Finite Group Databasebeta

← C2×#7102 #124044

Subgroups — C2×#7102 #124044

Order 512.

734 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 734 conjugacy classes of subgroups — too many to draw. Only its 202 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×#1954C211yesmaximal, characteristic
2C2×#1374C212yesmaximal, complemented
3#7102C214yesmaximal, direct product factor
4C22×C4×D4C411yescharacteristic
5C23×C16—22no—
6C22×C4⋊C8C411yescharacteristic
7C22×C22⋊C8—22no—
8C22×C4×C8C2211yescharacteristic
9#1954—22no—
10#1954C412yes—
11#1954C2212yes—
12#1374—22no—
13#1374C2214yescomplemented
14C23×D4C811yescharacteristic
15C24×C4—21no—
16C23×C8—21no—
17C23×C8D412yes—
18C2×C4×D4—21no—
19C2×C4×D4—21no—
20C2×C4×D4—42no—
21C2×C4×D4—42no—
22C2×C4×D4C812yes—
23C2×C4×D4C2×C412yes—
24C22×C4⋊C4C811yescharacteristic
25C22×C22⋊C4—21no—
26C22×C42C2×C411yescharacteristic
27C22×C16—28no—
28C22×C16—24no—
29C22×C16—44no—
30C22×C16—44no—
31C2×C4⋊C8—21no—
32C2×C4⋊C8—42no—
33C2×C4⋊C8C2×C412yes—
34C2×C22⋊C8—24no—
35C2×C22⋊C8—24no—
36C2×C22⋊C8—44no—
37C2×C22⋊C8—44no—
38C2×C4×C8—42no—
39C2×C4×C8C2×C411yescharacteristic
40C2×C4×C8D412yes—
41C2×C4×C8C2311yescharacteristic
42C25—21no—
43C22×D4—21no—
44C22×D4—22no—
45C22×D4—42no—
46C22×D4—42no—
47C22×D4C1614yescomplemented
48C22×D4C2×C812yes—
49C23×C4—21no—
50C23×C4—22no—
51C23×C4—22no—
52C23×C4—42no—
53C23×C4—42no—
54C23×C4—44no—
55C23×C4—44no—
56C23×C4C22⋊C411yescharacteristic
57C23×C4C2×C811yescharacteristic
58C23×C4M1611yescharacteristic
59C22×C8—24no—
60C22×C8—24no—
61C22×C8—22no—
62C22×C8—42no—
63C22×C8—44no—
64C22×C8C22⋊C412yes—
65C22×C8D814yes—
66C22×C8SD1614yes—
67C22×C8C2×D412yes—
68C4×D4—44no—
69C4×D4—42no—
70C4×D4—42no—
71C4×D4—44no—
72C4×D4—42no—
73C4×D4—42no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—42no—
76C2×C4⋊C4C2×C812yes—
77C2×C22⋊C4—22no—
78C2×C22⋊C4—22no—
79C2×C22⋊C4—44no—
80C2×C42—22no—
81C2×C42—22no—
82C2×C42C22⋊C411yescharacteristic
83C2×C42C2×C811yescharacteristic
84C2×C42M1611yescharacteristic
85C2×C42C22×C411yescharacteristic
86C2×C16—28no—
87C2×C16—416no—
88C2×C16—44no—
89C2×C16—44no—
90C4⋊C8—42no—
91C4⋊C8—42no—
92C22⋊C8—48no—
93C22⋊C8—44no—
94C22⋊C8—44no—
95C4×C8—41no—
96C4×C8—42no—
97C4×C8—41no—
98C24—24no—
99C24—22no—
100C24—42no—
101C24—44no—
102C24—42no—
103C24—44no—
104C24C22⋊C811yescharacteristic
105C2×D4—24no—
106C2×D4—22no—
107C2×D4—42no—
108C2×D4—48no—
109C2×D4—42no—
110C2×D4—42no—
111C2×D4—48no—
112C2×D4—42no—
113C2×D4C2×C1614yescomplemented
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—24no—
118C22×C4—22no—
119C22×C4—42no—
120C22×C4—44no—
121C22×C4—44no—
122C22×C4—48no—
123C22×C4—42no—
124C22×C4—42no—
125C22×C4—44no—
126C22×C4—44no—
127C22×C4—88no—
128C22×C4—84no—
129C22×C4—84no—
130C22×C4C22⋊C811yescharacteristic
131C22×C4C22⋊C811yescharacteristic
132C22×C4C22⋊C811yescharacteristic
133C22×C4#9912yes—
134C22×C4#10012yes—
135C22×C4#10214yes—
136C22×C4#10414yes—
137C22×C4C2×C1612yes—
138C22×C4M3212yes—
139C22×C4C2×C22⋊C411yescharacteristic
140C22×C4C22×C811yescharacteristic
141C22×C4C2×M1611yescharacteristic
142C2×C8—28no—
143C2×C8—28no—
144C2×C8—22no—
145C2×C8—22no—
146C2×C8—22no—
147C2×C8—22no—
148C2×C8—42no—
149C2×C8—44no—
150C2×C8—84no—
151C2×C8#10214yes—
152C2×C8C2×D812yes—
153C2×C8C2×SD1612yes—
154C4⋊C4—42no—
155C4⋊C4—42no—
156C22⋊C4—44no—
157C22⋊C4—44no—
158C42—22no—
159C42—22no—
160C42—21no—
161C42—21no—
162C42—21no—
163C42—21no—
164C16—48no—
165C23—24no—
166C23—48no—
167C23—416no—
168C23—42no—
169C23—44no—
170C23—42no—
171C23—44no—
172C23—84no—
173C23—88no—
174C23—84no—
175C23#32312yes—
176C23D4⋊C814yes—
177C23C22⋊C1612yes—
178C23#34912yes—
179C23#35014yes—
180C23C2×C22⋊C811yescharacteristic
181D4—44no—
182D4—48no—
183D4—44no—
184C2×C4—24no—
185C2×C4—24no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—44no—
195C2×C4—42no—
196C2×C4—44no—
197C2×C4—84no—
198C2×C4—82no—
199C2×C4—84no—
200C2×C4—82no—
201C2×C4—88no—
202C2×C4#32312yes—
203C2×C4D4⋊C814yes—
204C2×C4#32718yes—
205C2×C4#32914yes—
206C2×C4#33114yes—
207C2×C4C22⋊C1612yes—
208C2×C4C2×#9911yescharacteristic
209C2×C4C2×#10011yescharacteristic
210C2×C4C2×#10212yes—
211C2×C4C2×#10412yes—
212C2×C4C22×C1611yescharacteristic
213C2×C4C2×M3211yescharacteristic
214C8—22no—
215C8—24no—
216C8—22no—
217C8—82no—
218C22—48no—
219C22—44no—
220C22—84no—
221C22—82no—
222C22—82no—
223C22—816no—
224C22#127014yes—
225C22#131414yes—
226C22#131514yes—
227C22D4⋊C1618yes—
228C22#133018yes—
229C22C2×#32311yescharacteristic
230C22C2×D4⋊C812yes—
231C22C2×C22⋊C1611yescharacteristic
232C22C2×#34911yescharacteristic
233C22C2×#35012yes—
234C4—22no—
235C4—21no—
236C4—21no—
237C4—41no—
238C4—41no—
239C4—82no—
240C4#127014yes—
241C4C2×#32712yes—
242C4C2×#32911yescharacteristic
243C4C2×#33111yescharacteristic
244C2—84no—
245C2#710218yesdirect product factor
246C2C2×#127011yescharacteristic
247C2C2×#131411yescharacteristic
248C2C2×#131511yescharacteristic
249C2C2×D4⋊C1612yes—
250C2C2×#133012yes—
subgroupG/H
centerC23×C4C22⋊C4
commutatorC2×C4C22×C16
Frattini, cosocleC2×C4×C8C23
socleC24C22⋊C8
prime radicalC23×D4C8
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C4 ◁ C22×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1374C2semidirect
3. #7102C2direct (N × H)
13. #1374C22semidirect
47. C22×D4C16semidirect
113. C2×D4C2×C16semidirect
245. C2#7102direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#19542
C2×#13742
#71022