Finite Group Databasebeta

← C8×#356 #128458

Subgroups — C8×#356 #128458

Order 512.

384 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 122 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
1C8×C4⋊Q8C211yesmaximal, characteristic
2C4×#356C211yesmaximal, characteristic
3#11478C211yesmaximal, characteristic
4C8×#101C212yesmaximal
5#7206C212yesmaximal
6C2×C8×Q8C411yescharacteristic
7C2×C8×Q8C2211yescharacteristic
8C4×C4⋊Q8C2211yescharacteristic
9C2×#356C412yes—
10#1984C411yescharacteristic
11#1984C2211yescharacteristic
12#1939—22no—
13#1842C411yescharacteristic
14C8×C4⋊C4—22no—
15C4×#101C2212yes—
16C42×C8C411yescharacteristic
17#1461C2212yes—
18C8×M16—22no—
19#1394C412yes—
20#1325C412yes—
21#1319C412yes—
22#1283—22no—
23C2×C4⋊Q8C2×C411yescharacteristic
24C2×C4×Q8C2×C411yescharacteristic
25C2×C4×Q8C2311yescharacteristic
26C8×Q8—41no—
27C8×Q8—41no—
28C8×Q8—41no—
29C8×Q8—41no—
30#431—22no—
31C2×C4⋊C8—22no—
32C2×C4⋊C8—21no—
33C2×C4⋊C8C2×C411yescharacteristic
34C2×C4⋊C8C2×C411yescharacteristic
35C2×C4⋊C8D412yes—
36C2×#101C2×C414yes—
37C4×M16—22no—
38C2×C4×C8—21no—
39C2×C4×C8—21no—
40C2×C4×C8—21no—
41C2×C4×C8C2×C411yescharacteristic
42C2×C4×C8C2×C411yescharacteristic
43C2×C4×C8D412yes—
44#391C2×C411yescharacteristic
45#391C2×C411yescharacteristic
46#389—22no—
47#384C2×C411yescharacteristic
48C4×C4⋊C4—22no—
49C43C2×C411yescharacteristic
50#356C818yesdirect product factor
51#343—22no—
52#336—22no—
53#336—22no—
54#324—22no—
55#324C2×C414yes—
56#322—42no—
57C82—42no—
58C22×Q8C4211yescharacteristic
59C22×Q8C22×C411yescharacteristic
60C2×M16—24no—
61C22×C8—21no—
62C22×C8—21no—
63C22×C8C22⋊C412yes—
64C22×C8C22⋊C411yescharacteristic
65C22×C8C22⋊C411yescharacteristic
66C4⋊Q8—22no—
67C4⋊Q8C2×C814yes—
68C4×Q8—41no—
69C4×Q8—41no—
70C4×Q8—41no—
71C4×Q8—41no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4C4211yescharacteristic
76C2×C4⋊C4C22×C411yescharacteristic
77C2×C4⋊C4#4112yes—
78C2×C42—21no—
79C2×C42—21no—
80C2×C42—21no—
81C2×C42C4211yescharacteristic
82C2×C42C4211yescharacteristic
83C2×C42C22×C411yescharacteristic
84C2×C42C2×D412yes—
85C4⋊C8—41no—
86C4⋊C8—42no—
87C4⋊C8—41no—
88C4⋊C8—44no—
89C4⋊C8—42no—
90C4⋊C8—42no—
91#101—24no—
92#101C2×C818yes—
93C22⋊C8—24no—
94#97—42no—
95C4×C8—24no—
96C4×C8—22no—
97C4×C8—21no—
98C4×C8—22no—
99C4×C8—21no—
100C4×C8—41no—
101C4×C8—42no—
102C4×C8—42no—
103C4×C8—42no—
104C4×C8—41no—
105C4×C8—41no—
106#95—22no—
107#95—22no—
108C2×Q8—21no—
109C2×Q8—41no—
110C2×Q8—41no—
111C2×Q8—41no—
112C2×Q8—41no—
113C2×Q8C4×C812yes—
114C2×Q8C22×C812yes—
115C2×Q8#13112yes—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4C2×C4211yescharacteristic
121C22×C4C2×C22⋊C411yescharacteristic
122C22×C4#11711yescharacteristic
123C22×C4C4×D412yes—
124C22×C4C4×D412yes—
125M16—28no—
126M16—44no—
127C2×C8—22no—
128C2×C8—22no—
129C2×C8—21no—
130C2×C8—22no—
131C2×C8—41no—
132C2×C8—44no—
133C2×C8—41no—
134C2×C8—41no—
135C2×C8—42no—
136C2×C8—44no—
137C2×C8—42no—
138C2×C8—41no—
139C2×C8#9911yescharacteristic
140C2×C8#9911yescharacteristic
141C4⋊C4—28no—
142C4⋊C4—41no—
143C4⋊C4—42no—
144C4⋊C4—41no—
145C4⋊C4—44no—
146C4⋊C4—42no—
147C42—22no—
148C42—22no—
149C42—21no—
150C42—22no—
151C42—21no—
152C42—21no—
153C42—22no—
154C42—41no—
155C42—42no—
156C42—41no—
157C42—41no—
158C42C4×C812yes—
159C23C4×C22⋊C411yescharacteristic
160Q8—42no—
161Q8—42no—
162Q8—42no—
163Q8—42no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—21no—
168C2×C4—22no—
169C2×C4—21no—
170C2×C4—22no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—41no—
185C2×C4C2×C4×C811yescharacteristic
186C2×C4#40511yescharacteristic
187C2×C4C2×#9911yescharacteristic
188C2×C4#41011yescharacteristic
189C2×C4C8×D414yes—
190C8—21no—
191C8—42no—
192C8—48no—
193C8—41no—
194C8—42no—
195C8—84no—
196C8—81no—
197C8#35612yesdirect product factor
198C22—21no—
199C22—21no—
200C22C8×C22⋊C411yescharacteristic
201C22C4×#9911yescharacteristic
202C22#175711yescharacteristic
203C4—24no—
204C4—21no—
205C4—24no—
206C4—21no—
207C4—21no—
208C4—42no—
209C4—42no—
210C4—41no—
211C4—42no—
212C4—81no—
213C4C2×#35611yescharacteristic
214C4#213211yescharacteristic
215C2—21no—
216C2—21no—
217C2C8×#9911yescharacteristic
218C2C4×#35611yescharacteristic
219C2#1263311yescharacteristic
subgroupG/H
centerC2×C8#99
commutatorC2×C4C2×C4×C8
Frattini, cosocleC2×C4×Q8C23
socleC22C4×#99
prime radicalC23C4×C22⋊C4
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C8 ◁ C22×C8 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
50. #356C8direct (N × H)
197. C8#356direct (N × H)

Maximal subgroups

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

subgroupindex
C8×C4⋊Q82
C4×#3562
#114782
C8×#1012
#72062