Finite Group Databasebeta

← #25324

Subgroups — #25324

Order 256.

403 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 403 conjugacy classes of subgroups — too many to draw. Only its 139 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×#487C211yesmaximal, characteristic
2C2×#462C211yesmaximal, characteristic
3#2567C211yesmaximal, characteristic
4#2365C211yesmaximal, characteristic
5#2072C212yesmaximal
6#2056C212yesmaximal
7#1987C211yesmaximal, characteristic
8#1939C211yesmaximal, characteristic
9#1878C212yesmaximal
10#1803C212yesmaximal
11C4×#103C211yesmaximal, characteristic
12C22×Q16C2212yes—
13C2×C4⋊Q8C2211yescharacteristic
14C2×C4⋊Q8C2211yescharacteristic
15C2×C4×Q8C2211yescharacteristic
16C2×C4×Q8C2212yes—
17#487—24no—
18#462—24no—
19C2×C4⋊C8C2211yescharacteristic
20C2×C4⋊C8C2212yes—
21C2×#103C2211yescharacteristic
22C2×#103C2212yes—
23C2×#103C2212yes—
24C2×#103C2212yes—
25C2×#103C2211yescharacteristic
26C2×C4×C8C2211yescharacteristic
27#395—22no—
28#391C2212yes—
29#391C2212yes—
30#389—21no—
31#389—21no—
32#389—21no—
33#387—22no—
34#384C2212yes—
35C4×C4⋊C4—21no—
36C4×C4⋊C4C2211yescharacteristic
37C4×C4⋊C4C2211yescharacteristic
38#340C2212yes—
39#340C2211yescharacteristic
40#340C2212yes—
41#340C2211yescharacteristic
42#336C2212yes—
43C22×Q8—21no—
44C22×Q8C2312yes—
45C22×Q8C2311yescharacteristic
46C2×Q16—24no—
47C2×Q16—24no—
48C2×Q16—42no—
49C22×C8C2312yes—
50C22×C8C2311yescharacteristic
51C22×C8C2311yescharacteristic
52C4⋊Q8—22no—
53C4⋊Q8—22no—
54C4⋊Q8—22no—
55C4⋊Q8D414yes—
56C4×Q8—24no—
57C4×Q8—22no—
58C4×Q8—22no—
59C4×Q8—42no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4D411yescharacteristic
68C2×C4⋊C4D411yescharacteristic
69C2×C4⋊C4C2311yescharacteristic
70C2×C4⋊C4C2312yes—
71C2×C4⋊C4C2311yescharacteristic
72C2×C4⋊C4C2311yescharacteristic
73C2×C42—22no—
74C2×C42—21no—
75C2×C42—21no—
76C2×C42C2312yes—
77C2×C42C2311yescharacteristic
78C4⋊C8—22no—
79C4⋊C8—42no—
80#103—24no—
81#103—24no—
82#103—24no—
83#103—22no—
84#103—22no—
85C4×C8—22no—
86#95—22no—
87#95—21no—
88#95—22no—
89#95—22no—
90#95—22no—
91#95D412yes—
92C2×Q8—22no—
93C2×Q8—21no—
94C2×Q8—21no—
95C2×Q8—21no—
96C2×Q8—21no—
97C2×Q8—22no—
98C2×Q8—22no—
99C2×Q8—22no—
100C2×Q8—21no—
101C2×Q8—22no—
102C2×Q8—42no—
103C2×Q8—81no—
104C2×Q8—81no—
105C2×Q8C2×D414yes—
106C2×Q8#4112yes—
107C22×C4—21no—
108C22×C4—22no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—22no—
112C22×C4—21no—
113C22×C4C2×D412yes—
114C22×C4C2×D411yescharacteristic
115C22×C4C2×D412yes—
116C22×C4C2×D411yescharacteristic
117C22×C4C2411yescharacteristic
118Q16—48no—
119C2×C8—21no—
120C2×C8—21no—
121C2×C8—22no—
122C2×C8—21no—
123C2×C8—21no—
124C2×C8—22no—
125C2×C8—21no—
126C2×C8—22no—
127C2×C8—21no—
128C4⋊C4—24no—
129C4⋊C4—24no—
130C4⋊C4—22no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—22no—
134C4⋊C4—21no—
135C4⋊C4—21no—
136C4⋊C4—22no—
137C4⋊C4—21no—
138C4⋊C4—44no—
139C4⋊C4—42no—
140C4⋊C4—42no—
141C4⋊C4—82no—
142C4⋊C4—82no—
143C4⋊C4Q1614yescomplemented
144C4⋊C4#4112yes—
145C42—22no—
146C42—22no—
147C42—44no—
148C42—42no—
149C42—42no—
150C42C2×D412yes—
151C42#4112yes—
152C23C22×D411yescharacteristic
153Q8—22no—
154Q8—22no—
155Q8—24no—
156Q8—24no—
157Q8—42no—
158Q8—42no—
159Q8—41no—
160Q8—41no—
161Q8—84no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—24no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—41no—
173C2×C4—42no—
174C2×C4—41no—
175C2×C4—42no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—42no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—41no—
187C2×C4#12112yes—
188C2×C4#12112yes—
189C2×C4C2×Q1614yes—
190C2×C4C2×Q1612yes—
191C2×C4#13712yes—
192C2×C4C22×D411yescharacteristic
193C2×C4C2×#4111yescharacteristic
194C2×C4C2×#4111yescharacteristic
195C2×C4C2×#4111yescharacteristic
196C2×C4H(5,𝔽2)11yescharacteristic
197C2×C42−1+411yescharacteristic
198C8—41no—
199C8—42no—
200C8—41no—
201C22C2×#12111yescharacteristic
202C22#54611yescharacteristic
203C22#54811yescharacteristic
204C22C22×Q1611yescharacteristic
205C22C2×#13711yescharacteristic
206C22#57711yescharacteristic
207C22#57711yescharacteristic
208C4—22no—
209C4—24no—
210C4—42no—
211C4—42no—
212C4—44no—
213C4—42no—
214C4—81no—
215C4—81no—
216C4—82no—
217C4#46214yes—
218C4#52911yescharacteristic
219C4#54211yescharacteristic
220C4#54611yescharacteristic
221C4#54711yescharacteristic
222C2#242711yescharacteristic
223C2C2×#46211yescharacteristic
224C2#304311yescharacteristic
225C2#329911yescharacteristic
226C2#333111yescharacteristic
227C2#336311yescharacteristic
228C2#336911yescharacteristic
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
143. C4⋊C4Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#4872
C2×#4622
#25672
#23652
#20722
#20562
#19872
#19392
#18782
#18032
C4×#1032