Finite Group Databasebeta

← #29303

Subgroups — #29303

Order 256.

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

normal subgroups only

This group has 399 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
1#2851C211yesmaximal, characteristic
2C2×#389C211yesmaximal, characteristic
3#2085C212yesmaximal, complemented
4#2059C212yesmaximal, complemented
5#1944C212yesmaximal, complemented
6#1943C212yesmaximal, complemented
7#1853C211yesmaximal, characteristic
8#1853C211yesmaximal, characteristic
9#1852C211yesmaximal, characteristic
10#1852C211yesmaximal, characteristic
11#1749C211yesmaximal, characteristic
12C2×#125C2211yescharacteristic
13C22×C4⋊C4—21no—
14C22×C4⋊C4C2211yescharacteristic
15C22×C4⋊C4C2211yescharacteristic
16C22×C42C2211yescharacteristic
17C2×#107C2212yes—
18C2×#107C2212yes—
19C2×#106C2212yes—
20C2×#106C2212yes—
21C2×C22⋊C8C2211yescharacteristic
22C2×C22⋊C8C2211yescharacteristic
23C2×#97C2212yes—
24#389—21no—
25#389—21no—
26#389C2212yes—
27#389C2211yescharacteristic
28#389C2211yescharacteristic
29#389C2212yes—
30#387—21no—
31#387—21no—
32#385—21no—
33#385—21no—
34#382C2211yescharacteristic
35#380C2211yescharacteristic
36#380C2211yescharacteristic
37C2×#95—21no—
38#340C2212yes—
39#340C2212yes—
40#340C2212yes—
41#340C2212yes—
42#336C2212yes—
43C23×C4—21no—
44C23×C4—21no—
45C23×C4D411yescharacteristic
46C23×C4D411yescharacteristic
47C23×C4C2311yescharacteristic
48C22×C8C2312yes—
49C22×C8C2312yes—
50#125—42no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—24no—
53C2×C4⋊C4—22no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—42no—
61C2×C4⋊C4—44no—
62C2×C4⋊C4C2312yes—
63C2×C4⋊C4C2311yescharacteristic
64C2×C4⋊C4C2311yescharacteristic
65C2×C4⋊C4C2311yescharacteristic
66C2×C4⋊C4C2311yescharacteristic
67C2×C4⋊C4C2312yes—
68C2×C22⋊C4—21no—
69C2×C22⋊C4—21no—
70C2×C42—42no—
71C2×C42D411yescharacteristic
72C2×C42D411yescharacteristic
73C2×C42C2311yescharacteristic
74C2×C42C2311yescharacteristic
75#107—24no—
76#107—42no—
77#106—24no—
78#106—42no—
79C22⋊C8Q814yes—
80C22⋊C8Q814yes—
81#97—42no—
82#95—22no—
83#95—22no—
84#95—21no—
85#95—21no—
86#95—21no—
87#95—21no—
88#95—21no—
89#95—21no—
90C24C2×D411yescharacteristic
91C22×C4—22no—
92C22×C4—22no—
93C22×C4—24no—
94C22×C4—22no—
95C22×C4—22no—
96C22×C4—22no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—22no—
102C22×C4—41no—
103C22×C4—41no—
104C22×C4—41no—
105C22×C4—41no—
106C22×C4—42no—
107C22×C4—42no—
108C22×C4—42no—
109C22×C4—42no—
110C22×C4—42no—
111C22×C4C2×D411yescharacteristic
112C22×C4C2×D411yescharacteristic
113C22×C4C2×D411yescharacteristic
114C22×C4C2×D411yescharacteristic
115C22×C4C2×D411yescharacteristic
116C22×C4C2×Q812yes—
117C22×C4C2×Q812yes—
118C22×C4C2411yescharacteristic
119C2×C8—22no—
120C2×C8—22no—
121C2×C8—22no—
122C2×C8—22no—
123C2×C8C2×Q814yes—
124C2×C8C2×Q814yes—
125C4⋊C4—24no—
126C4⋊C4—22no—
127C4⋊C4—22no—
128C4⋊C4—22no—
129C4⋊C4—22no—
130C4⋊C4—22no—
131C4⋊C4—22no—
132C4⋊C4—22no—
133C4⋊C4—42no—
134C4⋊C4—42no—
135C4⋊C4—42no—
136C4⋊C4—44no—
137C4⋊C4—44no—
138C4⋊C4—42no—
139C4⋊C4—42no—
140C4⋊C4#4114yes—
141C22⋊C4—42no—
142C22⋊C4—42no—
143C42—22no—
144C42—22no—
145C42—41no—
146C42—41no—
147C23—21no—
148C23—21no—
149C23—21no—
150C23—21no—
151C23C4⋊Q812yes—
152C23AGL(1,ℤ/8)12yes—
153C23#13712yes—
154C23C22×D411yescharacteristic
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—24no—
158C2×C4—24no—
159C2×C4—24no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
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—21no—
172C2×C4—48no—
173C2×C4—44no—
174C2×C4—44no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—42no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—42no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—42no—
192C2×C4—84no—
193C2×C4C4⋊Q812yes—
194C2×C4C22×Q811yescharacteristic
195C2×C4C22×Q811yescharacteristic
196C2×C4C2×#4112yes—
197C2×C4H(5,𝔽2)12yes—
198C8—24no—
199C8—24no—
200C22—22no—
201C22—22no—
202C22—22no—
203C22—22no—
204C22—22no—
205C22—22no—
206C22#50114yes—
207C22#52511yescharacteristic
208C22C2×C4⋊Q811yescharacteristic
209C22#53411yescharacteristic
210C22C2×AGL(1,ℤ/8)11yescharacteristic
211C22C2×#13711yescharacteristic
212C22#57511yescharacteristic
213C22#57511yescharacteristic
214C4—22no—
215C4—44no—
216C4—41no—
217C4—41no—
218C4—41no—
219C4—41no—
220C4—81no—
221C4—81no—
222C4—81no—
223C4—82no—
224C4—81no—
225C4#55412yes—
226C4#55412yes—
227C2—24no—
228C2#258311yescharacteristic
229C2#311411yescharacteristic
230C2#311711yescharacteristic
231C2C2×#50111yescharacteristic
232C2#316611yescharacteristic
233C2#319411yescharacteristic
234C2#319611yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #2085C2semidirect
4. #2059C2semidirect
5. #1944C2semidirect
6. #1943C2semidirect

Maximal subgroups

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

subgroupindex
#28512
C2×#3892
#20852
#20592
#19442
#19432
#18532
#18522
#17492