Finite Group Databasebeta

← #29690

Subgroups — #29690

Order 256.

674 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 674 conjugacy classes of subgroups — too many to draw. Only its 145 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×#532C211yesmaximal, characteristic
2C2×#476C212yesmaximal, complemented
3#2857C211yesmaximal, characteristic, complemented
4#2058C212yesmaximal, complemented
5#2041C212yesmaximal, complemented
6#2034C214yesmaximal, complemented
7#1802C212yesmaximal, complemented
8#1797C211yesmaximal, characteristic, complemented
9C22×#41—21no—
10#532—28no—
11C2×C4⋊Q8C2211yescharacteristic
12C2×C4⋊Q8C2211yescharacteristic
13C2×#127C2211yescharacteristic
14C2×#124—21no—
15C2×C22⋊Q8—21no—
16C2×#121—21no—
17C2×#121C2212yes—
18C2×C4×D4—21no—
19C2×C4×D4C2212yes—
20C22×C42C2211yescharacteristic
21#476—28no—
22C2×#106C2214yes—
23C2×C4⋊C8C2212yes—
24C2×#102C2214yes—
25C2×#102C2214yes—
26C2×C22⋊C8C2212yes—
27#391—21no—
28#389—21no—
29#389C2212yes—
30#384C2211yescharacteristic
31#380—21no—
32#380C2212yes—
33#340C2214yes—
34#336C2212yes—
35C2×#41—28no—
36C2×#41—44no—
37C2×#41—44no—
38C22×Q8—21no—
39C22×Q8—21no—
40C22×D4—22no—
41C22×D4—21no—
42C22×D4—21no—
43C22×D4C2312yes—
44C23×C4—22no—
45C23×C4D411yescharacteristic
46C23×C4D411yescharacteristic
47C23×C4C2311yescharacteristic
48C22×C8C2314yes—
49C4⋊Q8—24no—
50C4⋊Q8—24no—
51#127—24no—
52#124—28no—
53C22⋊Q8—44no—
54#121—28no—
55#121—24no—
56#121—24no—
57C4×D4—28no—
58C4×D4—24no—
59C4×D4D418yes—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4C2312yes—
66C2×C4⋊C4C2312yes—
67C2×C4⋊C4C2312yes—
68C2×C22⋊C4—22no—
69C2×C22⋊C4—22no—
70C2×C22⋊C4—22no—
71C2×C42—24no—
72C2×C42D411yescharacteristic
73C2×C42D411yescharacteristic
74C2×C42C2311yescharacteristic
75C2×C42C2311yescharacteristic
76#106—28no—
77C4⋊C8—24no—
78#102—28no—
79#102—44no—
80C22⋊C8—42no—
81#95—22no—
82#95—22no—
83C24—22no—
84C24C2×D411yescharacteristic
85#41—432no—
86C2×Q8—24no—
87C2×Q8—42no—
88C2×Q8—42no—
89C2×Q8—42no—
90C2×Q8—42no—
91C2×Q8—42no—
92C2×D4—24no—
93C2×D4—28no—
94C2×D4—24no—
95C2×D4—22no—
96C2×D4—22no—
97C2×D4—22no—
98C2×D4—22no—
99C2×D4—22no—
100C2×D4—42no—
101C2×D4—42no—
102C2×D4—44no—
103C2×D4—44no—
104C2×D4—42no—
105C2×D4—42no—
106C2×D4C2×D414yes—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—28no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—21no—
115C22×C4—22no—
116C22×C4—21no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—41no—
120C22×C4—44no—
121C22×C4—44no—
122C22×C4SD1614yes—
123C22×C4C2×D411yescharacteristic
124C22×C4C2×D411yescharacteristic
125C22×C4C2×D411yescharacteristic
126C22×C4C2×D411yescharacteristic
127C22×C4C2×D411yescharacteristic
128C22×C4C2411yescharacteristic
129C2×C8—24no—
130C2×C8—24no—
131C2×C8—24no—
132C4⋊C4—24no—
133C4⋊C4—22no—
134C4⋊C4—24no—
135C4⋊C4—22no—
136C4⋊C4—22no—
137C4⋊C4—42no—
138C4⋊C4—42no—
139C4⋊C4—42no—
140C4⋊C4—44no—
141C4⋊C4C2×D414yes—
142C22⋊C4—28no—
143C22⋊C4—28no—
144C22⋊C4—44no—
145C42—22no—
146C42—22no—
147C42—22no—
148C42C2×D414yes—
149C23—21no—
150C23—21no—
151C23—21no—
152C23—21no—
153C23—21no—
154C23—21no—
155C23—24no—
156C23—42no—
157C23—42no—
158C23—42no—
159C23—42no—
160C23—42no—
161C23—42no—
162C23C2×SD1612yes—
163C23C22×D411yescharacteristic
164Q8—48no—
165Q8—84no—
166D4—24no—
167D4—24no—
168D4—48no—
169D4—48no—
170D4—416no—
171D4—42no—
172D4—42no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—28no—
180C2×C4—21no—
181C2×C4—24no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—44no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—416no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4#12012yes—
197C2×C4#12012yes—
198C2×C4C2×SD1612yes—
199C2×C4C2×SD1612yes—
200C2×C4AGL(1,ℤ/8)12yes—
201C2×C4AGL(1,ℤ/8)12yes—
202C2×C4C22×D412yes—
203C2×C4H(5,𝔽2)11yescharacteristic
204C2×C42−1+411yescharacteristic
205C2×C42−1+412yes—
206C8—44no—
207C22—22no—
208C22—22no—
209C22—22no—
210C22—41no—
211C22—41no—
212C22—41no—
213C22—41no—
214C22—44no—
215C22—44no—
216C22—44no—
217C22—82no—
218C22—82no—
219C22—82no—
220C22—82no—
221C22C2×#12011yescharacteristic
222C22#53611yescharacteristic
223C22#53711yescharacteristic
224C22C22×SD1611yescharacteristic
225C22C2×AGL(1,ℤ/8)11yescharacteristic
226C22C2×AGL(1,ℤ/8)11yescharacteristic
227C22#57511yescharacteristic
228C4—22no—
229C4—22no—
230C4—22no—
231C4—41no—
232C4—44no—
233C4—82no—
234C4—82no—
235C4#45014yes—
236C4#53612yes—
237C4#53712yes—
238C2—42no—
239C2—84no—
240C2#261611yescharacteristic
241C2C2×#45011yescharacteristic
242C2#300611yescharacteristic
243C2#322711yescharacteristic
244C2#323511yescharacteristic
245C2#324611yescharacteristic
246C2#325311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#532C2
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×#476C2semidirect
3. #2857C2semidirect
4. #2058C2semidirect
5. #2041C2semidirect
6. #2034C2semidirect
7. #1802C2semidirect
8. #1797C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5322
C2×#4762
#28572
#20582
#20412
#20342
#18022
#17972