Finite Group Databasebeta

← #138174

Subgroups — #138174

Order 512.

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

normal subgroups only

This group has 442 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#33172C211yesmaximal, characteristic
3C2×#1362C212yesmaximal, complemented
4#8596C211yesmaximal, characteristic
5#7274C212yesmaximal, complemented
6C2×#494C2211yescharacteristic
7C2×#467—21no—
8C2×#467C412yes—
9C22×C4×C8C2211yescharacteristic
10#2857—21no—
11#2163—24no—
12C2×C4⋊C16C2212yes—
13C2×C22⋊C16—21no—
14C2×#346—21no—
15#1969—21no—
16#1969C412yes—
17#1956—21no—
18#1845C2211yescharacteristic
19#1797—21no—
20C2×#335C2212yes—
21#1362—24no—
22#1277—21no—
23C22×Q16—21no—
24C22×Q16C2×C412yes—
25C23×C8C2×C411yescharacteristic
26C23×C8D411yescharacteristic
27C2×C4⋊Q8—21no—
28C2×C22⋊Q8—22no—
29C22×C42D411yescharacteristic
30C22×C16—22no—
31#494—22no—
32#494—22no—
33#467—44no—
34#467—44no—
35C2×#107—21no—
36C2×#107C2×C412yes—
37C2×C4⋊C8—22no—
38C2×#103—22no—
39C2×#103—22no—
40C2×C22⋊C8—41no—
41C2×C4×C8—22no—
42C2×C4×C8—22no—
43C2×C4×C8C2×C411yescharacteristic
44C2×C4×C8D411yescharacteristic
45C2×C4×C8D411yescharacteristic
46C2×C4×C8C2311yescharacteristic
47#391—22no—
48#389—41no—
49#389—41no—
50#384—21no—
51#380—41no—
52#380—41no—
53C4⋊C16—22no—
54C4⋊C16D414yes—
55C22⋊C16—24no—
56#346—24no—
57#336—41no—
58#335—22no—
59#335—22no—
60C22×Q8—22no—
61C23×C4C22⋊C411yescharacteristic
62C23×C4D811yescharacteristic
63C23×C4SD1611yescharacteristic
64C2×Q16—24no—
65C2×Q16—42no—
66C2×Q16—42no—
67C2×Q16—41no—
68C2×Q16—41no—
69C2×Q16—42no—
70C2×Q16—42no—
71C22×C8—22no—
72C22×C8—22no—
73C22×C8—21no—
74C22×C8—21no—
75C22×C8—22no—
76C22×C8—22no—
77C22×C8—41no—
78C22×C8—42no—
79C22×C8C22⋊C411yescharacteristic
80C22×C8C22⋊C411yescharacteristic
81C22×C8C22×C411yescharacteristic
82C22×C8C2×D411yescharacteristic
83C4⋊Q8—42no—
84C4⋊Q8—42no—
85C22⋊Q8—44no—
86C22⋊Q8—44no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—41no—
93C2×C22⋊C4—42no—
94C2×C42—22no—
95C2×C42—22no—
96C2×C42C22⋊C411yescharacteristic
97C2×C42D811yescharacteristic
98C2×C42SD1611yescharacteristic
99C2×C42C2×D411yescharacteristic
100C2×C16—24no—
101C2×C16—42no—
102C2×C16—42no—
103#107—42no—
104#107—42no—
105C4⋊C8—42no—
106C4⋊C8—42no—
107#103—44no—
108#103—44no—
109C22⋊C8—82no—
110C4×C8—22no—
111C4×C8—21no—
112C4×C8—21no—
113C4×C8—22no—
114C4×C8C2×D412yes—
115C4×C8#4112yes—
116#95—42no—
117#95—42no—
118C24#10211yescharacteristic
119C2×Q8—42no—
120C2×Q8—42no—
121C2×Q8—42no—
122C2×Q8—42no—
123C2×Q8—42no—
124C2×Q8—42no—
125C22×C4—22no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—22no—
137C22×C4—41no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4#10211yescharacteristic
141C22×C4#10211yescharacteristic
142C22×C4#10211yescharacteristic
143C22×C4C2×C22⋊C411yescharacteristic
144C22×C4C2×D811yescharacteristic
145C22×C4C2×SD1611yescharacteristic
146Q16—84no—
147Q16—82no—
148Q16—82no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—44no—
157C2×C8—42no—
158C2×C8—42no—
159C2×C8—82no—
160C2×C8—82no—
161C2×C8—82no—
162C2×C8#10012yes—
163C2×C8C4×D412yes—
164C2×C8#12011yescharacteristic
165C2×C8#12311yescharacteristic
166C2×C8#13512yes—
167C2×C8AGL(1,ℤ/8)12yes—
168C4⋊C4—42no—
169C4⋊C4—42no—
170C4⋊C4—82no—
171C4⋊C4—84no—
172C4⋊C4—82no—
173C4⋊C4—82no—
174C4⋊C4—81no—
175C4⋊C4—81no—
176C22⋊C4—84no—
177C42—21no—
178C42—21no—
179C42—22no—
180C42—22no—
181C42#12112yes—
182C42#13512yes—
183C16—44no—
184C23—21no—
185C23—21no—
186C23—21no—
187C23—21no—
188C23—21no—
189C23—21no—
190C23#36212yes—
191C23C2×#10211yescharacteristic
192Q8—82no—
193Q8—82no—
194Q8—82no—
195Q8—82no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—44no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—82no—
215C2×C4—82no—
216C2×C4—82no—
217C2×C4—82no—
218C2×C4—82no—
219C2×C4—82no—
220C2×C4#38611yescharacteristic
221C2×C4C2×#10011yescharacteristic
222C2×C4#41311yescharacteristic
223C2×C4#41611yescharacteristic
224C2×C4#41611yescharacteristic
225C2×C4#41811yescharacteristic
226C2×C4#46512yes—
227C2×C4#46612yes—
228C2×C4#50911yescharacteristic
229C2×C4#51011yescharacteristic
230C2×C4#51012yes—
231C8—22no—
232C8—22no—
233C8—41no—
234C8—162no—
235C8#44912yes—
236C8#48212yes—
237C22—22no—
238C22—22no—
239C22—22no—
240C22—41no—
241C22—41no—
242C22—41no—
243C22—41no—
244C22#154011yescharacteristic
245C22#154511yescharacteristic
246C22#189311yescharacteristic
247C22#213911yescharacteristic
248C22#214111yescharacteristic
249C22C2×#36211yescharacteristic
250C22#214811yescharacteristic
251C4—22no—
252C4—41no—
253C4—41no—
254C4—41no—
255C4—162no—
256C4—162no—
257C4#154311yescharacteristic
258C4#155311yescharacteristic
259C4#187511yescharacteristic
260C4#190011yescharacteristic
261C4#221912yes—
262C4#222012yes—
263C2—42no—
264C2#913411yescharacteristic
265C2#1195311yescharacteristic
266C2#1195411yescharacteristic
267C2#1196511yescharacteristic
268C2#1196611yescharacteristic
269C2#1306211yescharacteristic
270C2#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×#1362C2semidirect
5. #7274C2semidirect

Maximal subgroups

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

subgroupindex
C2×#21632
#331722
C2×#13622
#85962
#72742