Finite Group Databasebeta

← #24732

Subgroups — #24732

Order 256.

457 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 457 conjugacy classes of subgroups — too many to draw. Only its 151 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×#462C212yesmaximal
2C2×#462C211yesmaximal, characteristic
3C2×C4×Q16C211yesmaximal, characteristic
4#2546C211yesmaximal, characteristic
5Q8×C4⋊C4C211yesmaximal, characteristic
6#2068C212yesmaximal
7#2043C212yesmaximal
8#1956C211yesmaximal, characteristic
9#1939C211yesmaximal, characteristic
10#1878C212yesmaximal
11#1803C211yesmaximal, characteristic
12C22×Q16C2212yes—
13C22×Q16C2211yescharacteristic
14C2×C4⋊Q8C2211yescharacteristic
15C2×C4⋊Q8C2212yes—
16C2×C4×Q8C2211yescharacteristic
17C2×C4×Q8C2212yes—
18C2×C4×Q8C2211yescharacteristic
19#462—28no—
20#462—24no—
21C4×Q16—24no—
22C2×#107C2211yescharacteristic
23C2×C4⋊C8C2211yescharacteristic
24C2×C4⋊C8C2212yes—
25C2×#103C2212yes—
26C2×#103C2212yes—
27C2×#103C2212yes—
28C2×#103C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#103C2212yes—
31C2×C4×C8C2211yescharacteristic
32#398—22no—
33#391C2212yes—
34#389—21no—
35#389—21no—
36#389—22no—
37#389C2212yes—
38#384—21no—
39#384C2211yescharacteristic
40C4×C4⋊C4—21no—
41C4×C4⋊C4C2211yescharacteristic
42#340C2212yes—
43#336C2212yes—
44C22×Q8C2312yes—
45C22×Q8C2311yescharacteristic
46C22×Q8C2311yescharacteristic
47C2×Q16—22no—
48C2×Q16—22no—
49C2×Q16—24no—
50C2×Q16—24no—
51C2×Q16—24no—
52C2×Q16D414yes—
53C22×C8C2312yes—
54C22×C8C2311yescharacteristic
55C22×C8C2311yescharacteristic
56C4⋊Q8—24no—
57C4⋊Q8—22no—
58C4⋊Q8—22no—
59C4⋊Q8—42no—
60C4×Q8—22no—
61C4×Q8—24no—
62C4×Q8—22no—
63C4×Q8—22no—
64C4×Q8—42no—
65C4×Q8D414yes—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4D411yescharacteristic
77C2×C4⋊C4D411yescharacteristic
78C2×C4⋊C4C2311yescharacteristic
79C2×C4⋊C4C2312yes—
80C2×C4⋊C4C2311yescharacteristic
81C2×C42—21no—
82C2×C42—22no—
83C2×C42—21no—
84C2×C42C2312yes—
85C2×C42C2311yescharacteristic
86#107—22no—
87C4⋊C8—24no—
88C4⋊C8—22no—
89#103—22no—
90#103—24no—
91#103—24no—
92#103—22no—
93#103—24no—
94#103—42no—
95C4×C8—22no—
96#95—22no—
97#95—22no—
98#95—22no—
99#95D412yes—
100C2×Q8—21no—
101C2×Q8—21no—
102C2×Q8—21no—
103C2×Q8—21no—
104C2×Q8—21no—
105C2×Q8—22no—
106C2×Q8—22no—
107C2×Q8—21no—
108C2×Q8—21no—
109C2×Q8—22no—
110C2×Q8—22no—
111C2×Q8—22no—
112C2×Q8—21no—
113C2×Q8—22no—
114C2×Q8C2×D412yes—
115C2×Q8C2×D412yes—
116C2×Q8C2×D412yes—
117C2×Q8#4112yes—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4C2×D412yes—
125C22×C4C2×D411yescharacteristic
126C22×C4C2×D412yes—
127C22×C4C2×D411yescharacteristic
128C22×C4C2411yescharacteristic
129Q16—28no—
130Q16—48no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—21no—
134C2×C8—22no—
135C2×C8—22no—
136C2×C8—22no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8C2×D412yes—
140C4⋊C4—24no—
141C4⋊C4—22no—
142C4⋊C4—21no—
143C4⋊C4—22no—
144C4⋊C4—21no—
145C4⋊C4—21no—
146C4⋊C4—22no—
147C4⋊C4—44no—
148C4⋊C4—44no—
149C4⋊C4—44no—
150C4⋊C4—42no—
151C4⋊C4—42no—
152C4⋊C4—42no—
153C4⋊C4—42no—
154C4⋊C4—81no—
155C4⋊C4—82no—
156C4⋊C4Q1614yescomplemented
157C4⋊C4C2×D412yes—
158C42—24no—
159C42—22no—
160C42—22no—
161C42—44no—
162C42—42no—
163C42C2×D412yes—
164C42#4112yes—
165C23C22×D411yescharacteristic
166Q8—22no—
167Q8—22no—
168Q8—22no—
169Q8—22no—
170Q8—22no—
171Q8—41no—
172Q8—41no—
173Q8—42no—
174Q8—42no—
175Q8—42no—
176Q8—42no—
177Q8#12114yes—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—21no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4#12112yes—
204C2×C4#12112yes—
205C2×C4C2×Q1614yes—
206C2×C4C2×Q1612yes—
207C2×C4#13712yes—
208C2×C4C22×D411yescharacteristic
209C2×C4C22×D411yescharacteristic
210C2×C4C2×#4111yescharacteristic
211C2×C4C2×#4111yescharacteristic
212C2×C4H(5,𝔽2)11yescharacteristic
213C2×C42−1+411yescharacteristic
214C8—22no—
215C8—42no—
216C8—41no—
217C22C2×#12111yescharacteristic
218C22D4211yescharacteristic
219C22#54811yescharacteristic
220C22C22×Q1611yescharacteristic
221C22C2×#13711yescharacteristic
222C22#57711yescharacteristic
223C22#57811yescharacteristic
224C4—24no—
225C4—24no—
226C4—22no—
227C4—42no—
228C4—42no—
229C4—42no—
230C4—82no—
231C4—82no—
232C4—81no—
233C4#46214yescomplemented
234C4C2×#12111yescharacteristic
235C4#53711yescharacteristic
236C4#54711yescharacteristic
237C4#55311yescharacteristic
238C2#241611yescharacteristic
239C2C2×#46211yescharacteristic
240C2#304611yescharacteristic
241C2#327811yescharacteristic
242C2D4×Q1611yescharacteristic
243C2#336311yescharacteristic
244C2#337411yescharacteristic
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
156. C4⋊C4Q16semidirect
233. C4#462semidirect

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
Q8×C4⋊C42
#20682
#20432
#19562
#19392
#18782
#18032