Finite Group Databasebeta

← #24632

Subgroups — #24632

Order 256.

415 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 415 conjugacy classes of subgroups — too many to draw. Only its 139 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×#462C211yesmaximal, characteristic
2C2×C4×Q16C211yesmaximal, characteristic
3#2546C211yesmaximal, characteristic
4#2355C211yesmaximal, characteristic
5#2056C212yesmaximal
6#2016C212yesmaximal
7#1981C212yesmaximal
8#1956C211yesmaximal, characteristic
9#1939C211yesmaximal, characteristic
10#1928C212yesmaximal
11#1803C211yesmaximal, characteristic
12C22×Q16C2211yescharacteristic
13C22×Q16C2212yes—
14C2×C4⋊Q8C2211yescharacteristic
15C2×C4⋊Q8C2212yes—
16C2×C4×Q8C2211yescharacteristic
17C2×C4×Q8C2211yescharacteristic
18#462—24no—
19C4×Q16—24no—
20C2×#107C2211yescharacteristic
21C2×C4⋊C8C2211yescharacteristic
22C2×C4⋊C8C2212yes—
23C2×#103C2211yescharacteristic
24C2×#103C2211yescharacteristic
25C2×#103C2212yes—
26C2×#103C2212yes—
27C2×#103C2212yes—
28C2×#103C2212yes—
29C2×C4×C8C2211yescharacteristic
30#398—22no—
31#391C2212yes—
32#391C2212yes—
33#389—21no—
34#389—21no—
35#387—22no—
36#384C2211yescharacteristic
37#382—21no—
38C4×C4⋊C4—21no—
39C4×C4⋊C4C2212yes—
40C4×C4⋊C4C2211yescharacteristic
41#340C2212yes—
42#336C2212yes—
43C22×Q8C2312yes—
44C22×Q8C2311yescharacteristic
45C22×Q8C2311yescharacteristic
46C2×Q16—22no—
47C2×Q16—22no—
48C2×Q16—24no—
49C2×Q16—22no—
50C2×Q16—42no—
51C2×Q16—42no—
52C22×C8C2312yes—
53C22×C8C2311yescharacteristic
54C22×C8C2311yescharacteristic
55C4⋊Q8—24no—
56C4⋊Q8—22no—
57C4⋊Q8—22no—
58C4⋊Q8—42no—
59C4×Q8—22no—
60C4×Q8—22no—
61C4×Q8—22no—
62C4×Q8D414yes—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4D411yescharacteristic
72C2×C4⋊C4D411yescharacteristic
73C2×C4⋊C4C2311yescharacteristic
74C2×C4⋊C4C2312yes—
75C2×C4⋊C4C2311yescharacteristic
76C2×C42—21no—
77C2×C42—21no—
78C2×C42—22no—
79C2×C42C2312yes—
80C2×C42C2311yescharacteristic
81#107—22no—
82C4⋊C8—22no—
83C4⋊C8—42no—
84#103—22no—
85#103—24no—
86#103—22no—
87#103—24no—
88#103—42no—
89#103—42no—
90C4×C8—22no—
91#95—22no—
92#95—22no—
93#95—22no—
94#95—22no—
95#95—22no—
96#95D412yes—
97C2×Q8—21no—
98C2×Q8—21no—
99C2×Q8—21no—
100C2×Q8—22no—
101C2×Q8—21no—
102C2×Q8—21no—
103C2×Q8—21no—
104C2×Q8—21no—
105C2×Q8—22no—
106C2×Q8—22no—
107C2×Q8—22no—
108C2×Q8—21no—
109C2×Q8—21no—
110C2×Q8—21no—
111C2×Q8—22no—
112C2×Q8—22no—
113C2×Q8C2×D412yes—
114C2×Q8#4112yes—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4C2×D412yes—
122C22×C4C2×D411yescharacteristic
123C22×C4C2×D412yes—
124C22×C4C2×D411yescharacteristic
125C22×C4C2411yescharacteristic
126Q16—28no—
127Q16—48no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—21no—
131C2×C8—22no—
132C2×C8—21no—
133C2×C8—22no—
134C2×C8—21no—
135C2×C8—22no—
136C2×C8#4112yes—
137C4⋊C4—24no—
138C4⋊C4—22no—
139C4⋊C4—21no—
140C4⋊C4—21no—
141C4⋊C4—22no—
142C4⋊C4—21no—
143C4⋊C4—22no—
144C4⋊C4—44no—
145C4⋊C4—44no—
146C4⋊C4—42no—
147C4⋊C4—42no—
148C4⋊C4—42no—
149C4⋊C4—42no—
150C4⋊C4—81no—
151C4⋊C4Q1614yescomplemented
152C4⋊C4C2×D412yes—
153C42—24no—
154C42—22no—
155C42—22no—
156C42—42no—
157C42—82no—
158C42C2×D412yes—
159C42#4112yes—
160C23C22×D411yescharacteristic
161Q8—22no—
162Q8—22no—
163Q8—22no—
164Q8—22no—
165Q8—41no—
166Q8—41no—
167Q8—41no—
168Q8—41no—
169Q8—42no—
170Q8—42no—
171Q8—42no—
172Q8—42no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4#12112yes—
199C2×C4#12112yes—
200C2×C4C2×Q1614yes—
201C2×C4C2×Q1612yes—
202C2×C4#13712yes—
203C2×C4C22×D411yescharacteristic
204C2×C4C2×#4111yescharacteristic
205C2×C4C2×#4111yescharacteristic
206C2×C4C2×#4111yescharacteristic
207C2×C4H(5,𝔽2)11yescharacteristic
208C2×C4H(5,𝔽2)11yescharacteristic
209C8—22no—
210C8—41no—
211C8—42no—
212C22C2×#12111yescharacteristic
213C22#54611yescharacteristic
214C22#55011yescharacteristic
215C22C22×Q1611yescharacteristic
216C22C2×#13711yescharacteristic
217C22#57711yescharacteristic
218C22#57811yescharacteristic
219C4—24no—
220C4—22no—
221C4—42no—
222C4—42no—
223C4—42no—
224C4—42no—
225C4—82no—
226C4—82no—
227C4—81no—
228C4#46214yes—
229C4#52511yescharacteristic
230C4#53511yescharacteristic
231C4#55311yescharacteristic
232C4#55511yescharacteristic
233C2#241411yescharacteristic
234C2C2×#46211yescharacteristic
235C2#304611yescharacteristic
236C2#329911yescharacteristic
237C2#332811yescharacteristic
238C2#339511yescharacteristic
239C2#340011yescharacteristic
subgroupG/H
center, socle, prime radicalC23C22×D4
commutator, Frattini, cosocleC22×C4C24
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
151. C4⋊C4Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#4622
C2×C4×Q162
#25462
#23552
#20562
#20162
#19812
#19562
#19392
#19282
#18032