Finite Group Databasebeta

← #142193

Subgroups — #142193

Order 512.

390 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 102 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#33209C211yesmaximal, characteristic
2#12783C211yesmaximal, characteristic
3#10822C211yesmaximal, characteristic
4#7259C214yesmaximal
5C2×#500C411yescharacteristic
6C22×C4×C8C2211yescharacteristic
7#2857—21no—
8C2×C4⋊C16—21no—
9C2×C22⋊C16C2212yes—
10#1990—21no—
11#1942—21no—
12#1942C412yes—
13#1845C411yescharacteristic
14#1808—21no—
15#1808C2212yes—
16C2×#341—21no—
17C2×#341C2212yes—
18#1728—21no—
19#1374—21no—
20C22×M16—22no—
21C23×C8D411yescharacteristic
22C23×C8C2311yescharacteristic
23C2×C4⋊Q8—21no—
24C2×C22⋊Q8—41no—
25C22×C42D411yescharacteristic
26C22×C16—22no—
27#500—42no—
28C2×#107—21no—
29C2×#107—21no—
30C2×#107C2×C412yes—
31C2×#107C2×C412yes—
32C2×C4×C8—42no—
33C2×C4×C8C2×C411yescharacteristic
34C2×C4×C8C2×C411yescharacteristic
35C2×C4×C8D411yescharacteristic
36C2×C4×C8D411yescharacteristic
37#391—41no—
38#389—22no—
39#389—41no—
40#384—21no—
41#380—22no—
42#380—41no—
43C4⋊C16—42no—
44C22⋊C16—24no—
45#341—22no—
46#341—42no—
47#341Q814yescomplemented
48#340—22no—
49#340—22no—
50#336—41no—
51#336—41no—
52C22×Q8—41no—
53C23×C4SD1611yescharacteristic
54C23×C4SD1611yescharacteristic
55C23×C4C2×D411yescharacteristic
56C2×M16—24no—
57C2×M16—42no—
58C2×M16—42no—
59C2×M16—42no—
60C2×M16—42no—
61C2×M16—42no—
62C22×C8—22no—
63C22×C8—22no—
64C22×C8—41no—
65C22×C8—41no—
66C22×C8—41no—
67C22×C8—42no—
68C22×C8C22⋊C411yescharacteristic
69C22×C8C22⋊C411yescharacteristic
70C22×C8C22×C411yescharacteristic
71C22×C8C2×D411yescharacteristic
72C22×C8C2×Q812yes—
73C22×C8#4112yes—
74C4⋊Q8—42no—
75C4⋊Q8—42no—
76C22⋊Q8—84no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—41no—
83C2×C22⋊C4—42no—
84C2×C42—41no—
85C2×C42—41no—
86C2×C42C22⋊C411yescharacteristic
87C2×C42C22⋊C411yescharacteristic
88C2×C42D811yescharacteristic
89C2×C42D811yescharacteristic
90C2×C16—42no—
91C2×C16—42no—
92C2×C16—42no—
93#107—42no—
94#107—42no—
95#107—42no—
96#107—42no—
97C4×C8—22no—
98C4×C8—22no—
99C4×C8—41no—
100C4×C8—41no—
101#95—42no—
102#95—42no—
103C24C2×SD1611yescharacteristic
104C2×Q8—82no—
105C2×Q8—82no—
106C2×Q8—82no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—41no—
115C22×C4—41no—
116C22×C4—41no—
117C22×C4—42no—
118C22×C4—42no—
119C22×C4#10211yescharacteristic
120C22×C4#10211yescharacteristic
121C22×C4#10211yescharacteristic
122C22×C4#10211yescharacteristic
123C22×C4C2×C22⋊C411yescharacteristic
124C22×C4C22⋊Q812yes—
125C22×C4C2×D811yescharacteristic
126C22×C4#13712yes—
127M16—44no—
128M16—44no—
129M16—82no—
130M16—82no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—42no—
141C2×C8—42no—
142C2×C8—44no—
143C2×C8—44no—
144C2×C8—82no—
145C2×C8—82no—
146C2×C8#9912yes—
147C2×C8C4×Q812yes—
148C2×C8#12411yescharacteristic
149C2×C8C4⋊Q811yescharacteristic
150C4⋊C4—42no—
151C4⋊C4—42no—
152C4⋊C4—42no—
153C4⋊C4—42no—
154C4⋊C4—84no—
155C4⋊C4—82no—
156C4⋊C4—82no—
157C4⋊C4—82no—
158C22⋊C4—84no—
159C42—21no—
160C42—21no—
161C42—21no—
162C42—21no—
163C42—41no—
164C42—41no—
165C16—82no—
166C23—21no—
167C23—21no—
168C23—21no—
169C23—21no—
170C23—21no—
171C23—21no—
172C23C2×#10211yescharacteristic
173C23#47512yes—
174Q8—164no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—44no—
191C2×C4—44no—
192C2×C4—82no—
193C2×C4—82no—
194C2×C4—82no—
195C2×C4—82no—
196C2×C4—82no—
197C2×C4—82no—
198C2×C4#35912yes—
199C2×C4#36112yes—
200C2×C4#36212yes—
201C2×C4#39111yescharacteristic
202C2×C4C2×#9911yescharacteristic
203C2×C4#41011yescharacteristic
204C2×C4#41911yescharacteristic
205C2×C4#42111yescharacteristic
206C2×C4#42111yescharacteristic
207C2×C4#47412yes—
208C8—21no—
209C8—22no—
210C8—21no—
211C8—22no—
212C8—41no—
213C8—84no—
214C22—22no—
215C22—22no—
216C22—22no—
217C22—41no—
218C22—41no—
219C22—41no—
220C22—41no—
221C22#151711yescharacteristic
222C22#151911yescharacteristic
223C22#198811yescharacteristic
224C22C2×#35911yescharacteristic
225C22#214311yescharacteristic
226C22C2×#36111yescharacteristic
227C22C2×#36211yescharacteristic
228C4—22no—
229C4—22no—
230C4—22no—
231C4—22no—
232C4—41no—
233C4—162no—
234C4—162no—
235C4#151111yescharacteristic
236C4#151511yescharacteristic
237C4#199511yescharacteristic
238C4#199611yescharacteristic
239C2—42no—
240C2#991211yescharacteristic
241C2#1186211yescharacteristic
242C2#1186511yescharacteristic
243C2#1189211yescharacteristic
244C2#1189511yescharacteristic
245C2#1327411yescharacteristic
246C2#1327911yescharacteristic
subgroupG/H
center, socleC23C2×#102
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC24C2×SD16
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
47. #341Q8semidirect

Maximal subgroups

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

subgroupindex
#332092
#127832
#108222
#72592