Finite Group Databasebeta

← #345661

Subgroups — #345661

Order 512.

512 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 146 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#26512C212yesmaximal
2#26441C212yesmaximal
3#10597C212yesmaximal
4#10589C212yesmaximal
5#10580C212yesmaximal
6#9839C211yesmaximal, characteristic
7#9599C212yesmaximal
8#9066C211yesmaximal, characteristic
9#9066C211yesmaximal, characteristic
10C2×#494C2211yescharacteristic
11#2552C2212yes—
12#2369C2211yescharacteristic
13#2369C2211yescharacteristic
14#2100—22no—
15#2088—22no—
16#2087—22no—
17#2059—22no—
18#1990C2212yes—
19#1990C2212yes—
20#1989—22no—
21#1987C2212yes—
22#1987C2212yes—
23#1948C2212yes—
24#1947—22no—
25#1947C2212yes—
26#1939C2212yes—
27#1939C2212yes—
28#1921—22no—
29#1918—22no—
30#1844C2211yescharacteristic
31C4×#107C2212yes—
32C2×#334C2212yes—
33C2×#333C2211yescharacteristic
34C2×#333C2211yescharacteristic
35C2×#322C2211yescharacteristic
36#1296C2212yes—
37#1279C2212yes—
38#1279C2212yes—
39C22×Q16—41no—
40C2×C4⋊Q8C2311yescharacteristic
41C2×C4⋊Q8C2311yescharacteristic
42C2×#125—22no—
43#494—41no—
44#494Q814yes—
45C2×#107—22no—
46C2×#107D412yes—
47C2×#107D412yes—
48C2×#106—22no—
49C2×#106—41no—
50C2×C4⋊C8—22no—
51C2×C4⋊C8—22no—
52C2×C4⋊C8C2312yes—
53C2×C4⋊C8C2312yes—
54C2×C4×C8C2312yes—
55C2×C4×C8C2311yescharacteristic
56#398—42no—
57#391—41no—
58#391—41no—
59#389—22no—
60#389—22no—
61#389—22no—
62#389—22no—
63#384—22no—
64#384C2312yes—
65C4×C4⋊C4—22no—
66C4×C4⋊C4—22no—
67C4×C4⋊C4C2312yes—
68C4×C4⋊C4C2312yes—
69#340—22no—
70#340—22no—
71#340—22no—
72#340—22no—
73#340—22no—
74#340—22no—
75#340—22no—
76#340—22no—
77#340—22no—
78#340—22no—
79#340D412yes—
80#340D412yes—
81#336—22no—
82#336—22no—
83#334—42no—
84#333—41no—
85#333—41no—
86#322—41no—
87C22×Q8—21no—
88C22×Q8—21no—
89C2×Q16—44no—
90C2×Q16—82no—
91C2×Q16—82no—
92C22×C8—21no—
93C22×C8—22no—
94C22×C8—22no—
95C22×C8C2×D412yes—
96C22×C8C2×D412yes—
97C4⋊Q8—21no—
98C4⋊Q8—21no—
99C4⋊Q8—22no—
100C4⋊Q8—22no—
101C4⋊Q8C2×Q812yes—
102C4⋊Q8C2×Q812yes—
103#125—44no—
104#125—82no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—42no—
111C2×C4⋊C4—42no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4C2×D412yes—
116C2×C4⋊C4C2×D412yes—
117C2×C4⋊C4C2×D412yes—
118C2×C4⋊C4C2×D412yes—
119C2×C42—22no—
120C2×C42—22no—
121C2×C42—41no—
122C2×C42—41no—
123C2×C42C2411yescharacteristic
124#107—24no—
125#107—42no—
126#107—44no—
127#106—82no—
128#106—82no—
129C4⋊C8—22no—
130C4⋊C8—22no—
131C4⋊C8—22no—
132C4⋊C8—22no—
133C4⋊C8—44no—
134C4⋊C8—82no—
135C4×C8—22no—
136C4×C8—21no—
137C4×C8—22no—
138C4×C8C2×Q812yes—
139#95—22no—
140#95—22no—
141#95—42no—
142#95—42no—
143#95—42no—
144C2×Q8—22no—
145C2×Q8—22no—
146C2×Q8—41no—
147C2×Q8—41no—
148C2×Q8—81no—
149C2×Q8—81no—
150C2×Q8—81no—
151C2×Q8—81no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—42no—
159C22×C4C22×D412yes—
160C22×C4H(5,𝔽2)11yescharacteristic
161Q16—88no—
162C2×C8—22no—
163C2×C8—22no—
164C2×C8—22no—
165C2×C8—22no—
166C2×C8—22no—
167C2×C8—22no—
168C2×C8—22no—
169C2×C8—22no—
170C2×C8—22no—
171C2×C8—42no—
172C2×C8—42no—
173C2×C8—42no—
174C2×C8—42no—
175C2×C8—42no—
176C2×C8—42no—
177C4⋊C4—22no—
178C4⋊C4—24no—
179C4⋊C4—24no—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—22no—
183C4⋊C4—22no—
184C4⋊C4—22no—
185C4⋊C4—22no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—82no—
193C4⋊C4—84no—
194C4⋊C4—84no—
195C4⋊C4—162no—
196C4⋊C4—161no—
197C4⋊C4—161no—
198C4⋊C4#13714yes—
199C42—42no—
200C42—42no—
201C42—42no—
202C42—42no—
203C42—161no—
204C42—161no—
205C42C22×Q811yescharacteristic
206C42H(5,𝔽2)11yescharacteristic
207C422−1+412yes—
208C23D4211yescharacteristic
209Q8—82no—
210Q8—82no—
211Q8—161no—
212Q8—161no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—24no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—81no—
230C2×C4—81no—
231C2×C4—81no—
232C2×C4—81no—
233C2×C4—82no—
234C2×C4—82no—
235C2×C4—82no—
236C2×C4#53712yes—
237C2×C4#54311yescharacteristic
238C2×C4D4×Q812yes—
239C2×C4#56311yescharacteristic
240C2×C4C2×#13712yes—
241C2×C4#57512yes—
242C2×C4#57712yes—
243C2×C4#57812yes—
244C8—24no—
245C8—42no—
246C8—42no—
247C8—82no—
248C8—82no—
249C22#268311yescharacteristic
250C22#326712yes—
251C22#327012yes—
252C22#329111yescharacteristic
253C22#329211yescharacteristic
254C4—21no—
255C4—21no—
256C4—44no—
257C4—82no—
258C4—82no—
259C4—82no—
260C4—82no—
261C4—82no—
262C4—162no—
263C4#325212yes—
264C4#325812yes—
265C4#338712yes—
266C4#339012yes—
267C2#2009711yescharacteristic
268C2#2011412yes—
269C2#2014311yescharacteristic
270C2#3125312yes—
271C2#3125411yescharacteristic
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

Maximal subgroups

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

subgroupindex
#265122
#264412
#105972
#105892
#105802
#98392
#95992
#90662