Finite Group Databasebeta

← #28400

Subgroups — #28400

Order 256.

497 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 497 conjugacy classes of subgroups — too many to draw. Only its 155 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×#487C211yesmaximal, characteristic, complemented
2C2×#475C211yesmaximal, characteristic, complemented
3C2×#461C211yesmaximal, characteristic
4C2×C4×SD16C211yesmaximal, characteristic
5#2546C211yesmaximal, characteristic, complemented
6#2542C211yesmaximal, characteristic
7#2096C212yesmaximal, complemented
8#2013C212yesmaximal
9#1968C212yesmaximal
10#1956C212yesmaximal, complemented
11C8×C4⋊C4C211yesmaximal, characteristic, complemented
12C22×SD16C2211yescharacteristic
13C22×SD16C2212yes—
14C2×C4⋊Q8C2211yescharacteristic
15C2×C4⋊Q8C2212yes—
16C2×C4⋊Q8C2211yescharacteristic
17C2×C22⋊Q8—22no—
18C2×C4×Q8C2211yescharacteristic
19C2×C4×D4C2211yescharacteristic
20#487—24no—
21#475—24no—
22#461—24no—
23C4×SD16—24no—
24C2×#106C2212yes—
25C2×#106C2211yescharacteristic
26C2×C4⋊C8C2211yescharacteristic
27C2×#103C2212yes—
28C2×#103C2212yes—
29C2×#103C2212yes—
30C2×#103C2212yes—
31C2×#103C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×C4×C8C2212yes—
34C2×C4×C8C2211yescharacteristic
35#399—22no—
36#398—22no—
37#391C2212yes—
38#390C2212yes—
39#389—22no—
40#384C2212yes—
41C4×C4⋊C4C2211yescharacteristic
42#336C2212yes—
43C22×Q8—21no—
44C22×Q8C2312yes—
45C22×Q8C2311yescharacteristic
46C22×D4C2311yescharacteristic
47C23×C4—21no—
48C2×SD16—22no—
49C2×SD16—22no—
50C2×SD16—24no—
51C2×SD16—22no—
52C2×SD16—24no—
53C2×SD16—42no—
54C22×C8C2311yescharacteristic
55C22×C8C2312yes—
56C22×C8C2311yescharacteristic
57C4⋊Q8—24no—
58C4⋊Q8—22no—
59C4⋊Q8—22no—
60C4⋊Q8—22no—
61C4⋊Q8—22no—
62C4⋊Q8—42no—
63C22⋊Q8—48no—
64C4×Q8—22no—
65C4×Q8—22no—
66C4×D4—22no—
67C4×D4—22no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—24no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—24no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4D412yes—
75C2×C4⋊C4C2311yescharacteristic
76C2×C4⋊C4C2311yescharacteristic
77C2×C4⋊C4C2312yes—
78C2×C22⋊C4—21no—
79C2×C22⋊C4—24no—
80C2×C42—21no—
81C2×C42C2312yes—
82C2×C42C2311yescharacteristic
83#106—22no—
84#106—24no—
85C4⋊C8D414yes—
86#103—22no—
87#103—24no—
88#103—24no—
89#103—24no—
90#103—24no—
91#102—22no—
92C4×C8—42no—
93C4×C8D414yes—
94#95—24no—
95#95D412yes—
96C24—21no—
97C2×Q8—21no—
98C2×Q8—21no—
99C2×Q8—21no—
100C2×Q8—22no—
101C2×Q8—22no—
102C2×Q8—21no—
103C2×Q8—22no—
104C2×Q8—22no—
105C2×Q8—22no—
106C2×Q8—21no—
107C2×Q8—22no—
108C2×Q8—42no—
109C2×Q8—44no—
110C2×Q8#4112yes—
111C2×D4—21no—
112C2×D4—21no—
113C2×D4—21no—
114C2×D4—21no—
115C2×D4—21no—
116C2×D4#4112yes—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—21no—
122C22×C4—42no—
123C22×C4—44no—
124C22×C4C2×D411yescharacteristic
125C22×C4C2×D414yes—
126C22×C4C2×D411yescharacteristic
127C22×C4C2411yescharacteristic
128SD16—28no—
129SD16—48no—
130C2×C8—21no—
131C2×C8—21no—
132C2×C8—22no—
133C2×C8—21no—
134C2×C8—22no—
135C2×C8—21no—
136C2×C8C2×D412yes—
137C2×C8C2×D412yes—
138C2×C8C2×D414yes—
139C4⋊C4—21no—
140C4⋊C4—21no—
141C4⋊C4—22no—
142C4⋊C4—21no—
143C4⋊C4—22no—
144C4⋊C4—21no—
145C4⋊C4—48no—
146C4⋊C4—42no—
147C4⋊C4—42no—
148C4⋊C4—42no—
149C4⋊C4—42no—
150C4⋊C4—48no—
151C4⋊C4SD1618yescomplemented
152C22⋊C4—42no—
153C22⋊C4—48no—
154C42—22no—
155C42—22no—
156C42—42no—
157C42C2×D412yes—
158C42C2×D412yes—
159C23—22no—
160C23—41no—
161C23—41no—
162C23—42no—
163C23—42no—
164C23C22×D411yescharacteristic
165Q8—22no—
166Q8—22no—
167Q8—41no—
168Q8—41no—
169Q8—42no—
170Q8—42no—
171Q8—42no—
172Q8—42no—
173Q8—84no—
174D4—22no—
175D4—22no—
176D4—41no—
177D4—41no—
178C2×C4—24no—
179C2×C4—22no—
180C2×C4—24no—
181C2×C4—22no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—44no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—84no—
194C2×C4#12712yes—
195C2×C4#12712yes—
196C2×C4C2×SD1618yes—
197C2×C4C2×SD1614yes—
198C2×C4C22×D411yescharacteristic
199C2×C4C22×D411yescharacteristic
200C2×C4C2×#4111yescharacteristic
201C2×C4C2×#4111yescharacteristic
202C2×C4H(5,𝔽2)11yescharacteristic
203C2×C42−1+411yescharacteristic
204C8—22no—
205C8—22no—
206C8—24no—
207C22—44no—
208C22—42no—
209C22—81no—
210C22—81no—
211C22—82no—
212C22C2×#12711yescharacteristic
213C22#54611yescharacteristic
214C22#54811yescharacteristic
215C22C22×SD1612yes—
216C22#57812yes—
217C4—22no—
218C4—28no—
219C4—42no—
220C4—81no—
221C4—82no—
222C4—81no—
223C4—82no—
224C4#49214yes—
225C4#52511yescharacteristic
226C4#53711yescharacteristic
227C4#54611yescharacteristic
228C4#54811yescharacteristic
229C2—82no—
230C2#254011yescharacteristic
231C2C2×#49211yescharacteristic
232C2#315611yescharacteristic
233C2#329812yes—
234C2#335912yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#487C2semidirect
2. C2×#475C2semidirect
5. #2546C2semidirect
7. #2096C2semidirect
10. #1956C2semidirect
11. C8×C4⋊C4C2semidirect
151. C4⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#4872
C2×#4752
C2×#4612
C2×C4×SD162
#25462
#25422
#20962
#20132
#19682
#19562
C8×C4⋊C42