Finite Group Databasebeta

← #28798

Subgroups — #28798

Order 256.

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

normal subgroups only

This group has 325 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#2552C211yesmaximal, characteristic
2#2552C211yesmaximal, characteristic
3#2083C212yesmaximal
4#2059C212yesmaximal
5#1992C212yesmaximal
6#1948C211yesmaximal, characteristic
7#1947C211yesmaximal, characteristic
8#1943C212yesmaximal
9#1834C211yesmaximal, characteristic
10C4×#107C211yesmaximal, characteristic
11#1770C211yesmaximal, characteristic
12C2×#125—21no—
13C2×#125C2212yes—
14C2×#107C2211yescharacteristic
15C2×#107C2211yescharacteristic
16C2×#107C2212yes—
17C2×#106C2212yes—
18C2×C4⋊C8C2211yescharacteristic
19C2×#97C2212yes—
20C2×C4×C8C2211yescharacteristic
21#398—22no—
22#398—22no—
23#389—21no—
24#389—21no—
25#389C2212yes—
26#389C2212yes—
27#384—21no—
28#384C2212yes—
29#384C2211yescharacteristic
30#384C2211yescharacteristic
31C4×C4⋊C4C2211yescharacteristic
32C4×C4⋊C4C2211yescharacteristic
33C4×C4⋊C4C2211yescharacteristic
34#340C2212yes—
35#340C2212yes—
36#340C2212yes—
37#340C2212yes—
38#340C2211yescharacteristic
39#340C2211yescharacteristic
40#336C2212yes—
41C22×C8C2311yescharacteristic
42C22×C8C2312yes—
43C22×C8C2311yescharacteristic
44#125—24no—
45#125—42no—
46#125—44no—
47C2×C4⋊C4—21no—
48C2×C4⋊C4—22no—
49C2×C4⋊C4—22no—
50C2×C4⋊C4—22no—
51C2×C4⋊C4—22no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—22no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4D411yescharacteristic
58C2×C4⋊C4D411yescharacteristic
59C2×C4⋊C4C2312yes—
60C2×C4⋊C4C2311yescharacteristic
61C2×C4⋊C4C2312yes—
62C2×C4⋊C4C2311yescharacteristic
63C2×C4⋊C4C2311yescharacteristic
64C2×C4⋊C4C2311yescharacteristic
65C2×C42—21no—
66C2×C42—21no—
67C2×C42C2312yes—
68C2×C42C2311yescharacteristic
69#107—22no—
70#107—24no—
71#107—22no—
72#106—42no—
73C4⋊C8Q814yes—
74#97—42no—
75C4×C8Q814yes—
76#95—21no—
77#95—21no—
78#95—22no—
79#95—22no—
80#95D412yes—
81C22×C4—21no—
82C22×C4—21no—
83C22×C4—21no—
84C22×C4—21no—
85C22×C4—22no—
86C22×C4—22no—
87C22×C4C2×D412yes—
88C22×C4C2×D412yes—
89C22×C4C2×D411yescharacteristic
90C22×C4C2×D411yescharacteristic
91C22×C4C2411yescharacteristic
92C2×C8—21no—
93C2×C8—21no—
94C2×C8—22no—
95C2×C8—21no—
96C2×C8—22no—
97C2×C8—21no—
98C2×C8C2×Q812yes—
99C2×C8C2×Q812yes—
100C2×C8C2×Q814yes—
101C4⋊C4—22no—
102C4⋊C4—22no—
103C4⋊C4—21no—
104C4⋊C4—21no—
105C4⋊C4—21no—
106C4⋊C4—21no—
107C4⋊C4—22no—
108C4⋊C4—22no—
109C4⋊C4—21no—
110C4⋊C4—22no—
111C4⋊C4—21no—
112C4⋊C4—22no—
113C4⋊C4—44no—
114C4⋊C4—44no—
115C4⋊C4—44no—
116C4⋊C4—44no—
117C4⋊C4—42no—
118C4⋊C4—42no—
119C4⋊C4—81no—
120C4⋊C4—81no—
121C4⋊C4—82no—
122C4⋊C4—82no—
123C4⋊C4#4112yes—
124C4⋊C4#4112yes—
125C42—22no—
126C42—22no—
127C42—42no—
128C42—42no—
129C42C2×Q812yes—
130C42C2×Q812yes—
131C23C22×D411yescharacteristic
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—22no—
135C2×C4—22no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—22no—
141C2×C4—22no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—42no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—42no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—42no—
153C2×C4—42no—
154C2×C4—42no—
155C2×C4—41no—
156C2×C4—42no—
157C2×C4—41no—
158C2×C4C4⋊Q812yes—
159C2×C4C4⋊Q812yes—
160C2×C4AGL(1,ℤ/8)12yes—
161C2×C4#13712yes—
162C2×C4C22×Q811yescharacteristic
163C2×C4C22×Q811yescharacteristic
164C2×C4C2×#4111yescharacteristic
165C2×C4C2×#4111yescharacteristic
166C2×C4H(5,𝔽2)11yescharacteristic
167C2×C42−1+411yescharacteristic
168C8—22no—
169C8—22no—
170C8—24no—
171C22C2×C4⋊Q811yescharacteristic
172C22#54611yescharacteristic
173C22#54811yescharacteristic
174C22C2×AGL(1,ℤ/8)11yescharacteristic
175C22C2×#13711yescharacteristic
176C22#57611yescharacteristic
177C22#57811yescharacteristic
178C4—22no—
179C4—42no—
180C4—42no—
181C4—42no—
182C4—42no—
183C4—81no—
184C4—82no—
185C4—81no—
186C4—82no—
187C4#50114yes—
188C4#53311yescharacteristic
189C4#54411yescharacteristic
190C4#55411yescharacteristic
191C4#55711yescharacteristic
192C2#255011yescharacteristic
193C2C2×#50111yescharacteristic
194C2#316411yescharacteristic
195C2#333011yescharacteristic
196C2#333311yescharacteristic
197C2#336611yescharacteristic
198C2#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
#25522
#20832
#20592
#19922
#19482
#19472
#19432
#18342
C4×#1072
#17702