Finite Group Databasebeta

← #29559

Subgroups — #29559

Order 256.

360 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 117 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
1#2862C211yesmaximal, characteristic
2#2295C211yesmaximal, characteristic
3#2058C211yesmaximal, characteristic
4#2043C211yesmaximal, characteristic
5#2024C211yesmaximal, characteristic
6#2016C211yesmaximal, characteristic
7#1971C212yesmaximal
8#1935C212yesmaximal
9#1927C212yesmaximal
10#1878C212yesmaximal
11#1745C211yesmaximal, characteristic
12C22×Q16C2212yes—
13C2×C4⋊Q8C2211yescharacteristic
14C2×C4⋊Q8C2211yescharacteristic
15C2×C4×Q8C2211yescharacteristic
16C2×#106C2212yes—
17C2×#103C2212yes—
18C2×#103C2212yes—
19C2×#103C2212yes—
20C2×#103C2212yes—
21C2×#97C2212yes—
22#391—21no—
23#391—21no—
24#391C2212yes—
25#391C2212yes—
26#391C2211yescharacteristic
27#391C2211yescharacteristic
28#389C2211yescharacteristic
29#389C2211yescharacteristic
30#387—21no—
31#387—21no—
32#382—21no—
33#382—21no—
34C4×C4⋊C4—21no—
35C4×C4⋊C4C2211yescharacteristic
36#376—21no—
37C43C2211yescharacteristic
38#340C2212yes—
39#340C2212yes—
40#336C2211yescharacteristic
41#336C2212yes—
42#336C2211yescharacteristic
43C22×Q8—21no—
44C22×Q8C2312yes—
45C22×Q8C2311yescharacteristic
46C2×Q16—24no—
47C2×Q16—24no—
48C2×Q16—42no—
49C22×C8C2312yes—
50C22×C8C2312yes—
51C4⋊Q8—42no—
52C4⋊Q8—42no—
53C4×Q8—22no—
54C4×Q8—22no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4C2312yes—
62C2×C4⋊C4C2311yescharacteristic
63C2×C4⋊C4C2311yescharacteristic
64C2×C4⋊C4C2311yescharacteristic
65C2×C42—21no—
66C2×C42—21no—
67C2×C42D411yescharacteristic
68C2×C42D411yescharacteristic
69C2×C42D411yescharacteristic
70C2×C42D411yescharacteristic
71C2×C42C2311yescharacteristic
72C2×C42C2311yescharacteristic
73C2×C42C2311yescharacteristic
74#106—42no—
75#103—24no—
76#103—24no—
77#103—42no—
78#103—42no—
79#97—24no—
80#95—21no—
81#95—21no—
82#95—21no—
83#95—21no—
84#95—21no—
85#95—21no—
86#95—21no—
87#95—22no—
88#95—22no—
89#95—22no—
90#95—22no—
91C2×Q8—21no—
92C2×Q8—22no—
93C2×Q8—22no—
94C2×Q8—22no—
95C2×Q8—22no—
96C2×Q8—22no—
97C2×Q8—22no—
98C2×Q8—22no—
99C2×Q8—22no—
100C2×Q8—42no—
101C2×Q8—42no—
102C2×Q8—42no—
103C2×Q8#4112yes—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4C2×D411yescharacteristic
111C22×C4C2×D411yescharacteristic
112C22×C4C2×D411yescharacteristic
113C22×C4C2×D411yescharacteristic
114C22×C4C2×D411yescharacteristic
115C22×C4C2×D411yescharacteristic
116C22×C4C2411yescharacteristic
117Q16—48no—
118C2×C8—22no—
119C2×C8—22no—
120C2×C8—22no—
121C2×C8—22no—
122C2×C8—22no—
123C2×C8#4114yes—
124C4⋊C4—24no—
125C4⋊C4—21no—
126C4⋊C4—22no—
127C4⋊C4—22no—
128C4⋊C4—22no—
129C4⋊C4—22no—
130C4⋊C4—42no—
131C4⋊C4—42no—
132C4⋊C4—42no—
133C4⋊C4—42no—
134C4⋊C4—42no—
135C4⋊C4#4112yes—
136C42—24no—
137C42—22no—
138C42—22no—
139C42—22no—
140C42—22no—
141C42—21no—
142C42—21no—
143C42—22no—
144C42—41no—
145C42—42no—
146C23C22×D411yescharacteristic
147Q8—24no—
148Q8—42no—
149Q8—42no—
150Q8—42no—
151Q8—42no—
152Q8—42no—
153Q8—84no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—42no—
176C2×C4—41no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—41no—
180C2×C4—42no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—42no—
185C2×C4—41no—
186C2×C4—42no—
187C2×C4#13712yes—
188C2×C4#13712yes—
189C2×C4#13712yes—
190C2×C4C2×#4111yescharacteristic
191C2×C4C2×#4112yes—
192C2×C4C2×#4111yescharacteristic
193C2×C4H(5,𝔽2)12yes—
194C8—24no—
195C8—42no—
196C22#52511yescharacteristic
197C22#53211yescharacteristic
198C22#53511yescharacteristic
199C22C2×#13711yescharacteristic
200C22C2×#13711yescharacteristic
201C22C2×#13711yescharacteristic
202C22#57511yescharacteristic
203C4—22no—
204C4—22no—
205C4—22no—
206C4—41no—
207C4—42no—
208C4—42no—
209C4—41no—
210C4—41no—
211C4—82no—
212C4—81no—
213C4—82no—
214C4—81no—
215C4#55112yes—
216C4#55512yes—
217C2#261011yescharacteristic
218C2#311311yescharacteristic
219C2#311611yescharacteristic
220C2#317211yescharacteristic
221C2#318511yescharacteristic
222C2#321511yescharacteristic
223C2#321811yescharacteristic
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
#28622
#22952
#20582
#20432
#20242
#20162
#19712
#19352
#19272
#18782
#17452