Finite Group Databasebeta

← #23781

Subgroups — #23781

Order 256.

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

normal subgroups only

This group has 404 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
1C4×C4⋊Q8C211yesmaximal, characteristic, complemented
2#2293C211yesmaximal, characteristic
3#2099C211yesmaximal, characteristic, complemented
4#2093C211yesmaximal, characteristic
5#2087C212yesmaximal, complemented
6#2065C212yesmaximal
7#2058C211yesmaximal, characteristic, complemented
8#2023C211yesmaximal, characteristic
9#1776C212yesmaximal, complemented
10#1767C211yesmaximal, characteristic, complemented
11#1762C212yesmaximal
12C2×C4⋊Q8C2211yescharacteristic
13C2×C4×Q8—21no—
14C2×C4×D4C2211yescharacteristic
15C2×#117—21no—
16C2×#107C2212yes—
17C2×#106C2212yes—
18C2×C4⋊C8C2212yes—
19C2×#102C2212yes—
20C2×#102C2212yes—
21C2×#97C2212yes—
22#391—21no—
23#390C2211yescharacteristic
24#390C2211yescharacteristic
25#389—21no—
26#389C2212yes—
27#389C2211yescharacteristic
28#389C2211yescharacteristic
29#388—21no—
30#388—21no—
31#384C2211yescharacteristic
32C4×C4⋊C4—21no—
33C4×C4⋊C4C2212yes—
34C4×C4⋊C4C2211yescharacteristic
35C4×C22⋊C4—21no—
36C43C2211yescharacteristic
37#340C2212yes—
38#340C2212yes—
39#340C2212yes—
40#340C2212yes—
41#336C2211yescharacteristic
42#336C2211yescharacteristic
43C22×Q8—21no—
44C22×D4C2311yescharacteristic
45C23×C4—21no—
46C22×C8C2312yes—
47C22×C8C2312yes—
48C4⋊Q8—24no—
49C4×Q8—44no—
50C4×D4—22no—
51C4×D4Q814yescomplemented
52#117—44no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4C2312yes—
60C2×C4⋊C4C2311yescharacteristic
61C2×C4⋊C4C2311yescharacteristic
62C2×C4⋊C4C2311yescharacteristic
63C2×C4⋊C4C2312yes—
64C2×C22⋊C4—21no—
65C2×C22⋊C4—21no—
66C2×C22⋊C4—21no—
67C2×C22⋊C4—21no—
68C2×C22⋊C4—21no—
69C2×C42—21no—
70C2×C42—22no—
71C2×C42D411yescharacteristic
72C2×C42D411yescharacteristic
73C2×C42D411yescharacteristic
74C2×C42D411yescharacteristic
75C2×C42C2311yescharacteristic
76C2×C42C2311yescharacteristic
77C2×C42C2311yescharacteristic
78#107—24no—
79#106—24no—
80C4⋊C8—24no—
81#102—24no—
82#102—24no—
83#97—24no—
84#95—21no—
85#95—21no—
86#95—22no—
87#95—21no—
88#95—22no—
89C24—21no—
90C2×Q8—24no—
91C2×Q8—42no—
92C2×Q8—42no—
93C2×D4—22no—
94C2×D4—22no—
95C2×D4—21no—
96C2×D4C2×Q812yes—
97C22×C4—22no—
98C22×C4—21no—
99C22×C4—24no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—22no—
103C22×C4—42no—
104C22×C4—42no—
105C22×C4C2×D411yescharacteristic
106C22×C4C2×D411yescharacteristic
107C22×C4C2×D411yescharacteristic
108C22×C4C2×D411yescharacteristic
109C22×C4C2×D411yescharacteristic
110C22×C4C2×D411yescharacteristic
111C22×C4C2411yescharacteristic
112C2×C8—22no—
113C2×C8—22no—
114C2×C8—22no—
115C2×C8—22no—
116C2×C8—22no—
117C2×C8#4114yes—
118C4⋊C4—24no—
119C4⋊C4—21no—
120C4⋊C4—22no—
121C4⋊C4—22no—
122C4⋊C4—22no—
123C4⋊C4—22no—
124C4⋊C4—22no—
125C4⋊C4—42no—
126C4⋊C4—44no—
127C4⋊C4—42no—
128C4⋊C4—42no—
129C4⋊C4—42no—
130C4⋊C4C2×Q812yes—
131C4⋊C4#4114yes—
132C22⋊C4—24no—
133C22⋊C4—42no—
134C22⋊C4—42no—
135C22⋊C4—42no—
136C22⋊C4—42no—
137C42—22no—
138C42—22no—
139C42—22no—
140C42—22no—
141C42—22no—
142C42—41no—
143C42—42no—
144C42—44no—
145C42C2×Q812yes—
146C42#4112yes—
147C23—22no—
148C23—41no—
149C23—41no—
150C23—41no—
151C23—41no—
152C23—41no—
153C23—41no—
154C23C22×D411yescharacteristic
155Q8—48no—
156D4—24no—
157D4—42no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—22no—
169C2×C4—24no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—48no—
174C2×C4—42no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—42no—
178C2×C4—41no—
179C2×C4—42no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—42no—
184C2×C4—41no—
185C2×C4—42no—
186C2×C4—41no—
187C2×C4C22⋊Q812yes—
188C2×C4C22⋊Q812yes—
189C2×C4AGL(1,ℤ/8)12yes—
190C2×C4AGL(1,ℤ/8)12yes—
191C2×C4#13712yes—
192C2×C4C22×Q811yescharacteristic
193C2×C4C2×#4111yescharacteristic
194C2×C4C2×#4112yes—
195C2×C4C2×#4112yes—
196C8—24no—
197C8—42no—
198C22—42no—
199C22—42no—
200C22—42no—
201C22—81no—
202C22—81no—
203C22—81no—
204C22—81no—
205C22C2×C22⋊Q811yescharacteristic
206C22#52511yescharacteristic
207C22#53211yescharacteristic
208C22C2×AGL(1,ℤ/8)11yescharacteristic
209C22C2×AGL(1,ℤ/8)11yescharacteristic
210C22C2×#13711yescharacteristic
211C22#57511yescharacteristic
212C4—22no—
213C4—22no—
214C4—22no—
215C4—22no—
216C4—22no—
217C4—42no—
218C4—44no—
219C4—41no—
220C4—81no—
221C4—81no—
222C4—82no—
223C4#52912yes—
224C4#53312yes—
225C2—82no—
226C2#239611yescharacteristic
227C2#307711yescharacteristic
228C2#307911yescharacteristic
229C2#311211yescharacteristic
230C2#311411yescharacteristic
231C2#318111yescharacteristic
232C2#318411yescharacteristic
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
1. C4×C4⋊Q8C2semidirect
3. #2099C2semidirect
5. #2087C2semidirect
7. #2058C2semidirect
9. #1776C2semidirect
10. #1767C2semidirect
51. C4×D4Q8semidirect

Maximal subgroups

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

subgroupindex
C4×C4⋊Q82
#22932
#20992
#20932
#20872
#20652
#20582
#20232
#17762
#17672
#17622