22C2×#49611yescharacteristic
288C22#315112yes—
289C22#315311yescharacteristic
290C22#315412yes—
291C4—22no—
292C4—42no—
293C4—162no—
294C4—162no—
295C4—162no—
296C4#309412yes—
297C4#323412yes—
298C4#334514yes—
299C4#335414yes—
300C2—42no—
301C2—322no—
302C2#1967112yes—
303C2#1972411yescharacteristic
304C2#1974711yescharacteristic
305C2#2791811yescharacteristic
306C2#2794112yes—
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radical#33175C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #27906C2semidirect
3. #27891C2semidirect
4. #20764C2semidirect
5. #10677C2semidirect
6. #10672C2semidirect
7. #9832C2semidirect
8. #8819C2semidirect
9. #8147C2semidirect

Maximal subgroups

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

subgroupindex
#331752
#279062
#278912
#207642
#106772
#106722
#98322
#88192
#81472