Finite Group Databasebeta

← C8×#355 #128457

Subgroups — C8×#355 #128457

Order 512.

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

normal subgroups only

This group has 406 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×#124C211yesmaximal, characteristic
2C4×#355C211yesmaximal, characteristic
3#11476C211yesmaximal, characteristic
4C8×#101C212yesmaximal, complemented
5#7206C212yesmaximal, complemented
6C2×C8×Q8C2211yescharacteristic
7C2×C8×D4C411yescharacteristic
8C4×#124C2211yescharacteristic
9C2×#355C412yes—
10#1984C2211yescharacteristic
11#1954C411yescharacteristic
12#1923—22no—
13#1831C411yescharacteristic
14C4×#101C2212yes—
15C8×C22⋊C4—22no—
16C42×C8C411yescharacteristic
17#1461C2212yes—
18C8×M16—22no—
19#1394C412yes—
20#1324C412yes—
21#1318C412yes—
22#1283—22no—
23C23×C8—21no—
24C2×#124C2×C411yescharacteristic
25C2×C4×Q8C2311yescharacteristic
26C2×C4×D4C2×C411yescharacteristic
27C8×Q8—41no—
28C8×Q8—41no—
29C8×D4—41no—
30C8×D4—41no—
31#431—22no—
32C2×C4⋊C8C2×C411yescharacteristic
33C2×C4⋊C8C2×C411yescharacteristic
34C2×C4⋊C8D412yes—
35C2×#101C2×C414yes—
36C2×C22⋊C8—21no—
37C2×C22⋊C8—22no—
38C4×M16—22no—
39C2×C4×C8—21no—
40C2×C4×C8—21no—
41C2×C4×C8C2×C411yescharacteristic
42C2×C4×C8C2×C411yescharacteristic
43C2×C4×C8D412yes—
44#391C2×C411yescharacteristic
45#390C2×C411yescharacteristic
46#388—22no—
47#382C2×C411yescharacteristic
48C4×C22⋊C4—22no—
49C43C2×C411yescharacteristic
50#355C818yesdirect product factor
51#343—22no—
52#336—22no—
53#336—22no—
54#324—22no—
55#324C2×C414yes—
56#322—42no—
57C82—42no—
58C22×Q8C22×C411yescharacteristic
59C22×D4C4211yescharacteristic
60C23×C4—21no—
61C2×M16—24no—
62C22×C8—21no—
63C22×C8—21no—
64C22×C8—44no—
65C22×C8—41no—
66C22×C8—41no—
67C22×C8C22⋊C412yes—
68C22×C8C22⋊C411yescharacteristic
69C22×C8C22⋊C411yescharacteristic
70#124—22no—
71#124C2×C814yes—
72C4×Q8—41no—
73C4×Q8—41no—
74C4×D4—41no—
75C4×D4—41no—
76C2×C4⋊C4C4211yescharacteristic
77C2×C4⋊C4C22×C411yescharacteristic
78C2×C4⋊C4#4112yes—
79C2×C22⋊C4—22no—
80C2×C22⋊C4—21no—
81C2×C22⋊C4—22no—
82C2×C42—21no—
83C2×C42—21no—
84C2×C42C4211yescharacteristic
85C2×C42C4211yescharacteristic
86C2×C42C22×C411yescharacteristic
87C2×C42C2×D412yes—
88C4⋊C8—41no—
89C4⋊C8—42no—
90C4⋊C8—41no—
91C4⋊C8—42no—
92#101—24no—
93#101C2×C818yescomplemented
94C22⋊C8—24no—
95C22⋊C8—44no—
96C22⋊C8—42no—
97#97—42no—
98C4×C8—22no—
99C4×C8—21no—
100C4×C8—22no—
101C4×C8—21no—
102C4×C8—41no—
103C4×C8—42no—
104C4×C8—42no—
105C4×C8—42no—
106C4×C8—42no—
107#95—22no—
108#95—22no—
109C24—21no—
110C2×Q8—41no—
111C2×Q8—41no—
112C2×Q8C22×C812yes—
113C2×Q8#13112yes—
114C2×D4—21no—
115C2×D4—41no—
116C2×D4—41no—
117C2×D4C4×C812yes—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—42no—
122C22×C4—42no—
123C22×C4—41no—
124C22×C4—41no—
125C22×C4C2×C4211yescharacteristic
126C22×C4C2×C22⋊C411yescharacteristic
127C22×C4#11711yescharacteristic
128C22×C4C4×D412yes—
129C22×C4C4×D412yes—
130M16—28no—
131M16—44no—
132C2×C8—22no—
133C2×C8—22no—
134C2×C8—21no—
135C2×C8—22no—
136C2×C8—41no—
137C2×C8—44no—
138C2×C8—41no—
139C2×C8—41no—
140C2×C8—42no—
141C2×C8—44no—
142C2×C8—42no—
143C2×C8—41no—
144C2×C8—82no—
145C2×C8—82no—
146C2×C8#9911yescharacteristic
147C2×C8#9911yescharacteristic
148C4⋊C4—41no—
149C4⋊C4—42no—
150C4⋊C4—41no—
151C22⋊C4—28no—
152C22⋊C4—44no—
153C22⋊C4—42no—
154C42—21no—
155C42—22no—
156C42—21no—
157C42—21no—
158C42—22no—
159C42—41no—
160C42—41no—
161C42—41no—
162C42—42no—
163C42C4×C812yes—
164C23—22no—
165C23—42no—
166C23—42no—
167C23—41no—
168C23—41no—
169C23C4×C22⋊C411yescharacteristic
170Q8—42no—
171Q8—42no—
172D4—42no—
173D4—42no—
174C2×C4—21no—
175C2×C4—22no—
176C2×C4—21no—
177C2×C4—22no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—41no—
192C2×C4—82no—
193C2×C4—81no—
194C2×C4—81no—
195C2×C4C2×C4×C811yescharacteristic
196C2×C4#40511yescharacteristic
197C2×C4C2×#9911yescharacteristic
198C2×C4#41011yescharacteristic
199C2×C4C8×D414yes—
200C8—21no—
201C8—42no—
202C8—48no—
203C8—41no—
204C8—42no—
205C8—84no—
206C8—81no—
207C8#35512yesdirect product factor
208C22—21no—
209C22—21no—
210C22—44no—
211C22—42no—
212C22—82no—
213C22—81no—
214C22—81no—
215C22C8×C22⋊C411yescharacteristic
216C22C4×#9911yescharacteristic
217C22#175711yescharacteristic
218C4—21no—
219C4—24no—
220C4—21no—
221C4—21no—
222C4—42no—
223C4—42no—
224C4—42no—
225C4—41no—
226C4—81no—
227C4C2×#35511yescharacteristic
228C4#213211yescharacteristic
229C2—21no—
230C2—21no—
231C2—82no—
232C2C8×#9911yescharacteristic
233C2C4×#35511yescharacteristic
234C2#1263311yescharacteristic
subgroupG/H
centerC2×C8#99
commutatorC2×C4C2×C4×C8
Frattini, cosocleC2×C4×Q8C23
socleC22C4×#99
prime radicalC22×D4C42
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
4. C8×#101C2semidirect
5. #7206C2semidirect
50. #355C8direct (N × H)
93. #101C2×C8semidirect
207. C8#355direct (N × H)

Maximal subgroups

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

subgroupindex
C8×#1242
C4×#3552
#114762
C8×#1012
#72062