Finite Group Databasebeta

← #28615

Subgroups — #28615

Order 256.

575 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 575 conjugacy classes of subgroups — too many to draw. Only its 147 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×#493C211yesmaximal, characteristic
2C2×#486C211yesmaximal, characteristic, complemented
3#2858C211yesmaximal, characteristic, complemented
4C4×#127C211yesmaximal, characteristic, complemented
5#2054C214yesmaximal, complemented
6#1990C211yesmaximal, characteristic, complemented
7#1957C211yesmaximal, characteristic, complemented
8#1879C214yesmaximal, complemented
9C4×C4⋊C8C211yesmaximal, characteristic, complemented
10C22×D8C2214yes—
11C2×C4⋊Q8C2211yescharacteristic
12C2×#127C2211yescharacteristic
13C2×#127C2211yescharacteristic
14C2×#124—21no—
15C2×#124—21no—
16C2×C4×D4—21no—
17C2×C4×D4—21no—
18C2×C4×D4C2212yes—
19#493—24no—
20#486—24no—
21C2×C4⋊C8C2212yes—
22C2×#102C2214yes—
23C2×#102C2214yes—
24C2×C4×C8C2211yescharacteristic
25C2×C4×C8C2211yescharacteristic
26#390—21no—
27#390—21no—
28#390—21no—
29#390—21no—
30#390C2212yes—
31#390C2212yes—
32#390C2212yes—
33#384C2211yescharacteristic
34C43C2211yescharacteristic
35#340C2214yes—
36#336C2212yes—
37C22×Q8—21no—
38C22×D4—21no—
39C22×D4—21no—
40C22×D4—21no—
41C22×D4—21no—
42C22×D4C2312yes—
43C22×D4C2312yes—
44C23×C4—22no—
45C2×D8—28no—
46C2×D8—28no—
47C2×D8—44no—
48C22×C8C2312yes—
49C22×C8C2312yes—
50C4⋊Q8—22no—
51C4⋊Q8—22no—
52#127—22no—
53#127—22no—
54#127—22no—
55#127D414yes—
56#124—44no—
57#124—44no—
58C4×D4—44no—
59C4×D4—44no—
60C4×D4—44no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4C2312yes—
65C2×C4⋊C4C2312yes—
66C2×C22⋊C4—22no—
67C2×C22⋊C4—22no—
68C2×C22⋊C4—22no—
69C2×C22⋊C4—22no—
70C2×C22⋊C4—22no—
71C2×C22⋊C4—22no—
72C2×C22⋊C4—22no—
73C2×C42D411yescharacteristic
74C2×C42D411yescharacteristic
75C2×C42D411yescharacteristic
76C2×C42D411yescharacteristic
77C2×C42C2311yescharacteristic
78C2×C42C2311yescharacteristic
79C2×C42C2311yescharacteristic
80C4⋊C8—24no—
81#102—28no—
82#102—44no—
83C4×C8—22no—
84C4×C8—22no—
85C24—22no—
86C24—22no—
87C2×Q8—42no—
88C2×Q8—42no—
89C2×Q8—42no—
90C2×D4—24no—
91C2×D4—24no—
92C2×D4—22no—
93C2×D4—24no—
94C2×D4—22no—
95C2×D4—22no—
96C2×D4—24no—
97C2×D4—24no—
98C2×D4—24no—
99C2×D4—42no—
100C2×D4—42no—
101C2×D4—42no—
102C2×D4—42no—
103C2×D4—42no—
104C2×D4—42no—
105C2×D4—42no—
106C2×D4—42no—
107C2×D4—42no—
108C2×D4—42no—
109C2×D4C2×D414yes—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—44no—
113C22×C4—44no—
114C22×C4—44no—
115C22×C4C2×D411yescharacteristic
116C22×C4C2×D411yescharacteristic
117C22×C4C2×D411yescharacteristic
118C22×C4C2×D411yescharacteristic
119C22×C4C2×D411yescharacteristic
120C22×C4C2×D411yescharacteristic
121C22×C4C2411yescharacteristic
122D8—416no—
123C2×C8—22no—
124C2×C8—22no—
125C2×C8—22no—
126C2×C8—22no—
127C2×C8—22no—
128C2×C8#4114yes—
129C4⋊C4—22no—
130C4⋊C4—22no—
131C4⋊C4—24no—
132C4⋊C4—42no—
133C4⋊C4—42no—
134C4⋊C4—42no—
135C22⋊C4—44no—
136C22⋊C4—44no—
137C22⋊C4—44no—
138C22⋊C4—44no—
139C22⋊C4—44no—
140C22⋊C4—44no—
141C22⋊C4—44no—
142C42—22no—
143C42—22no—
144C42—22no—
145C42—22no—
146C42—22no—
147C42D814yescomplemented
148C42C2×D412yes—
149C42#4112yes—
150C23—24no—
151C23—42no—
152C23—42no—
153C23—42no—
154C23—42no—
155C23—42no—
156C23—42no—
157C23—42no—
158C23—42no—
159C23—42no—
160C23—42no—
161C23—42no—
162C23—42no—
163C23—42no—
164C23C22×D411yescharacteristic
165Q8—84no—
166D4—28no—
167D4—44no—
168D4—44no—
169D4—44no—
170D4—48no—
171D4—48no—
172D4—84no—
173D4—84no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—88no—
191C2×C4#12112yes—
192C2×C4#12112yes—
193C2×C4C2×D812yes—
194C2×C4C2×D812yes—
195C2×C4C2×D812yes—
196C2×C4#13512yes—
197C2×C4AGL(1,ℤ/8)12yes—
198C2×C4AGL(1,ℤ/8)12yes—
199C2×C4C22×D411yescharacteristic
200C2×C4C2×#4112yes—
201C2×C4C2×#4111yescharacteristic
202C2×C4H(5,𝔽2)12yes—
203C8—24no—
204C8—42no—
205C22—44no—
206C22—44no—
207C22—44no—
208C22—82no—
209C22—82no—
210C22—82no—
211C22—82no—
212C22—82no—
213C22—82no—
214C22—82no—
215C22—82no—
216C22—82no—
217C22—82no—
218C22—82no—
219C22C2×#12111yescharacteristic
220C22#53211yescharacteristic
221C22#53511yescharacteristic
222C22C22×D811yescharacteristic
223C22C2×#13511yescharacteristic
224C22C2×AGL(1,ℤ/8)11yescharacteristic
225C22C2×AGL(1,ℤ/8)11yescharacteristic
226C4—22no—
227C4—22no—
228C4—41no—
229C4—41no—
230C4—82no—
231C4—82no—
232C4#45914yes—
233C4#46314yes—
234C4#54612yes—
235C4#55912yes—
236C2—84no—
237C2—162no—
238C2#254411yescharacteristic
239C2C2×#45911yescharacteristic
240C2C2×#46311yescharacteristic
241C2#316911yescharacteristic
242C2#318411yescharacteristic
243C2#320511yescharacteristic
244C2#320711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#493C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#486C2semidirect
3. #2858C2semidirect
4. C4×#127C2semidirect
5. #2054C2semidirect
6. #1990C2semidirect
7. #1957C2semidirect
8. #1879C2semidirect
9. C4×C4⋊C8C2semidirect
147. C42D8semidirect

Maximal subgroups

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

subgroupindex
C2×#4932
C2×#4862
#28582
C4×#1272
#20542
#19902
#19572
#18792
C4×C4⋊C82