Finite Group Databasebeta

← #24302

Subgroups — #24302

Order 256.

647 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 647 conjugacy classes of subgroups — too many to draw. Only its 167 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
1C22×C4⋊Q8C211yesmaximal, characteristic
2C2×#477C212yesmaximal, complemented
3C4×C22⋊Q8C211yesmaximal, characteristic
4#2057C212yesmaximal, complemented
5#2024C212yesmaximal, complemented
6#1987C212yesmaximal, complemented
7#1872C212yesmaximal
8#1852C212yesmaximal
9#1797C211yesmaximal, characteristic
10C23×Q8—21no—
11C2×C4⋊Q8—28no—
12C2×C4⋊Q8—21no—
13C2×C4⋊Q8—21no—
14C2×C4⋊Q8C2212yes—
15C2×C4⋊Q8C2212yes—
16C2×C22⋊Q8C2211yescharacteristic
17C2×C4×Q8C2211yescharacteristic
18C22×C4⋊C4—21no—
19C22×C4⋊C4C2212yes—
20C22×C42C2211yescharacteristic
21#477—28no—
22C2×#107C2214yes—
23C2×C4⋊C8C2212yes—
24C2×#103C2214yes—
25C2×C22⋊C8C2212yes—
26#391C2211yescharacteristic
27#389C2211yescharacteristic
28#387—21no—
29#385—21no—
30#380C2211yescharacteristic
31C4×C4⋊C4—21no—
32C4×C4⋊C4C2211yescharacteristic
33C4×C22⋊C4—21no—
34#340C2214yes—
35#340C2214yes—
36#336C2212yes—
37C22×Q8—28no—
38C22×Q8—22no—
39C22×Q8—22no—
40C22×Q8—44no—
41C22×Q8—44no—
42C22×Q8C2311yescharacteristic
43C23×C4—22no—
44C23×C4D411yescharacteristic
45C23×C4D411yescharacteristic
46C23×C4C2311yescharacteristic
47C22×C8C2314yes—
48C4⋊Q8—28no—
49C4⋊Q8—28no—
50C4⋊Q8—28no—
51C4⋊Q8—24no—
52C4⋊Q8D418yes—
53C22⋊Q8—22no—
54C22⋊Q8Q814yescomplemented
55C4×Q8—24no—
56C2×C4⋊C4—28no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—28no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4C2311yescharacteristic
65C2×C4⋊C4C2311yescharacteristic
66C2×C4⋊C4C2314yes—
67C2×C4⋊C4C2311yescharacteristic
68C2×C22⋊C4—21no—
69C2×C22⋊C4—21no—
70C2×C42—24no—
71C2×C42—21no—
72C2×C42—21no—
73C2×C42D411yescharacteristic
74C2×C42D411yescharacteristic
75C2×C42C2311yescharacteristic
76C2×C42C2311yescharacteristic
77#107—28no—
78C4⋊C8—24no—
79#103—28no—
80C22⋊C8—24no—
81#95—21no—
82#95—21no—
83#95—21no—
84#95—21no—
85#95—21no—
86C24C2×D411yescharacteristic
87C2×Q8—28no—
88C2×Q8—28no—
89C2×Q8—21no—
90C2×Q8—22no—
91C2×Q8—22no—
92C2×Q8—432no—
93C2×Q8—44no—
94C2×Q8—44no—
95C2×Q8—44no—
96C2×Q8—44no—
97C2×Q8C2×Q812yescomplemented
98C22×C4—22no—
99C22×C4—22no—
100C22×C4—22no—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—28no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—24no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—44no—
112C22×C4—44no—
113C22×C4Q1614yes—
114C22×C4C2×D411yescharacteristic
115C22×C4C2×D411yescharacteristic
116C22×C4C2×D411yescharacteristic
117C22×C4C2×D411yescharacteristic
118C22×C4C2×D411yescharacteristic
119C22×C4C2×Q812yes—
120C22×C4#4112yes—
121C22×C4C2411yescharacteristic
122C2×C8—24no—
123C2×C8—24no—
124C2×C8—24no—
125C4⋊C4—28no—
126C4⋊C4—28no—
127C4⋊C4—28no—
128C4⋊C4—24no—
129C4⋊C4—24no—
130C4⋊C4—21no—
131C4⋊C4—22no—
132C4⋊C4—22no—
133C4⋊C4—42no—
134C4⋊C4—42no—
135C4⋊C4C2×D418yes—
136C4⋊C4C2×Q812yescomplemented
137C22⋊C4—24no—
138C22⋊C4—42no—
139C42—22no—
140C42—22no—
141C42—22no—
142C42—42no—
143C42—42no—
144C42C2×D414yes—
145C23—21no—
146C23—21no—
147C23—21no—
148C23—21no—
149C23C22⋊Q812yes—
150C23C2×Q1612yes—
151C23AGL(1,ℤ/8)12yes—
152C23C22×D411yescharacteristic
153Q8—24no—
154Q8—416no—
155Q8—416no—
156Q8—42no—
157C2×C4—28no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—24no—
167C2×C4—24no—
168C2×C4—28no—
169C2×C4—28no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—416no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—44no—
177C2×C4—44no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4#12012yes—
186C2×C4#12012yes—
187C2×C4C22⋊Q812yes—
188C2×C4C2×Q1612yes—
189C2×C4C2×Q1612yes—
190C2×C4AGL(1,ℤ/8)12yes—
191C2×C4#13712yes—
192C2×C4C22×D412yes—
193C2×C4C22×Q811yescharacteristic
194C2×C4C2×#4111yescharacteristic
195C2×C42−1+412yes—
196C8—44no—
197C22—22no—
198C22—22no—
199C22—22no—
200C22—22no—
201C22—22no—
202C22—22no—
203C22#47714yescomplemented
204C22C2×#12011yescharacteristic
205C22C2×C22⋊Q811yescharacteristic
206C22#53611yescharacteristic
207C22C22×Q1611yescharacteristic
208C22C2×AGL(1,ℤ/8)11yescharacteristic
209C22C2×AGL(1,ℤ/8)11yescharacteristic
210C22C2×#13711yescharacteristic
211C4—22no—
212C4—22no—
213C4—22no—
214C4—22no—
215C4—42no—
216C4—42no—
217C4—48no—
218C4—81no—
219C4—81no—
220C4#45114yes—
221C4#54812yes—
222C4D4×Q812yes—
223C2—24no—
224C2#240911yescharacteristic
225C2C2×#45111yescharacteristic
226C2#300311yescharacteristic
227C2C2×#47711yescharacteristic
228C2#307911yescharacteristic
229C2#323011yescharacteristic
230C2#323811yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#477C2semidirect
4. #2057C2semidirect
5. #2024C2semidirect
6. #1987C2semidirect
54. C22⋊Q8Q8semidirect
97. C2×Q8C2×Q8semidirect
136. C4⋊C4C2×Q8semidirect
203. C22#477semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊Q82
C2×#4772
C4×C22⋊Q82
#20572
#20242
#19872
#18722
#18522
#17972