Finite Group Databasebeta

← #23683

Subgroups — #23683

Order 256.

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

normal subgroups only

This group has 456 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
2C4×C4⋊Q8C211yesmaximal, characteristic, complemented
3#2293C211yesmaximal, characteristic
4#2083C211yesmaximal, characteristic, complemented
5#2071C211yesmaximal, characteristic
6#2068C212yesmaximal, complemented
7#2055C212yesmaximal
8#2043C211yesmaximal, characteristic, complemented
9#2013C211yesmaximal, characteristic
10#1767C211yesmaximal, characteristic, complemented
11#1763C212yesmaximal, complemented
12C22×SD16C2212yes—
13C2×C4⋊Q8C2211yescharacteristic
14C2×C4×Q8C2212yes—
15C2×C4×D4C2211yescharacteristic
16C2×#117—21no—
17#440—28no—
18C2×#107C2212yes—
19C2×C4⋊C8C2212yes—
20C2×#103C2212yes—
21C2×#103C2212yes—
22C2×#103C2212yes—
23C2×#102C2212yes—
24C2×#97C2212yes—
25#391C2212yes—
26#390C2211yescharacteristic
27#390C2211yescharacteristic
28#389—21no—
29#389—21no—
30#389C2211yescharacteristic
31#389C2211yescharacteristic
32#388—21no—
33#388—21no—
34#384C2211yescharacteristic
35C4×C4⋊C4—21no—
36C4×C4⋊C4—21no—
37C4×C4⋊C4C2211yescharacteristic
38C4×C22⋊C4—21no—
39C43C2211yescharacteristic
40#340C2212yes—
41#340C2212yes—
42#336C2211yescharacteristic
43#336C2211yescharacteristic
44C22×Q8C2312yes—
45C22×D4C2311yescharacteristic
46C23×C4—21no—
47C2×SD16—24no—
48C2×SD16—24no—
49C2×SD16—24no—
50C22×C8C2312yes—
51C22×C8C2312yes—
52C4⋊Q8—24no—
53C4×Q8—24no—
54C4×Q8—24no—
55C4×D4—22no—
56C4×D4D414yes—
57#117—44no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4C2312yes—
66C2×C4⋊C4C2311yescharacteristic
67C2×C4⋊C4C2311yescharacteristic
68C2×C4⋊C4C2311yescharacteristic
69C2×C22⋊C4—21no—
70C2×C22⋊C4—21no—
71C2×C22⋊C4—21no—
72C2×C22⋊C4—21no—
73C2×C22⋊C4—21no—
74C2×C42—21no—
75C2×C42—22no—
76C2×C42D411yescharacteristic
77C2×C42D411yescharacteristic
78C2×C42D411yescharacteristic
79C2×C42D411yescharacteristic
80C2×C42C2311yescharacteristic
81C2×C42C2311yescharacteristic
82C2×C42C2311yescharacteristic
83#107—24no—
84C4⋊C8—24no—
85#103—24no—
86#103—24no—
87#103—24no—
88#102—24no—
89#97—24no—
90#95—21no—
91#95—21no—
92#95—22no—
93#95—22no—
94#95—21no—
95C24—21no—
96C2×Q8—22no—
97C2×Q8—22no—
98C2×Q8—22no—
99C2×Q8—22no—
100C2×Q8—22no—
101C2×Q8#4114yes—
102C2×D4—22no—
103C2×D4—22no—
104C2×D4—21no—
105C2×D4C2×D412yes—
106C22×C4—22no—
107C22×C4—21no—
108C22×C4—24no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—22no—
112C22×C4—42no—
113C22×C4—42no—
114C22×C4C2×D411yescharacteristic
115C22×C4C2×D411yescharacteristic
116C22×C4C2×D411yescharacteristic
117C22×C4C2×D411yescharacteristic
118C22×C4C2×D411yescharacteristic
119C22×C4C2×D411yescharacteristic
120C22×C4C2411yescharacteristic
121SD16—216no—
122C2×C8—22no—
123C2×C8—22no—
124C2×C8—22no—
125C2×C8—22no—
126C2×C8—22no—
127C2×C8#4114yes—
128C4⋊C4—24no—
129C4⋊C4—24no—
130C4⋊C4—21no—
131C4⋊C4—22no—
132C4⋊C4—22no—
133C4⋊C4—22no—
134C4⋊C4—22no—
135C4⋊C4—42no—
136C4⋊C4—42no—
137C4⋊C4—44no—
138C4⋊C4—42no—
139C4⋊C4—42no—
140C4⋊C4C2×D412yes—
141C22⋊C4—24no—
142C22⋊C4—42no—
143C22⋊C4—42no—
144C22⋊C4—42no—
145C22⋊C4—42no—
146C42—22no—
147C42—22no—
148C42—22no—
149C42—22no—
150C42—22no—
151C42—41no—
152C42—42no—
153C42—44no—
154C42C2×D412yes—
155C42#4112yes—
156C23—22no—
157C23—41no—
158C23—41no—
159C23—41no—
160C23—41no—
161C23—41no—
162C23—41no—
163C23C22×D411yescharacteristic
164Q8—24no—
165Q8—24no—
166Q8—42no—
167Q8—42no—
168D4—24no—
169D4—42no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—24no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—48no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—42no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—42no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—42no—
199C2×C4#12112yes—
200C2×C4#12112yes—
201C2×C4AGL(1,ℤ/8)12yes—
202C2×C4#13712yes—
203C2×C4#13712yes—
204C2×C4C22×D411yescharacteristic
205C2×C4C2×#4112yes—
206C2×C4C2×#4112yes—
207C2×C4C2×#4111yescharacteristic
208C8—24no—
209C8—42no—
210C22—42no—
211C22—42no—
212C22—42no—
213C22—81no—
214C22—81no—
215C22—81no—
216C22—81no—
217C22C2×#12111yescharacteristic
218C22#52511yescharacteristic
219C22#53211yescharacteristic
220C22C2×AGL(1,ℤ/8)11yescharacteristic
221C22C2×#13711yescharacteristic
222C22C2×#13711yescharacteristic
223C22#57511yescharacteristic
224C4—22no—
225C4—22no—
226C4—22no—
227C4—22no—
228C4—22no—
229C4—44no—
230C4—41no—
231C4—42no—
232C4—81no—
233C4—81no—
234C4—82no—
235C4#52512yes—
236C4#52912yes—
237C2—82no—
238C2#239211yescharacteristic
239C2#303611yescharacteristic
240C2#303911yescharacteristic
241C2#311211yescharacteristic
242C2#311711yescharacteristic
243C2#318111yescharacteristic
244C2#318211yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
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. C4×C4⋊Q8C2semidirect
4. #2083C2semidirect
6. #2068C2semidirect
8. #2043C2semidirect
10. #1767C2semidirect
11. #1763C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4402
C4×C4⋊Q82
#22932
#20832
#20712
#20682
#20552
#20432
#20132
#17672
#17632