Finite Group Databasebeta

← #102795

Subgroups — #102795

Order 512.

701 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 701 conjugacy classes of subgroups — too many to draw. Only its 112 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#30239C211yesmaximal, characteristic
2#27909C212yesmaximal, complemented
3#11373C211yesmaximal, characteristic, complemented
4#9680C212yesmaximal, complemented
5#7016C211yesmaximal, characteristic, complemented
6C2×#480—24no—
7C2×#466—22no—
8#2856C2212yes—
9C2×#384C2211yescharacteristic
10#2093—24no—
11#2093—22no—
12#2011—22no—
13#1957—22no—
14#1845—22no—
15C4×C22⋊C8C2212yes—
16C2×#340—22no—
17C2×#336C2211yescharacteristic
18C2×#327—24no—
19C4⋊#95C2211yescharacteristic
20C22×D8—22no—
21C23×C8D412yes—
22C2×#127D412yes—
23C2×#121—22no—
24C2×#121—22no—
25C2×#121—22no—
26C22×C4⋊C4—22no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4—21no—
29C22×C42C2311yescharacteristic
30#480—48no—
31#480—48no—
32#466—44no—
33#466—44no—
34C2×#107—28no—
35C2×#107—22no—
36C2×#102—28no—
37C2×#102—24no—
38C2×C22⋊C8D414yes—
39C2×C4×C8—22no—
40#390—22no—
41#390—22no—
42#390—22no—
43#384—22no—
44#384—21no—
45#384D412yes—
46#384D412yes—
47C2×#95—22no—
48#340—44no—
49#336—22no—
50#336—22no—
51#327—48no—
52C22×D4—22no—
53C22×D4—22no—
54C22×D4—22no—
55C22×D4—42no—
56C23×C4—41no—
57C23×C4—41no—
58C23×C4C2×D411yescharacteristic
59C23×C4C2×D412yes—
60C2×D8—44no—
61C2×D8—82no—
62C2×D8—82no—
63C22×C8—22no—
64C22×C8—24no—
65C22×C8—22no—
66C22×C8—42no—
67C22×C8—42no—
68C22×C8D814yes—
69#127—44no—
70#121—44no—
71#121—44no—
72#121—44no—
73#121—44no—
74#121—44no—
75#121—44no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—24no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4—44no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—44no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—42no—
91C2×C22⋊C4—42no—
92C2×C22⋊C4—42no—
93C2×C22⋊C4—42no—
94C2×C42—42no—
95C2×C42C2×D412yes—
96C2×C42C2×D411yescharacteristic
97C2×C42#4111yescharacteristic
98#107—44no—
99#107—416no—
100#102—48no—
101#102—416no—
102C22⋊C8—24no—
103C22⋊C8D818yescomplemented
104C4×C8—44no—
105#95—44no—
106C24—42no—
107C24#12711yescharacteristic
108C2×D4—44no—
109C2×D4—44no—
110C2×D4—44no—
111C2×D4—44no—
112C2×D4—42no—
113C2×D4—42no—
114C2×D4—44no—
115C2×D4—44no—
116C2×D4—82no—
117C2×D4—82no—
118C2×D4—82no—
119C2×D4—82no—
120C2×D4—82no—
121C2×D4—82no—
122C2×D4—82no—
123C2×D4—82no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—22no—
127C22×C4—42no—
128C22×C4—42no—
129C22×C4—42no—
130C22×C4—42no—
131C22×C4—42no—
132C22×C4—81no—
133C22×C4—82no—
134C22×C4—82no—
135C22×C4—81no—
136C22×C4—81no—
137C22×C4—82no—
138C22×C4—81no—
139C22×C4—82no—
140C22×C4#12012yes—
141C22×C4#12011yescharacteristic
142C22×C4#12111yescharacteristic
143C22×C4#12112yes—
144C22×C4C2×D812yes—
145C22×C4C2×D814yes—
146C22×C4AGL(1,ℤ/8)12yes—
147C22×C4AGL(1,ℤ/8)14yes—
148D8—88no—
149C2×C8—28no—
150C2×C8—24no—
151C2×C8—44no—
152C2×C8—44no—
153C2×C8—42no—
154C2×C8—42no—
155C2×C8—44no—
156C2×C8—82no—
157C4⋊C4—44no—
158C4⋊C4—48no—
159C4⋊C4—44no—
160C4⋊C4—44no—
161C4⋊C4—44no—
162C4⋊C4—44no—
163C4⋊C4—82no—
164C4⋊C4—82no—
165C4⋊C4—84no—
166C22⋊C4—84no—
167C22⋊C4—84no—
168C22⋊C4—84no—
169C42—24no—
170C42—41no—
171C42—41no—
172C23—21no—
173C23—22no—
174C23—21no—
175C23—82no—
176C23—82no—
177C23—82no—
178C23—82no—
179C23—82no—
180C23—82no—
181C23—82no—
182C23#39211yescharacteristic
183C23#49312yes—
184C23#49311yescharacteristic
185C23#49612yes—
186C23#49611yescharacteristic
187D4—88no—
188D4—84no—
189D4—84no—
190D4—164no—
191D4—162no—
192D4—162no—
193D4—162no—
194D4—162no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—44no—
206C2×C4—44no—
207C2×C4—82no—
208C2×C4—84no—
209C2×C4—82no—
210C2×C4—82no—
211C2×C4—82no—
212C2×C4—84no—
213C2×C4—82no—
214C2×C4—162no—
215C2×C4—161no—
216C2×C4—161no—
217C2×C4—161no—
218C2×C4—161no—
219C2×C4—162no—
220C2×C4#44714yes—
221C2×C4#44712yes—
222C2×C4#45511yescharacteristic
223C2×C4#45611yescharacteristic
224C2×C4#45812yes—
225C2×C4#45912yes—
226C2×C4#45914yes—
227C8—48no—
228C8—44no—
229C22—22no—
230C22—22no—
231C22—22no—
232C22—22no—
233C22—21no—
234C22—21no—
235C22—21no—
236C22—21no—
237C22—162no—
238C22—162no—
239C22—162no—
240C22—162no—
241C22—162no—
242C22—162no—
243C22—162no—
244C22#171211yescharacteristic
245C22#172112yes—
246C22#172111yescharacteristic
247C22#200711yescharacteristic
248C22#200912yes—
249C22#201112yes—
250C22#201111yescharacteristic
251C22#202211yescharacteristic
252C4—24no—
253C4—24no—
254C4—22no—
255C4—162no—
256C4—162no—
257C4—162no—
258C4#160712yes—
259C4#161012yes—
260C2—21no—
261C2—22no—
262C2—21no—
263C2—322no—
264C2#995212yes—
265C2#997011yescharacteristic
266C2#1020412yes—
267C2#1022511yescharacteristic
268C2#1030411yescharacteristic
subgroupG/H
center, socleC23#392
commutator, Frattini, cosocleC22×C42C23
prime radical#30239C2
LCSG ▷ C22×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #27909C2semidirect
3. #11373C2semidirect
4. #9680C2semidirect
5. #7016C2semidirect
103. C22⋊C8D8semidirect

Maximal subgroups

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

subgroupindex
#302392
#279092
#113732
#96802
#70162