Finite Group Databasebeta

← #138378

Subgroups — #138378

Order 512.

494 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 114 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#33130C211yesmaximal, characteristic
2C2×#1834C211yesmaximal, characteristic
3#12844C211yesmaximal, characteristic
4C2×#1348C212yesmaximal
5#7258C212yesmaximal
6C2×#467C2211yescharacteristic
7C22×C4×C8C2211yescharacteristic
8#2855—21no—
9C2×#384—21no—
10C2×C22⋊C16C2212yes—
11#1969C412yescomplemented
12#1942C412yescomplemented
13#1933—22no—
14#1892—21no—
15#1834—24no—
16#1808C2211yescharacteristic
17C2×#340—21no—
18C2×#340C2212yes—
19#1374—22no—
20#1348—24no—
21C22×Q16C2×C411yescharacteristic
22C23×C8D411yescharacteristic
23C23×C8C2311yescharacteristic
24C2×C22⋊Q8—21no—
25C22×C4⋊C4—22no—
26C22×C4⋊C4—41no—
27C22×C4⋊C4—41no—
28C22×C42D411yescharacteristic
29C22×C16—22no—
30#467—22no—
31#467—22no—
32C2×#107C2×C411yescharacteristic
33C2×#107C2×C412yes—
34C2×#103—22no—
35C2×#103—21no—
36C2×C4×C8—22no—
37C2×C4×C8—22no—
38C2×C4×C8C2×C412yes—
39C2×C4×C8D412yes—
40#391—22no—
41#389—22no—
42#387—42no—
43#385—41no—
44#384—24no—
45#384—42no—
46#381—41no—
47#380—21no—
48C22⋊C16—22no—
49C22⋊C16—22no—
50#340—24no—
51#340—22no—
52#340—21no—
53#340—22no—
54#340D414yes—
55C22×Q8—21no—
56C23×C4—42no—
57C23×C4D811yescharacteristic
58C23×C4D811yescharacteristic
59C23×C4C2×D411yescharacteristic
60C2×Q16—22no—
61C2×Q16—42no—
62C22×C8—21no—
63C22×C8—21no—
64C22×C8—42no—
65C22×C8—42no—
66C22×C8C22⋊C412yes—
67C22×C8C22×C411yescharacteristic
68C22×C8C2×D411yescharacteristic
69C22×C8C2×D412yes—
70C22×C8#4112yes—
71C22⋊Q8—41no—
72C22⋊Q8—41no—
73C22⋊Q8—41no—
74C22⋊Q8—41no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—24no—
78C2×C4⋊C4—44no—
79C2×C4⋊C4—44no—
80C2×C4⋊C4—44no—
81C2×C4⋊C4—44no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4—82no—
86C2×C4⋊C4—84no—
87C2×C4⋊C4—82no—
88C2×C22⋊C4—41no—
89C2×C22⋊C4—41no—
90C2×C42—21no—
91C2×C42—21no—
92C2×C42—21no—
93C2×C42—21no—
94C2×C42C22⋊C412yes—
95C2×C42SD1612yes—
96C2×C16—44no—
97C2×C16—42no—
98#107—22no—
99#107—42no—
100#103—44no—
101#103—42no—
102C4×C8—24no—
103C4×C8—24no—
104#95—42no—
105#95—42no—
106#95—42no—
107#95—41no—
108C24C2×D811yescharacteristic
109C2×Q8—41no—
110C2×Q8—42no—
111C2×Q8—41no—
112C2×Q8—42no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—24no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—42no—
123C22×C4—41no—
124C22×C4—41no—
125C22×C4—44no—
126C22×C4—84no—
127C22×C4—84no—
128C22×C4—82no—
129C22×C4—82no—
130C22×C4#10212yes—
131C22×C4#10212yes—
132C22×C4SD3212yes—
133C22×C4Q3212yes—
134C22×C4C2×C22⋊C411yescharacteristic
135C22×C4#12011yescharacteristic
136C22×C4#12311yescharacteristic
137C22×C4C2×SD1611yescharacteristic
138C22×C4AGL(1,ℤ/8)11yescharacteristic
139C22×C4#13711yescharacteristic
140Q16—82no—
141C2×C8—22no—
142C2×C8—22no—
143C2×C8—22no—
144C2×C8—22no—
145C2×C8—22no—
146C2×C8—42no—
147C2×C8—42no—
148C2×C8—42no—
149C2×C8—42no—
150C2×C8C4×D412yes—
151C2×C8#12112yes—
152C4⋊C4—44no—
153C4⋊C4—44no—
154C4⋊C4—41no—
155C4⋊C4—42no—
156C4⋊C4—41no—
157C4⋊C4—42no—
158C4⋊C4—84no—
159C4⋊C4—88no—
160C4⋊C4—88no—
161C4⋊C4—84no—
162C4⋊C4—81no—
163C4⋊C4—81no—
164C22⋊C4—81no—
165C22⋊C4—81no—
166C22⋊C4—81no—
167C22⋊C4—81no—
168C42—22no—
169C42—22no—
170C42—22no—
171C42—22no—
172C16—82no—
173C23—22no—
174C23—22no—
175C23C2×#10211yescharacteristic
176C23#44711yescharacteristic
177C23#48011yescharacteristic
178C23C2×SD3211yescharacteristic
179C23C2×Q3211yescharacteristic
180C23#50911yescharacteristic
181C23#51011yescharacteristic
182Q8—81no—
183Q8—81no—
184Q8—82no—
185C2×C4—24no—
186C2×C4—24no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—84no—
203C2×C4—84no—
204C2×C4—81no—
205C2×C4—81no—
206C2×C4—82no—
207C2×C4—82no—
208C2×C4—82no—
209C2×C4—84no—
210C2×C4—82no—
211C2×C4—82no—
212C2×C4—164no—
213C2×C4—164no—
214C2×C4#35814yes—
215C2×C4#36014yes—
216C2×C4#38611yescharacteristic
217C2×C4#41012yes—
218C2×C4#41711yescharacteristic
219C2×C4#42112yes—
220C2×C4#44811yescharacteristic
221C2×C4#48111yescharacteristic
222C8—22no—
223C8—22no—
224C8—42no—
225C22—22no—
226C22—22no—
227C22—22no—
228C22—22no—
229C22—21no—
230C22—21no—
231C22—21no—
232C22—21no—
233C22#151712yes—
234C22#187411yescharacteristic
235C22C2×#35811yescharacteristic
236C22#213911yescharacteristic
237C22C2×#36012yes—
238C22#218711yescharacteristic
239C22#218811yescharacteristic
240C22#223511yescharacteristic
241C22#223611yescharacteristic
242C4—24no—
243C4—21no—
244C4—21no—
245C4—24no—
246C4—164no—
247C4—162no—
248C4—161no—
249C4—161no—
250C4#151212yes—
251C4#189711yescharacteristic
252C4#189911yescharacteristic
253C2—21no—
254C2—21no—
255C2—21no—
256C2—21no—
257C2#915611yescharacteristic
258C2#1185712yes—
259C2#1186012yes—
260C2#1299311yescharacteristic
261C2#1300111yescharacteristic
subgroupG/H
center, socleC23C2×#102
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
11. #1969C4semidirect
12. #1942C4semidirect

Maximal subgroups

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

subgroupindex
#331302
C2×#18342
#128442
C2×#13482
#72582