Finite Group Databasebeta

← #126014

Subgroups — #126014

Order 512.

453 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 453 conjugacy classes of subgroups — too many to draw. Only its 64 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#17306C211yesmaximal, characteristic
2#10343C212yesmaximal
3#9967C211yesmaximal, characteristic
4#7132C212yesmaximal
5#7131C211yesmaximal, characteristic
6C2×#451—21no—
7#2659C2211yescharacteristic
8#1824—22no—
9C2×#328—21no—
10#1446—21no—
11#1446C2211yescharacteristic
12#1445C2212yes—
13#1444—21no—
14#1441—22no—
15#1441—22no—
16#1294—22no—
17#1280C2212yes—
18#1280C2211yescharacteristic
19C23×Q8D411yescharacteristic
20C22×Q16—41no—
21C2×C22⋊Q8—21no—
22C22×C4⋊C4—22no—
23C22×C4⋊C4—21no—
24C22×C4⋊C4D412yes—
25#451—42no—
26#451—42no—
27C2×#103—41no—
28C2×C22⋊C8—21no—
29C2×C22⋊C8—22no—
30#398—41no—
31#395—41no—
32#380—22no—
33C2×#95—21no—
34C2×#95—21no—
35C2×#95—22no—
36C2×#95—21no—
37C2×#95—22no—
38C2×#95—22no—
39C2×#95—22no—
40C2×#95D412yes—
41C2×#95D411yescharacteristic
42C2×#95C2311yescharacteristic
43#340—42no—
44#340—42no—
45#328—42no—
46C22×Q8—21no—
47C22×Q8—21no—
48C22×Q8—41no—
49C22×Q8—41no—
50C22×Q8—82no—
51C22×Q8SD1612yes—
52C23×C4—21no—
53C23×C4—22no—
54C23×C4—21no—
55C23×C4—22no—
56C23×C4C2×D411yescharacteristic
57C23×C4C2×D412yes—
58C2×Q16—44no—
59C2×Q16—84no—
60C22×C8—41no—
61C22×C8—42no—
62C22⋊Q8—41no—
63C22⋊Q8—41no—
64C22⋊Q8—41no—
65C22⋊Q8—41no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—42no—
71C2×C4⋊C4—42no—
72C2×C4⋊C4—44no—
73C2×C4⋊C4—41no—
74C2×C4⋊C4—82no—
75C2×C4⋊C4—82no—
76C2×C4⋊C4—81no—
77C2×C4⋊C4—82no—
78C2×C4⋊C4Q1614yescomplemented
79C2×C22⋊C4—41no—
80#103—44no—
81C22⋊C8—22no—
82C22⋊C8—41no—
83C22⋊C8—42no—
84C22⋊C8—42no—
85#95—41no—
86#95—41no—
87#95—42no—
88#95—42no—
89#95—42no—
90#95—82no—
91#95—82no—
92#95—82no—
93#95—82no—
94#95—81no—
95#95—81no—
96C24#12011yescharacteristic
97C2×Q8—42no—
98C2×Q8—41no—
99C2×Q8—41no—
100C2×Q8—44no—
101C2×Q8—81no—
102C2×Q8—82no—
103C2×Q8—82no—
104C2×Q8—82no—
105C2×Q8—82no—
106C2×Q8—84no—
107C22×C4—22no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—24no—
114C22×C4—42no—
115C22×C4—42no—
116C22×C4—42no—
117C22×C4—42no—
118C22×C4—42no—
119C22×C4—42no—
120C22×C4—42no—
121C22×C4—42no—
122C22×C4—42no—
123C22×C4—42no—
124C22×C4—41no—
125C22×C4—41no—
126C22×C4—41no—
127C22×C4—42no—
128C22×C4—42no—
129C22×C4—42no—
130C22×C4—41no—
131C22×C4—42no—
132C22×C4—42no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4C2×SD1611yescharacteristic
141C22×C4C2×Q1612yes—
142C22×C4#13711yescharacteristic
143C22×C4#13712yes—
144Q16—88no—
145C2×C8—42no—
146C2×C8—82no—
147C2×C8—82no—
148C2×C8—82no—
149C2×C8—82no—
150C4⋊C4—44no—
151C4⋊C4—41no—
152C4⋊C4—41no—
153C4⋊C4—82no—
154C4⋊C4—81no—
155C4⋊C4—81no—
156C4⋊C4—82no—
157C4⋊C4—82no—
158C4⋊C4—88no—
159C4⋊C4—84no—
160C22⋊C4—81no—
161C22⋊C4—81no—
162C23—22no—
163C23—21no—
164C23—21no—
165C23#44811yescharacteristic
166C23#45112yes—
167C23#45311yescharacteristic
168C23#45512yes—
169C23B(4,𝔽2)11yescharacteristic
170Q8—44no—
171Q8—82no—
172Q8—84no—
173Q8—82no—
174Q8—161no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—41no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—41no—
186C2×C4—44no—
187C2×C4—42no—
188C2×C4—44no—
189C2×C4—44no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—44no—
194C2×C4—82no—
195C2×C4—81no—
196C2×C4—82no—
197C2×C4—81no—
198C2×C4—82no—
199C2×C4—82no—
200C2×C4—82no—
201C2×C4—81no—
202C2×C4—82no—
203C2×C4—82no—
204C2×C4—82no—
205C2×C4—82no—
206C2×C4—82no—
207C2×C4—82no—
208C2×C4—82no—
209C2×C4—82no—
210C2×C4—82no—
211C2×C4—81no—
212C2×C4—81no—
213C2×C4—82no—
214C2×C4—82no—
215C2×C4—81no—
216C2×C4—81no—
217C8—82no—
218C8—162no—
219C22—21no—
220C22—21no—
221C22—21no—
222C22—21no—
223C22—22no—
224C22—22no—
225C22—22no—
226C22—22no—
227C22#159611yescharacteristic
228C22#160212yes—
229C22#163311yescharacteristic
230C22#164512yes—
231C22#165911yescharacteristic
232C22#220012yes—
233C22#220111yescharacteristic
234C22#220411yescharacteristic
235C4—41no—
236C4—41no—
237C4—42no—
238C4—42no—
239C4—82no—
240C4—82no—
241C4—84no—
242C4—162no—
243C4—161no—
244C4—161no—
245C2—21no—
246C2—21no—
247C2—22no—
248C2#775911yescharacteristic
249C2#1212411yescharacteristic
250C2#1213111yescharacteristic
251C2#1221912yes—
252C2#1222312yes—
subgroupG/H
center, socleC23B(4,𝔽2)
commutator, Frattini, cosocleC2×#95C23
prime radicalC24#120
LCSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24 ◁ C2×#95 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
78. C2×C4⋊C4Q16semidirect

Maximal subgroups

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

subgroupindex
#173062
#103432
#99672
#71322
#71312