Finite Group Databasebeta

← #23684

Subgroups — #23684

Order 256.

502 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 502 conjugacy classes of subgroups — too many to draw. Only its 133 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×#440C212yesmaximal, complemented
2C4×#127C211yesmaximal, characteristic
3#2295C211yesmaximal, characteristic, complemented
4#2083C211yesmaximal, characteristic, complemented
5#2072C211yesmaximal, characteristic, complemented
6#2067C212yesmaximal, complemented
7#2055C212yesmaximal, complemented
8#2041C211yesmaximal, characteristic
9#2014C211yesmaximal, characteristic
10#1767C211yesmaximal, characteristic, complemented
11#1762C212yesmaximal, complemented
12C22×SD16C2212yes—
13C2×#127C2211yescharacteristic
14C2×C4×Q8C2211yescharacteristic
15C2×C4×D4—21no—
16C2×C4×D4—21no—
17C2×C4×D4C2212yes—
18#440—28no—
19C2×#107C2212yes—
20C2×C4⋊C8C2212yes—
21C2×#103C2212yes—
22C2×#102C2212yes—
23C2×#102C2212yes—
24C2×#102C2212yes—
25C2×#97C2212yes—
26#391C2211yescharacteristic
27#391C2211yescharacteristic
28#390—21no—
29#390—21no—
30#390C2212yes—
31#389C2211yescharacteristic
32#389C2211yescharacteristic
33#387—21no—
34#387—21no—
35#384C2211yescharacteristic
36C4×C4⋊C4—21no—
37C4×C4⋊C4C2211yescharacteristic
38#376—21no—
39C43C2211yescharacteristic
40#340C2212yes—
41#340C2212yes—
42#336C2211yescharacteristic
43#336C2211yescharacteristic
44C22×Q8C2311yescharacteristic
45C22×D4—21no—
46C22×D4—21no—
47C22×D4C2312yes—
48C23×C4—22no—
49C2×SD16—24no—
50C2×SD16—24no—
51C2×SD16—24no—
52C22×C8C2312yes—
53C22×C8C2312yes—
54#127—24no—
55C4×Q8—22no—
56C4×Q8D414yescomplemented
57C4×D4—24no—
58C4×D4—24no—
59C4×D4—44no—
60C4×D4—44no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4C2312yes—
67C2×C4⋊C4C2311yescharacteristic
68C2×C4⋊C4C2311yescharacteristic
69C2×C4⋊C4C2311yescharacteristic
70C2×C22⋊C4—22no—
71C2×C22⋊C4—22no—
72C2×C22⋊C4—22no—
73C2×C42—21no—
74C2×C42—21no—
75C2×C42D411yescharacteristic
76C2×C42D411yescharacteristic
77C2×C42D411yescharacteristic
78C2×C42D411yescharacteristic
79C2×C42C2311yescharacteristic
80C2×C42C2311yescharacteristic
81C2×C42C2311yescharacteristic
82#107—24no—
83C4⋊C8—24no—
84#103—24no—
85#102—24no—
86#102—24no—
87#102—24no—
88#97—24no—
89#95—21no—
90#95—21no—
91#95—21no—
92#95—21no—
93#95—21no—
94#95—21no—
95#95—21no—
96C24—22no—
97C2×Q8—21no—
98C2×Q8—22no—
99C2×Q8—22no—
100C2×Q8C2×D412yes—
101C2×D4—24no—
102C2×D4—24no—
103C2×D4—22no—
104C2×D4—22no—
105C2×D4—22no—
106C2×D4—22no—
107C2×D4—22no—
108C2×D4—42no—
109C2×D4—42no—
110C2×D4—42no—
111C2×D4—42no—
112C2×D4#4114yes—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—22no—
118C22×C4—44no—
119C22×C4—44no—
120C22×C4—44no—
121C22×C4C2×D411yescharacteristic
122C22×C4C2×D411yescharacteristic
123C22×C4C2×D411yescharacteristic
124C22×C4C2×D411yescharacteristic
125C22×C4C2×D411yescharacteristic
126C22×C4C2×D411yescharacteristic
127C22×C4C2411yescharacteristic
128SD16—216no—
129C2×C8—22no—
130C2×C8—22no—
131C2×C8—22no—
132C2×C8—22no—
133C2×C8—22no—
134C2×C8#4114yes—
135C4⋊C4—24no—
136C4⋊C4—21no—
137C4⋊C4—22no—
138C4⋊C4—22no—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4—42no—
142C4⋊C4—42no—
143C4⋊C4—42no—
144C4⋊C4—42no—
145C4⋊C4C2×D412yes—
146C22⋊C4—44no—
147C22⋊C4—44no—
148C22⋊C4—44no—
149C42—24no—
150C42—22no—
151C42—22no—
152C42—22no—
153C42—22no—
154C42—22no—
155C42—41no—
156C42—42no—
157C42C2×D412yes—
158C42#4112yes—
159C23—24no—
160C23—42no—
161C23—42no—
162C23—42no—
163C23—42no—
164C23—42no—
165C23—42no—
166C23C22×D411yescharacteristic
167Q8—24no—
168Q8—42no—
169D4—24no—
170D4—24no—
171D4—48no—
172D4—48no—
173D4—42no—
174D4—42no—
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—21no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—42no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—88no—
204C2×C4#12112yes—
205C2×C4#12112yes—
206C2×C4AGL(1,ℤ/8)12yes—
207C2×C4AGL(1,ℤ/8)12yes—
208C2×C4#13712yes—
209C2×C4C22×D411yescharacteristic
210C2×C4C2×#4112yes—
211C2×C4C2×#4112yes—
212C2×C4C2×#4111yescharacteristic
213C8—24no—
214C8—42no—
215C22—44no—
216C22—44no—
217C22—44no—
218C22—82no—
219C22—82no—
220C22—82no—
221C22—82no—
222C22C2×#12111yescharacteristic
223C22#52511yescharacteristic
224C22#53211yescharacteristic
225C22C2×AGL(1,ℤ/8)11yescharacteristic
226C22C2×AGL(1,ℤ/8)11yescharacteristic
227C22C2×#13711yescharacteristic
228C22#57511yescharacteristic
229C4—22no—
230C4—22no—
231C4—22no—
232C4—22no—
233C4—22no—
234C4—42no—
235C4—41no—
236C4—42no—
237C4—81no—
238C4—81no—
239C4—82no—
240C4#52512yes—
241C4#52912yes—
242C2—84no—
243C2#239211yescharacteristic
244C2#303711yescharacteristic
245C2#303811yescharacteristic
246C2#311011yescharacteristic
247C2#311411yescharacteristic
248C2#318011yescharacteristic
249C2#318211yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#127C22
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
1. C2×#440C2semidirect
3. #2295C2semidirect
4. #2083C2semidirect
5. #2072C2semidirect
6. #2067C2semidirect
7. #2055C2semidirect
10. #1767C2semidirect
11. #1762C2semidirect
56. C4×Q8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4402
C4×#1272
#22952
#20832
#20722
#20672
#20552
#20412
#20142
#17672
#17622