Finite Group Databasebeta

← #320495

Subgroups — #320495

Order 512.

874 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 874 conjugacy classes of subgroups — too many to draw. Only its 190 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#33231C211yesmaximal, characteristic
2#29177C212yesmaximal, complemented
3#28441C212yesmaximal
4#21727C211yesmaximal, characteristic, complemented
5#10672C212yesmaximal, complemented
6#10315C212yesmaximal, complemented
7#9829C212yesmaximal, complemented
8#9820C212yesmaximal, complemented
9#8262C211yesmaximal, characteristic, complemented
10C2×#498C2211yescharacteristic
11C2×#492C2212yes—
12C2×#492C2212yes—
13C2×#490—22no—
14C2×#460—24no—
15#2867C2211yescharacteristic
16#2859C2211yescharacteristic
17C4×C4⋊Q8C2212yes—
18#2088—24no—
19#2014—24no—
20#1990C2212yes—
21#1990C2212yes—
22#1990C2212yes—
23#1961—22no—
24#1957C2212yes—
25#1957C2212yes—
26#1957C2212yes—
27#1957C2212yes—
28#1944—22no—
29#1919—24no—
30#1845C2211yescharacteristic
31#1844C2212yes—
32#1834C2212yes—
33#1776—22no—
34C4×C4⋊C8C2212yes—
35C4×#97C2211yescharacteristic
36C2×#322C2211yescharacteristic
37C2×C82C2211yescharacteristic
38#1273C2212yes—
39C22×SD16—24no—
40C22×SD16—24no—
41C2×C4⋊Q8C2312yes—
42C2×#127—21no—
43C2×#127—21no—
44C2×#127—22no—
45C2×#127—22no—
46C2×#127C2312yes—
47C2×#125—21no—
48C2×C4×Q8—22no—
49#498—24no—
50#492—24no—
51#492—24no—
52#492—42no—
53#492—42no—
54#490—48no—
55#460—416no—
56C2×#107—22no—
57C2×#106—22no—
58C2×#106—24no—
59C2×C4⋊C8D414yes—
60C2×#102—24no—
61C2×#102—24no—
62C2×#102—24no—
63C2×#102—24no—
64C2×#97D412yes—
65C2×#97D412yes—
66C2×C4×C8C2312yes—
67C2×C4×C8C2312yes—
68C2×C4×C8C2311yescharacteristic
69C2×C4×C8C2311yescharacteristic
70#391—22no—
71#389—22no—
72#389—22no—
73#389—22no—
74#389—22no—
75#384C2312yes—
76#384C2312yes—
77#382—21no—
78C4×C4⋊C4—22no—
79C4×C4⋊C4—22no—
80C43C2311yescharacteristic
81#340—24no—
82#340—24no—
83#340—24no—
84#340—24no—
85#336D414yes—
86#322—41no—
87C82—41no—
88C22×Q8—22no—
89C22×D4—22no—
90C22×D4—22no—
91C22×D4—22no—
92C22×D4—41no—
93C22×D4—41no—
94C22×D4—42no—
95C2×SD16—48no—
96C2×SD16—48no—
97C2×SD16—48no—
98C2×SD16—48no—
99C2×SD16—84no—
100C2×SD16—84no—
101C22×C8C2×D414yes—
102C22×C8C2×D412yes—
103C22×C8C2×D412yes—
104C22×C8C2×D414yes—
105C4⋊Q8—22no—
106C4⋊Q8—22no—
107C4⋊Q8—22no—
108C4⋊Q8#4114yes—
109#127—22no—
110#127—22no—
111#127—22no—
112#127—22no—
113#127—44no—
114#127—48no—
115#127—44no—
116#127—84no—
117#125—44no—
118C4×Q8—48no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4—42no—
129C2×C42—42no—
130C2×C42C2×D412yes—
131C2×C42C2×D412yes—
132C2×C42C2×D411yescharacteristic
133C2×C42C2×D411yescharacteristic
134C2×C42C2411yescharacteristic
135#107—44no—
136#106—28no—
137#106—48no—
138C4⋊C8—28no—
139#102—48no—
140#102—48no—
141#102—48no—
142#102—48no—
143#97—42no—
144#97SD1618yescomplemented
145C4×C8—22no—
146C4×C8—22no—
147C4×C8—22no—
148C4×C8—21no—
149C4×C8—22no—
150C4×C8—22no—
151C4×C8—21no—
152#95—42no—
153#95—42no—
154#95—41no—
155#95—41no—
156C24—42no—
157C2×Q8—24no—
158C2×Q8—42no—
159C2×Q8—44no—
160C2×Q8—44no—
161C2×D4—42no—
162C2×D4—42no—
163C2×D4—42no—
164C2×D4—42no—
165C2×D4—42no—
166C2×D4—42no—
167C2×D4—44no—
168C2×D4—44no—
169C2×D4—44no—
170C2×D4—44no—
171C2×D4—44no—
172C2×D4—44no—
173C2×D4—84no—
174C2×D4—84no—
175C2×D4—82no—
176C2×D4—82no—
177C2×D4—84no—
178C2×D4—84no—
179C2×D4—84no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4#12711yescharacteristic
184C22×C4#12711yescharacteristic
185C22×C4#12712yes—
186C22×C4C22×D411yescharacteristic
187C22×C4C22×D412yes—
188SD16—816no—
189SD16—816no—
190C2×C8—22no—
191C2×C8—22no—
192C2×C8—24no—
193C2×C8—22no—
194C2×C8—22no—
195C2×C8—24no—
196C2×C8—24no—
197C2×C8—24no—
198C2×C8—24no—
199C2×C8C2×SD1614yes—
200C2×C8C2×SD1614yes—
201C2×C8AGL(1,ℤ/8)18yes—
202C4⋊C4—24no—
203C4⋊C4—24no—
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—84no—
213C4⋊C4—84no—
214C42—24no—
215C42—22no—
216C42—24no—
217C42—84no—
218C42C2×SD1614yes—
219C42C2×#4112yes—
220C42H(5,𝔽2)11yescharacteristic
221C422−1+411yescharacteristic
222C23—82no—
223C23—82no—
224C23—82no—
225C23—82no—
226C23—82no—
227C23—82no—
228C23—82no—
229C23C2×#12711yescharacteristic
230Q8—48no—
231Q8—84no—
232D4—84no—
233D4—84no—
234D4—84no—
235D4—84no—
236D4—84no—
237D4—84no—
238D4—168no—
239D4—164no—
240D4—164no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—44no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4—82no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4#49214yes—
257C2×C4#52511yescharacteristic
258C2×C4#53711yescharacteristic
259C2×C4#54612yes—
260C2×C4#54812yes—
261C2×C4C22×SD1612yes—
262C2×C4C2×AGL(1,ℤ/8)14yes—
263C2×C4#57512yes—
264C2×C4#57714yes—
265C8—24no—
266C8—24no—
267C8—28no—
268C8—44no—
269C22—162no—
270C22—162no—
271C22—162no—
272C22—162no—
273C22—162no—
274C22—162no—
275C22—162no—
276C22#254011yescharacteristic
277C22C2×#49211yescharacteristic
278C22#315112yes—
279C22#315311yescharacteristic
280C22#315412yes—
281C4—24no—
282C4—41no—
283C4—41no—
284C4—84no—
285C4—162no—
286C4—162no—
287C4#310312yes—
288C4#324612yes—
289C4#331014yes—
290C4#336814yes—
291C2—322no—
292C2#1962112yes—
293C2#1971711yescharacteristic
294C2#1974711yescharacteristic
295C2#2837411yescharacteristic
296C2#2843512yes—
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radical#2867C22
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
2. #29177C2semidirect
4. #21727C2semidirect
5. #10672C2semidirect
6. #10315C2semidirect
7. #9829C2semidirect
8. #9820C2semidirect
9. #8262C2semidirect
144. #97SD16semidirect

Maximal subgroups

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

subgroupindex
#332312
#291772
#284412
#217272
#106722
#103152
#98292
#98202
#82622