Finite Group Databasebeta

← #26440

Subgroups — #26440

Order 256.

369 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 369 conjugacy classes of subgroups — too many to draw. Only its 147 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#2551C211yesmaximal, characteristic
2#2365C211yesmaximal, characteristic
3#2086C212yesmaximal
4#2057C212yesmaximal
5#1990C211yesmaximal, characteristic
6#1988C212yesmaximal
7#1988C211yesmaximal, characteristic
8#1947C211yesmaximal, characteristic
9#1939C211yesmaximal, characteristic
10#1918C212yesmaximal
11C4×#107C211yesmaximal, characteristic
12C2×C4⋊Q8—21no—
13C2×C4⋊Q8C2212yes—
14C2×C4⋊Q8C2211yescharacteristic
15C2×C4×Q8—21no—
16C2×#107C2211yescharacteristic
17C2×#107C2212yes—
18C2×#107C2211yescharacteristic
19C2×C4⋊C8C2211yescharacteristic
20C2×C4⋊C8C2212yes—
21C2×C4×C8C2211yescharacteristic
22#395—22no—
23#391—21no—
24#389—21no—
25#389C2212yes—
26#389C2212yes—
27#387—22no—
28#384C2211yescharacteristic
29C4×C4⋊C4C2212yes—
30C4×C4⋊C4C2211yescharacteristic
31C4×C4⋊C4C2211yescharacteristic
32C4×C4⋊C4C2211yescharacteristic
33#340C2212yes—
34#340C2212yes—
35#340C2211yescharacteristic
36#340C2212yes—
37#340C2212yes—
38#340C2212yes—
39#340C2211yescharacteristic
40#336C2212yes—
41C22×Q8—21no—
42C22×Q8—22no—
43C22×C8C2312yes—
44C22×C8C2311yescharacteristic
45C22×C8C2311yescharacteristic
46C4⋊Q8—24no—
47C4⋊Q8—22no—
48C4⋊Q8—42no—
49C4⋊Q8—44no—
50C4⋊Q8Q814yescomplemented
51C4×Q8—44no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—22no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4D411yescharacteristic
59C2×C4⋊C4D411yescharacteristic
60C2×C4⋊C4C2312yes—
61C2×C4⋊C4C2311yescharacteristic
62C2×C4⋊C4C2312yes—
63C2×C4⋊C4C2311yescharacteristic
64C2×C4⋊C4C2311yescharacteristic
65C2×C4⋊C4C2311yescharacteristic
66C2×C42—22no—
67C2×C42—21no—
68C2×C42—21no—
69C2×C42C2312yes—
70C2×C42C2311yescharacteristic
71#107—22no—
72#107—42no—
73#107Q814yescomplemented
74C4⋊C8—22no—
75C4⋊C8—42no—
76C4×C8—22no—
77#95—21no—
78#95—22no—
79#95—21no—
80#95—22no—
81#95—22no—
82#95D412yes—
83C2×Q8—24no—
84C2×Q8—44no—
85C2×Q8—42no—
86C2×Q8—44no—
87C2×Q8—42no—
88C2×Q8—82no—
89C22×C4—21no—
90C22×C4—21no—
91C22×C4—22no—
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—22no—
95C22×C4C2×D412yes—
96C22×C4C2×D411yescharacteristic
97C22×C4C2×D412yes—
98C22×C4C2×D411yescharacteristic
99C22×C4C2411yescharacteristic
100C2×C8—21no—
101C2×C8—21no—
102C2×C8—21no—
103C2×C8—22no—
104C2×C8—21no—
105C2×C8—22no—
106C2×C8—22no—
107C2×C8—21no—
108C2×C8C2×Q812yes—
109C4⋊C4—24no—
110C4⋊C4—22no—
111C4⋊C4—22no—
112C4⋊C4—21no—
113C4⋊C4—21no—
114C4⋊C4—21no—
115C4⋊C4—21no—
116C4⋊C4—22no—
117C4⋊C4—21no—
118C4⋊C4—22no—
119C4⋊C4—44no—
120C4⋊C4—44no—
121C4⋊C4—44no—
122C4⋊C4—42no—
123C4⋊C4—81no—
124C4⋊C4D814yes—
125C4⋊C4C2×Q812yes—
126C4⋊C4C2×Q812yes—
127C4⋊C4C2×Q814yes—
128C4⋊C4#4112yes—
129C42—22no—
130C42—22no—
131C42—44no—
132C42—42no—
133C42—42no—
134C42C2×Q812yes—
135C42#4112yes—
136C23C22×D411yescharacteristic
137Q8—48no—
138Q8—88no—
139C2×C4—22no—
140C2×C4—21no—
141C2×C4—22no—
142C2×C4—21no—
143C2×C4—24no—
144C2×C4—22no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—42no—
153C2×C4—41no—
154C2×C4—42no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—42no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—41no—
164C2×C4C22⋊Q812yes—
165C2×C4C22⋊Q812yes—
166C2×C4C2×D814yes—
167C2×C4C2×D812yes—
168C2×C4#13712yes—
169C2×C4C22×Q811yescharacteristic
170C2×C4C22×Q811yescharacteristic
171C2×C4C2×#4111yescharacteristic
172C2×C4C2×#4111yescharacteristic
173C2×C4H(5,𝔽2)11yescharacteristic
174C2×C42−1+411yescharacteristic
175C8—22no—
176C8—41no—
177C8—42no—
178C22C2×C22⋊Q811yescharacteristic
179C22#54611yescharacteristic
180C22D4×Q811yescharacteristic
181C22C22×D811yescharacteristic
182C22C2×#13711yescharacteristic
183C22#57711yescharacteristic
184C22#57811yescharacteristic
185C4—22no—
186C4—24no—
187C4—42no—
188C4—42no—
189C4—42no—
190C4—42no—
191C4—44no—
192C4—81no—
193C4—82no—
194C4#47414yes—
195C4#53311yescharacteristic
196C4#54211yescharacteristic
197C4#55711yescharacteristic
198C4Q8211yescharacteristic
199C2#245111yescharacteristic
200C2C2×#47411yescharacteristic
201C2#308211yescharacteristic
202C2#329411yescharacteristic
203C2#332811yescharacteristic
204C2Q8×D811yescharacteristic
205C2#338711yescharacteristic
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

Complements

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

NH ≅ G/Ntype
50. C4⋊Q8Q8semidirect
73. #107Q8semidirect

Maximal subgroups

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

subgroupindex
#25512
#23652
#20862
#20572
#19902
#19882
#19472
#19392
#19182
C4×#1072