Finite Group Databasebeta

← #325957

Subgroups — #325957

Order 512.

496 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 170 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#28828C212yesmaximal
2#28797C212yesmaximal
3#10577C211yesmaximal, characteristic
4#10576C212yesmaximal
5#10574C211yesmaximal, characteristic
6#9621C212yesmaximal
7#9613C212yesmaximal
8#8820C211yesmaximal, characteristic
9#8288C212yesmaximal
10#2571C2212yes—
11#2571C2211yescharacteristic
12#2551C2211yescharacteristic
13#2085—22no—
14#2085—22no—
15#2085—22no—
16#2057—22no—
17#1993—22no—
18#1948C2212yes—
19#1948C2212yes—
20#1948C2212yes—
21#1947C2212yes—
22#1944—22no—
23#1942—22no—
24#1846—22no—
25#1845C2212yes—
26#1845C2211yescharacteristic
27#1844C2211yescharacteristic
28#1834C2212yes—
29C4×#107C2212yes—
30#1770C2212yes—
31C8×C4⋊C4C2212yes—
32C2×#334C2212yes—
33C2×#322C2211yescharacteristic
34#1296C2211yescharacteristic
35#1296C2212yes—
36#1296C2211yescharacteristic
37#1279C2212yes—
38#1279C2212yes—
39C2×C4⋊Q8—22no—
40C2×#107—22no—
41C2×#107—22no—
42C2×#107—22no—
43C2×#107—41no—
44C2×#107D412yes—
45C2×#107D412yes—
46C2×#106—22no—
47C2×#106—22no—
48C2×#106—22no—
49C2×#106—41no—
50C2×C4⋊C8C2312yes—
51C2×C4⋊C8C2312yes—
52C2×#97—22no—
53C2×C4×C8—22no—
54C2×C4×C8C2311yescharacteristic
55C2×C4×C8C2312yes—
56#400—42no—
57#400—41no—
58#395—41no—
59#389—22no—
60#389—22no—
61#389—22no—
62#389—22no—
63#384C2312yes—
64#384C2311yescharacteristic
65#384C2311yescharacteristic
66#382—22no—
67#382—22no—
68#382—22no—
69C4×C4⋊C4C2312yes—
70C4×C4⋊C4C2312yes—
71#340—22no—
72#340—22no—
73#340—22no—
74#340—22no—
75#340—22no—
76#340—22no—
77#340D412yes—
78#340D412yes—
79#336—22no—
80#336—22no—
81#334Q818yescomplemented
82#322Q814yescomplemented
83C22×Q8—41no—
84C22×C8—21no—
85C22×C8—22no—
86C22×C8—22no—
87C22×C8C2×D412yes—
88C22×C8C2×D412yes—
89C4⋊Q8—44no—
90C4⋊Q8—82no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4C2×D412yes—
102C2×C4⋊C4C2×D412yes—
103C2×C4⋊C4C2×D412yes—
104C2×C4⋊C4C2×D412yes—
105C2×C42—22no—
106C2×C42—22no—
107C2×C42C2411yescharacteristic
108#107—42no—
109#107—42no—
110#107—44no—
111#107—82no—
112#107—82no—
113#107—82no—
114#106—44no—
115#106—82no—
116#106—82no—
117#106—82no—
118C4⋊C8—22no—
119C4⋊C8—22no—
120C4⋊C8C2×Q814yes—
121C4⋊C8C2×Q814yes—
122#97—82no—
123C4×C8—21no—
124C4×C8—44no—
125C4×C8C2×Q812yes—
126C4×C8C2×Q814yes—
127C4×C8C2×Q814yes—
128#95—22no—
129#95—22no—
130#95—41no—
131#95—42no—
132#95—42no—
133#95—42no—
134#95—42no—
135#95—41no—
136C2×Q8—84no—
137C2×Q8—161no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—42no—
143C22×C4—41no—
144C22×C4—41no—
145C22×C4C22×D412yes—
146C22×C4H(5,𝔽2)11yescharacteristic
147C2×C8—22no—
148C2×C8—22no—
149C2×C8—22no—
150C2×C8—22no—
151C2×C8—22no—
152C2×C8—22no—
153C2×C8—22no—
154C2×C8—24no—
155C2×C8—24no—
156C2×C8—42no—
157C2×C8—42no—
158C2×C8—42no—
159C2×C8—42no—
160C2×C8C4⋊Q814yes—
161C2×C8C4⋊Q814yes—
162C4⋊C4—22no—
163C4⋊C4—22no—
164C4⋊C4—22no—
165C4⋊C4—22no—
166C4⋊C4—22no—
167C4⋊C4—22no—
168C4⋊C4—22no—
169C4⋊C4—22no—
170C4⋊C4—42no—
171C4⋊C4—42no—
172C4⋊C4—42no—
173C4⋊C4—42no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C4⋊C4—42no—
178C4⋊C4—81no—
179C4⋊C4—81no—
180C4⋊C4—82no—
181C4⋊C4—84no—
182C4⋊C4—161no—
183C4⋊C4—162no—
184C42—42no—
185C42—42no—
186C42—42no—
187C42—42no—
188C42C22×Q811yescharacteristic
189C42C22×Q812yes—
190C422−1+411yescharacteristic
191C23D4211yescharacteristic
192Q8—164no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—82no—
208C2×C4—82no—
209C2×C4—82no—
210C2×C4—82no—
211C2×C4—81no—
212C2×C4—82no—
213C2×C4—81no—
214C2×C4C2×C4⋊Q812yes—
215C2×C4#54411yescharacteristic
216C2×C4D4×Q812yes—
217C2×C4Q8211yescharacteristic
218C2×C4#57512yes—
219C2×C4#57512yes—
220C2×C4#57712yes—
221C2×C4#57712yes—
222C8—24no—
223C8—24no—
224C8—24no—
225C8—44no—
226C8—44no—
227C22#255111yescharacteristic
228C22#326712yes—
229C22#327112yes—
230C22#329211yescharacteristic
231C22#329311yescharacteristic
232C4—21no—
233C4—21no—
234C4—82no—
235C4—82no—
236C4—82no—
237C4—82no—
238C4—161no—
239C4—162no—
240C4—161no—
241C4#316312yes—
242C4#316612yes—
243C4#338812yes—
244C4#338812yes—
245C2#2013111yescharacteristic
246C2#2013512yes—
247C2#2015011yescharacteristic
248C2#2882212yes—
249C2#2882511yescharacteristic
subgroupG/H
center, socle, prime radicalC23D42
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
81. #334Q8semidirect
82. #322Q8semidirect

Maximal subgroups

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

subgroupindex
#288282
#287972
#105772
#105762
#105742
#96212
#96132
#88202
#82882