Finite Group Databasebeta

← #23685

Subgroups — #23685

Order 256.

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

normal subgroups only

This group has 418 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×#441C212yesmaximal
2C4×C4⋊Q8C211yesmaximal, characteristic
3#2295C211yesmaximal, characteristic
4#2084C211yesmaximal, characteristic
5#2072C211yesmaximal, characteristic
6#2066C212yesmaximal
7#2056C212yesmaximal
8#2043C211yesmaximal, characteristic
9#2016C211yesmaximal, characteristic
10#1767C211yesmaximal, characteristic
11#1763C212yesmaximal
12C22×Q16C2212yes—
13C2×C4⋊Q8C2211yescharacteristic
14C2×C4×Q8C2212yes—
15C2×C4×Q8C2211yescharacteristic
16#441—28no—
17C2×#106C2212yes—
18C2×C4⋊C8C2212yes—
19C2×#103C2212yes—
20C2×#103C2212yes—
21C2×#103C2212yes—
22C2×#103C2212yes—
23C2×#97C2212yes—
24#391C2212yes—
25#391C2211yescharacteristic
26#391C2211yescharacteristic
27#389—21no—
28#389—21no—
29#389C2211yescharacteristic
30#389C2211yescharacteristic
31#387—21no—
32#387—21no—
33#384C2211yescharacteristic
34C4×C4⋊C4—21no—
35C4×C4⋊C4—21no—
36C4×C4⋊C4—21no—
37C4×C4⋊C4C2211yescharacteristic
38#376—21no—
39C43C2211yescharacteristic
40#340C2212yes—
41#340C2212yes—
42#336C2211yescharacteristic
43#336C2211yescharacteristic
44C22×Q8C2311yescharacteristic
45C22×Q8C2312yes—
46C2×Q16—24no—
47C2×Q16—24no—
48C2×Q16—24no—
49C22×C8C2312yes—
50C22×C8C2312yes—
51C4⋊Q8—24no—
52C4×Q8—22no—
53C4×Q8—24no—
54C4×Q8—24no—
55C4×Q8D414yes—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4C2312yes—
65C2×C4⋊C4C2311yescharacteristic
66C2×C4⋊C4C2311yescharacteristic
67C2×C4⋊C4C2311yescharacteristic
68C2×C42—21no—
69C2×C42—22no—
70C2×C42—21no—
71C2×C42D411yescharacteristic
72C2×C42D411yescharacteristic
73C2×C42D411yescharacteristic
74C2×C42D411yescharacteristic
75C2×C42C2311yescharacteristic
76C2×C42C2311yescharacteristic
77C2×C42C2311yescharacteristic
78#106—24no—
79C4⋊C8—24no—
80#103—24no—
81#103—24no—
82#103—24no—
83#103—24no—
84#97—24no—
85#95—21no—
86#95—21no—
87#95—21no—
88#95—21no—
89#95—21no—
90#95—21no—
91#95—22no—
92#95—22no—
93#95—21no—
94C2×Q8—21no—
95C2×Q8—22no—
96C2×Q8—22no—
97C2×Q8—22no—
98C2×Q8—22no—
99C2×Q8—22no—
100C2×Q8—22no—
101C2×Q8—22no—
102C2×Q8C2×D412yes—
103C2×Q8#4114yes—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—22no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—22no—
110C22×C4C2×D411yescharacteristic
111C22×C4C2×D411yescharacteristic
112C22×C4C2×D411yescharacteristic
113C22×C4C2×D411yescharacteristic
114C22×C4C2×D411yescharacteristic
115C22×C4C2×D411yescharacteristic
116C22×C4C2411yescharacteristic
117Q16—216no—
118C2×C8—22no—
119C2×C8—22no—
120C2×C8—22no—
121C2×C8—22no—
122C2×C8—22no—
123C2×C8#4114yes—
124C4⋊C4—24no—
125C4⋊C4—24no—
126C4⋊C4—24no—
127C4⋊C4—21no—
128C4⋊C4—22no—
129C4⋊C4—22no—
130C4⋊C4—22no—
131C4⋊C4—22no—
132C4⋊C4—42no—
133C4⋊C4—44no—
134C4⋊C4—42no—
135C4⋊C4—42no—
136C4⋊C4—42no—
137C4⋊C4C2×D412yes—
138C42—24no—
139C42—22no—
140C42—22no—
141C42—22no—
142C42—22no—
143C42—22no—
144C42—41no—
145C42—42no—
146C42—44no—
147C42C2×D412yes—
148C42#4112yes—
149C23C22×D411yescharacteristic
150Q8—24no—
151Q8—24no—
152Q8—24no—
153Q8—42no—
154Q8—42no—
155Q8—42no—
156C2×C4—21no—
157C2×C4—21no—
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—22no—
167C2×C4—22no—
168C2×C4—24no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—42no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—42no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—42no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—42no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—42no—
187C2×C4#12112yes—
188C2×C4#12112yes—
189C2×C4#13712yes—
190C2×C4#13712yes—
191C2×C4#13712yes—
192C2×C4C22×D411yescharacteristic
193C2×C4C2×#4112yes—
194C2×C4C2×#4112yes—
195C2×C4C2×#4111yescharacteristic
196C8—24no—
197C8—42no—
198C22C2×#12111yescharacteristic
199C22#52511yescharacteristic
200C22#53211yescharacteristic
201C22C2×#13711yescharacteristic
202C22C2×#13711yescharacteristic
203C22C2×#13711yescharacteristic
204C22#57511yescharacteristic
205C4—22no—
206C4—22no—
207C4—22no—
208C4—22no—
209C4—22no—
210C4—42no—
211C4—44no—
212C4—41no—
213C4—42no—
214C4—81no—
215C4—81no—
216C4—82no—
217C4#52512yes—
218C4#52912yes—
219C2#239211yescharacteristic
220C2#303711yescharacteristic
221C2#303911yescharacteristic
222C2#311311yescharacteristic
223C2#311711yescharacteristic
224C2#318111yescharacteristic
225C2#318511yescharacteristic
subgroupG/H
center, socle, prime radicalC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#4412
C4×C4⋊Q82
#22952
#20842
#20722
#20662
#20562
#20432
#20162
#17672
#17632