Finite Group Databasebeta

← #24683

Subgroups — #24683

Order 256.

373 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 373 conjugacy classes of subgroups — too many to draw. Only its 131 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×#475C211yesmaximal, characteristic
2C2×C4×Q16C211yesmaximal, characteristic
3#2568C211yesmaximal, characteristic
4#2355C211yesmaximal, characteristic
5#2074C212yesmaximal
6#2056C212yesmaximal
7#1990C211yesmaximal, characteristic
8#1949C212yesmaximal
9#1939C211yesmaximal, characteristic
10#1927C212yesmaximal
11#1803C211yesmaximal, characteristic
12C22×Q16C2211yescharacteristic
13C2×C4⋊Q8C2211yescharacteristic
14C2×C4×Q8C2211yescharacteristic
15C2×C4×Q8C2211yescharacteristic
16#475—24no—
17C4×Q16—24no—
18C2×#107C2211yescharacteristic
19C2×#106C2212yes—
20C2×C4⋊C8C2211yescharacteristic
21C2×C4⋊C8C2212yes—
22C2×#103C2212yes—
23C2×#103C2212yes—
24C2×#103C2211yescharacteristic
25C2×#103C2211yescharacteristic
26C2×C4×C8C2211yescharacteristic
27#400—22no—
28#391—21no—
29#391—21no—
30#391C2212yes—
31#391C2212yes—
32#387—22no—
33#384C2211yescharacteristic
34#382—21no—
35#382C2212yes—
36C4×C4⋊C4—21no—
37C4×C4⋊C4C2212yes—
38C4×C4⋊C4C2211yescharacteristic
39#340C2212yes—
40#340C2212yes—
41#340C2212yes—
42#336C2212yes—
43C22×Q8—21no—
44C22×Q8C2311yescharacteristic
45C22×Q8C2311yescharacteristic
46C2×Q16—22no—
47C2×Q16—22no—
48C2×Q16—22no—
49C22×C8C2311yescharacteristic
50C22×C8C2312yes—
51C22×C8C2311yescharacteristic
52C4⋊Q8—22no—
53C4⋊Q8—22no—
54C4×Q8—22no—
55C4×Q8—22no—
56C4×Q8—22no—
57C4×Q8D414yes—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4D411yescharacteristic
64C2×C4⋊C4D411yescharacteristic
65C2×C4⋊C4C2311yescharacteristic
66C2×C4⋊C4C2312yes—
67C2×C4⋊C4C2312yes—
68C2×C4⋊C4C2311yescharacteristic
69C2×C42—21no—
70C2×C42—21no—
71C2×C42—22no—
72C2×C42C2312yes—
73C2×C42C2311yescharacteristic
74#107—22no—
75#106—24no—
76C4⋊C8—24no—
77C4⋊C8—22no—
78#103—22no—
79#103—24no—
80#103—24no—
81#103—22no—
82C4×C8—22no—
83#95—22no—
84#95—22no—
85#95—22no—
86#95—22no—
87#95—22no—
88#95—22no—
89#95—22no—
90#95D412yes—
91C2×Q8—21no—
92C2×Q8—21no—
93C2×Q8—21no—
94C2×Q8—21no—
95C2×Q8—21no—
96C2×Q8—21no—
97C2×Q8—21no—
98C2×Q8—21no—
99C2×Q8—21no—
100C2×Q8—21no—
101C2×Q8—42no—
102C2×Q8—42no—
103C2×Q8—42no—
104C2×Q8C2×D412yes—
105C2×Q8#4112yes—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4C2×D412yes—
113C22×C4C2×D412yes—
114C22×C4C2×D411yescharacteristic
115C22×C4C2×D411yescharacteristic
116C22×C4C2411yescharacteristic
117Q16—28no—
118C2×C8—21no—
119C2×C8—21no—
120C2×C8—21no—
121C2×C8—22no—
122C2×C8—22no—
123C2×C8—22no—
124C2×C8—21no—
125C2×C8—21no—
126C2×C8#4112yes—
127C4⋊C4—24no—
128C4⋊C4—22no—
129C4⋊C4—22no—
130C4⋊C4—21no—
131C4⋊C4—22no—
132C4⋊C4—21no—
133C4⋊C4—22no—
134C4⋊C4—21no—
135C4⋊C4—22no—
136C4⋊C4—22no—
137C4⋊C4—42no—
138C4⋊C4—42no—
139C4⋊C4—42no—
140C4⋊C4—81no—
141C4⋊C4C2×D412yes—
142C42—24no—
143C42—22no—
144C42—22no—
145C42—42no—
146C42—82no—
147C42C2×D412yes—
148C42#4112yes—
149C23C22×D411yescharacteristic
150Q8—22no—
151Q8—22no—
152Q8—22no—
153Q8—22no—
154Q8—41no—
155Q8—41no—
156Q8—41no—
157Q8—41no—
158Q8—84no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4#12112yes—
186C2×C4#12112yes—
187C2×C4#13512yes—
188C2×C4#13712yes—
189C2×C4C22×D411yescharacteristic
190C2×C4C2×#4111yescharacteristic
191C2×C4C2×#4111yescharacteristic
192C2×C4C2×#4111yescharacteristic
193C2×C42−1+411yescharacteristic
194C2×C42−1+411yescharacteristic
195C8—22no—
196C8—42no—
197C8—41no—
198C22C2×#12111yescharacteristic
199C22#54711yescharacteristic
200C22#54811yescharacteristic
201C22C2×#13511yescharacteristic
202C22C2×#13711yescharacteristic
203C22#57711yescharacteristic
204C22#57811yescharacteristic
205C4—22no—
206C4—42no—
207C4—42no—
208C4—42no—
209C4—42no—
210C4—42no—
211C4—82no—
212C4—82no—
213C4—81no—
214C4#46414yes—
215C4#52511yescharacteristic
216C4#53611yescharacteristic
217C4#55211yescharacteristic
218C4#55611yescharacteristic
219C2#241511yescharacteristic
220C2C2×#46411yescharacteristic
221C2#304611yescharacteristic
222C2#334211yescharacteristic
223C2#335611yescharacteristic
224C2#336511yescharacteristic
225C2#337411yescharacteristic
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

Maximal subgroups

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

subgroupindex
C2×#4752
C2×C4×Q162
#25682
#23552
#20742
#20562
#19902
#19492
#19392
#19272
#18032