Finite Group Databasebeta

← #25294

Subgroups — #25294

Order 256.

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

normal subgroups only

This group has 391 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×#462C211yesmaximal, characteristic
2C2×C8×Q8C211yesmaximal, characteristic
3#2582C211yesmaximal, characteristic
4#2568C211yesmaximal, characteristic
5#2068C212yesmaximal
6#2056C212yesmaximal
7#1969C212yesmaximal
8#1956C211yesmaximal, characteristic
9#1947C211yesmaximal, characteristic
10#1933C212yesmaximal
11C4×#103C211yesmaximal, characteristic
12C22×Q16C2212yes—
13C2×C4⋊Q8C2211yescharacteristic
14C2×C4×Q8C2211yescharacteristic
15C2×C4×Q8C2211yescharacteristic
16#462—24no—
17C8×Q8—24no—
18C2×#107C2212yes—
19C2×C4⋊C8C2212yes—
20C2×C4⋊C8C2211yescharacteristic
21C2×#103C2211yescharacteristic
22C2×#103C2212yes—
23C2×#103C2212yes—
24C2×#103C2211yescharacteristic
25C2×#103C2212yes—
26C2×C4×C8C2212yes—
27C2×C4×C8C2211yescharacteristic
28#400—22no—
29#398—22no—
30#391—21no—
31#391—21no—
32#391C2212yes—
33#391C2212yes—
34#391C2212yes—
35#389—21no—
36#389C2212yes—
37#384C2211yescharacteristic
38#382—21no—
39C4×C4⋊C4C2211yescharacteristic
40#340C2212yes—
41#340C2211yescharacteristic
42#340C2211yescharacteristic
43C22×Q8D411yescharacteristic
44C22×Q8C2312yes—
45C22×Q8C2311yescharacteristic
46C2×Q16—24no—
47C2×Q16—24no—
48C2×Q16—42no—
49C22×C8C2311yescharacteristic
50C22×C8C2312yes—
51C22×C8C2311yescharacteristic
52C4⋊Q8—22no—
53C4⋊Q8—22no—
54C4×Q8—22no—
55C4×Q8—22no—
56C4×Q8—22no—
57C4×Q8—22no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4D411yescharacteristic
65C2×C4⋊C4D412yes—
66C2×C4⋊C4C2311yescharacteristic
67C2×C4⋊C4C2311yescharacteristic
68C2×C4⋊C4C2311yescharacteristic
69C2×C4⋊C4C2312yes—
70C2×C42—21no—
71C2×C42—21no—
72C2×C42C2312yes—
73C2×C42C2311yescharacteristic
74#107—42no—
75C4⋊C8—24no—
76C4⋊C8D414yes—
77#103—22no—
78#103—24no—
79#103—22no—
80#103—24no—
81#103—42no—
82C4×C8—24no—
83C4×C8—22no—
84#95—21no—
85#95—22no—
86#95—22no—
87#95—22no—
88#95—22no—
89#95—22no—
90#95—22no—
91#95—22no—
92C2×Q8—22no—
93C2×Q8—22no—
94C2×Q8—21no—
95C2×Q8—21no—
96C2×Q8—22no—
97C2×Q8—21no—
98C2×Q8—21no—
99C2×Q8—22no—
100C2×Q8—22no—
101C2×Q8—22no—
102C2×Q8—21no—
103C2×Q8—22no—
104C2×Q8—22no—
105C2×Q8Q1614yescomplemented
106C2×Q8#4112yes—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—22no—
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—22no—
121C2×C8—22no—
122C2×C8—21no—
123C2×C8—21no—
124C2×C8—21no—
125C2×C8—21no—
126C2×C8C2×D414yes—
127C2×C8#4112yes—
128C4⋊C4—24no—
129C4⋊C4—22no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—22no—
134C4⋊C4—21no—
135C4⋊C4—22no—
136C4⋊C4—21no—
137C4⋊C4—21no—
138C4⋊C4—42no—
139C4⋊C4—42no—
140C4⋊C4—42no—
141C4⋊C4—42no—
142C4⋊C4—81no—
143C4⋊C4—82no—
144C42—22no—
145C42—22no—
146C42—42no—
147C42—42no—
148C42C2×D412yes—
149C42#4112yes—
150C23C22×D411yescharacteristic
151Q8—28no—
152Q8—22no—
153Q8—22no—
154Q8—41no—
155Q8—41no—
156Q8—42no—
157Q8—42no—
158Q8—42no—
159Q8—42no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—42no—
172C2×C4—41no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—41no—
176C2×C4—42no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4#12112yes—
186C2×C4#12112yes—
187C2×C4C2×Q1614yes—
188C2×C4C2×Q1612yes—
189C2×C4#13512yes—
190C2×C4C22×D411yescharacteristic
191C2×C4C2×#4111yescharacteristic
192C2×C4C2×#4111yescharacteristic
193C2×C4C2×#4111yescharacteristic
194C2×C4H(5,𝔽2)11yescharacteristic
195C2×C42−1+411yescharacteristic
196C8—24no—
197C8—22no—
198C8—41no—
199C22C2×#12111yescharacteristic
200C22#54811yescharacteristic
201C22#55011yescharacteristic
202C22C22×Q1611yescharacteristic
203C22C2×#13511yescharacteristic
204C22#57711yescharacteristic
205C22#57811yescharacteristic
206C4—24no—
207C4—22no—
208C4—42no—
209C4—42no—
210C4—81no—
211C4—81no—
212C4—82no—
213C4—82no—
214C4—81no—
215C4#46714yes—
216C4#52911yescharacteristic
217C4#54211yescharacteristic
218C4#54611yescharacteristic
219C4#54811yescharacteristic
220C2#242611yescharacteristic
221C2C2×#46711yescharacteristic
222C2#306311yescharacteristic
223C2#336311yescharacteristic
224C2#336411yescharacteristic
225C2#339411yescharacteristic
226C2#339511yescharacteristic
subgroupG/H
center, socle, prime radicalC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
105. C2×Q8Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#4622
C2×C8×Q82
#25822
#25682
#20682
#20562
#19692
#19562
#19472
#19332
C4×#1032