Finite Group Databasebeta

← #138116

Subgroups — #138116

Order 512.

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

normal subgroups only

This group has 398 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
1C2×#2163C211yesmaximal, characteristic
2#33209C211yesmaximal, characteristic
3C2×#1364C212yesmaximal, complemented
4#8596C211yesmaximal, characteristic
5#7274C212yesmaximal, complemented
6C2×#500C2211yescharacteristic
7C22×C4×C8C2211yescharacteristic
8#2857—21no—
9#2163—24no—
10C2×C4⋊C16C2212yes—
11C2×C22⋊C16—21no—
12C2×#346—21no—
13#1990—21no—
14#1942—21no—
15#1942C412yes—
16#1845C2211yescharacteristic
17#1808—21no—
18#1808C412yes—
19#1797—21no—
20C2×#335C2212yes—
21#1364—24no—
22#1277—21no—
23C23×C8C2×C411yescharacteristic
24C23×C8D411yescharacteristic
25C2×C4⋊Q8—21no—
26C2×C22⋊Q8—41no—
27C22×C42D411yescharacteristic
28C22×C16—22no—
29#500—22no—
30#500—22no—
31C2×#107—21no—
32C2×#107—21no—
33C2×#107C2×C412yes—
34C2×#107C2×C412yes—
35C2×C4⋊C8—22no—
36C2×C22⋊C8—41no—
37C2×C4×C8—22no—
38C2×C4×C8—22no—
39C2×C4×C8C2×C411yescharacteristic
40C2×C4×C8D411yescharacteristic
41C2×C4×C8D411yescharacteristic
42C2×C4×C8C2311yescharacteristic
43#391—41no—
44#389—22no—
45#389—41no—
46#384—21no—
47#380—22no—
48#380—41no—
49C4⋊C16—22no—
50C4⋊C16D414yes—
51C22⋊C16—24no—
52#346—24no—
53#340—22no—
54#340—22no—
55#336—41no—
56#335—22no—
57#335—22no—
58C22×Q8—41no—
59C23×C4C22⋊C411yescharacteristic
60C23×C4D811yescharacteristic
61C23×C4SD1611yescharacteristic
62C22×C8—22no—
63C22×C8—22no—
64C22×C8—21no—
65C22×C8—21no—
66C22×C8—22no—
67C22×C8—22no—
68C22×C8—41no—
69C22×C8—42no—
70C22×C8C22⋊C411yescharacteristic
71C22×C8C22⋊C411yescharacteristic
72C22×C8C22×C411yescharacteristic
73C22×C8C2×D411yescharacteristic
74C4⋊Q8—42no—
75C4⋊Q8—42no—
76C22⋊Q8—84no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—42no—
80C2×C4⋊C4—41no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—41no—
83C2×C22⋊C4—42no—
84C2×C42—22no—
85C2×C42—22no—
86C2×C42C22⋊C411yescharacteristic
87C2×C42D811yescharacteristic
88C2×C42SD1611yescharacteristic
89C2×C42C2×D411yescharacteristic
90C2×C16—24no—
91C2×C16—42no—
92C2×C16—42no—
93#107—24no—
94#107—41no—
95#107—41no—
96#107—42no—
97#107—42no—
98C4⋊C8—42no—
99C4⋊C8—42no—
100C22⋊C8—82no—
101C4×C8—22no—
102C4×C8—21no—
103C4×C8—21no—
104C4×C8—22no—
105C4×C8C2×D412yes—
106C4×C8#4112yes—
107#95—42no—
108#95—42no—
109C24#10211yescharacteristic
110C2×Q8—82no—
111C2×Q8—82no—
112C2×Q8—81no—
113C2×Q8—81no—
114C22×C4—22no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—21no—
124C22×C4—22no—
125C22×C4—21no—
126C22×C4—41no—
127C22×C4—42no—
128C22×C4—42no—
129C22×C4#10211yescharacteristic
130C22×C4#10211yescharacteristic
131C22×C4#10211yescharacteristic
132C22×C4C2×C22⋊C411yescharacteristic
133C22×C4C2×D811yescharacteristic
134C22×C4C2×SD1611yescharacteristic
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—44no—
143C2×C8—42no—
144C2×C8—42no—
145C2×C8—82no—
146C2×C8—82no—
147C2×C8—82no—
148C2×C8#10112yes—
149C2×C8C4×D412yes—
150C2×C8#12011yescharacteristic
151C2×C8#12311yescharacteristic
152C2×C8#13512yes—
153C2×C8#13712yes—
154C4⋊C4—42no—
155C4⋊C4—42no—
156C4⋊C4—42no—
157C4⋊C4—42no—
158C4⋊C4—82no—
159C4⋊C4—84no—
160C4⋊C4—82no—
161C4⋊C4—82no—
162C22⋊C4—84no—
163C42—21no—
164C42—21no—
165C42—22no—
166C42—22no—
167C42#12112yes—
168C42#13512yes—
169C16—44no—
170C23—21no—
171C23—21no—
172C23—21no—
173C23—21no—
174C23—21no—
175C23—21no—
176C23#36212yes—
177C23C2×#10211yescharacteristic
178Q8—162no—
179Q8—162no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—44no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—82no—
199C2×C4—82no—
200C2×C4—82no—
201C2×C4—82no—
202C2×C4—82no—
203C2×C4—82no—
204C2×C4#38611yescharacteristic
205C2×C4C2×#10111yescharacteristic
206C2×C4#41311yescharacteristic
207C2×C4#41611yescharacteristic
208C2×C4#41611yescharacteristic
209C2×C4#41911yescharacteristic
210C2×C4#46512yes—
211C2×C4#46612yes—
212C2×C4#50911yescharacteristic
213C2×C4#50912yes—
214C2×C4#51011yescharacteristic
215C8—22no—
216C8—22no—
217C8—41no—
218C8—162no—
219C8#45212yes—
220C8#48512yes—
221C22—22no—
222C22—22no—
223C22—22no—
224C22—41no—
225C22—41no—
226C22—41no—
227C22—41no—
228C22#154011yescharacteristic
229C22#154611yescharacteristic
230C22#189311yescharacteristic
231C22#213911yescharacteristic
232C22#214011yescharacteristic
233C22C2×#36211yescharacteristic
234C22#214811yescharacteristic
235C4—22no—
236C4—41no—
237C4—41no—
238C4—41no—
239C4—162no—
240C4—162no—
241C4#154411yescharacteristic
242C4#155411yescharacteristic
243C4#187311yescharacteristic
244C4#190111yescharacteristic
245C4#221812yes—
246C4#222012yes—
247C2—42no—
248C2#913011yescharacteristic
249C2#1195111yescharacteristic
250C2#1195411yescharacteristic
251C2#1197411yescharacteristic
252C2#1197611yescharacteristic
253C2#1306111yescharacteristic
254C2#1306611yescharacteristic
subgroupG/H
center, socleC23C2×#102
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC24#102
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1364C2semidirect
5. #7274C2semidirect

Maximal subgroups

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

subgroupindex
C2×#21632
#332092
C2×#13642
#85962
#72742