Finite Group Databasebeta

← #29976

Subgroups — #29976

Order 256.

337 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 337 conjugacy classes of subgroups — too many to draw. Only its 119 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#2628C211yesmaximal, characteristic
2#2317C211yesmaximal, characteristic
3#2101C212yesmaximal, complemented
4#2087C212yesmaximal, complemented
5#2086C212yesmaximal, complemented
6#2057C212yesmaximal, complemented
7#1852C212yesmaximal
8#1825C212yesmaximal
9#1798C211yesmaximal, characteristic
10C2×C4⋊Q8C2211yescharacteristic
11C2×#117C2211yescharacteristic
12C2×#117C2211yescharacteristic
13C22×C4⋊C4C2211yescharacteristic
14C2×#107C2212yes—
15C2×#107C2212yes—
16C2×C4⋊C8C2211yescharacteristic
17C2×C4⋊C8C2211yescharacteristic
18C2×C22⋊C8C2212yes—
19#400—21no—
20#399—21no—
21#395—21no—
22#393—21no—
23#389C2212yes—
24#389C2212yes—
25#389C2212yes—
26#385—21no—
27#385—21no—
28#384—21no—
29#382—21no—
30#382C2211yescharacteristic
31#380C2212yes—
32#340C2212yes—
33#340C2212yes—
34#340C2212yes—
35#340C2212yes—
36#340C2212yes—
37#340C2212yes—
38#336C2212yes—
39C22×Q8—21no—
40C23×C4—21no—
41C23×C4C2311yescharacteristic
42C22×C8C2312yes—
43C22×C8C2312yes—
44C4⋊Q8—41no—
45C4⋊Q8—41no—
46#117—22no—
47#117—41no—
48#117—41no—
49#117Q814yescomplemented
50C2×C4⋊C4—21no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—22no—
53C2×C4⋊C4—22no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—41no—
56C2×C4⋊C4—41no—
57C2×C4⋊C4D411yescharacteristic
58C2×C4⋊C4D411yescharacteristic
59C2×C4⋊C4C2312yes—
60C2×C4⋊C4C2312yes—
61C2×C4⋊C4C2312yes—
62C2×C4⋊C4C2312yes—
63C2×C22⋊C4—21no—
64C2×C22⋊C4—22no—
65C2×C22⋊C4D411yescharacteristic
66C2×C22⋊C4D411yescharacteristic
67C2×C42—21no—
68C2×C42C2311yescharacteristic
69C2×C42C2311yescharacteristic
70#107—42no—
71#107—42no—
72C4⋊C8—41no—
73C4⋊C8—41no—
74C22⋊C8—24no—
75#95—22no—
76#95—22no—
77#95—22no—
78#95—21no—
79#95—21no—
80#95—22no—
81C24C2×D411yescharacteristic
82C2×Q8—42no—
83C2×Q8—82no—
84C22×C4—21no—
85C22×C4—21no—
86C22×C4—21no—
87C22×C4—21no—
88C22×C4—22no—
89C22×C4—22no—
90C22×C4—22no—
91C22×C4—22no—
92C22×C4—42no—
93C22×C4—82no—
94C22×C4C2×D411yescharacteristic
95C22×C4C2×D411yescharacteristic
96C22×C4C2×D411yescharacteristic
97C22×C4C2×D411yescharacteristic
98C22×C4C2×D411yescharacteristic
99C22×C4C2×Q812yes—
100C22×C4#4112yes—
101C22×C4C2411yescharacteristic
102C2×C8—22no—
103C2×C8—22no—
104C2×C8—22no—
105C2×C8—22no—
106C2×C8—22no—
107C2×C8—22no—
108C4⋊C4—22no—
109C4⋊C4—22no—
110C4⋊C4—22no—
111C4⋊C4—22no—
112C4⋊C4—22no—
113C4⋊C4—22no—
114C4⋊C4—22no—
115C4⋊C4—41no—
116C4⋊C4—41no—
117C4⋊C4—42no—
118C4⋊C4—42no—
119C4⋊C4—82no—
120C4⋊C4—82no—
121C4⋊C4—82no—
122C4⋊C4C2×Q814yes—
123C22⋊C4—24no—
124C22⋊C4—41no—
125C22⋊C4—41no—
126C22⋊C4—82no—
127C42—24no—
128C42—21no—
129C42—21no—
130C42—21no—
131C42—21no—
132C23—21no—
133C23—21no—
134C23—22no—
135C23—22no—
136C23C22⋊Q812yes—
137C23C22×D411yescharacteristic
138Q8—84no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—22no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—24no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—42no—
155C2×C4—42no—
156C2×C4—42no—
157C2×C4—42no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—84no—
166C2×C4C22⋊Q812yes—
167C2×C4C22×Q811yescharacteristic
168C2×C4C2×#4111yescharacteristic
169C2×C4H(5,𝔽2)11yescharacteristic
170C2×C4H(5,𝔽2)11yescharacteristic
171C2×C42−1+411yescharacteristic
172C2×C42−1+411yescharacteristic
173C8—42no—
174C8—42no—
175C22—24no—
176C22—22no—
177C22—41no—
178C22—41no—
179C22—42no—
180C22C2×C22⋊Q811yescharacteristic
181C22#53411yescharacteristic
182C22#53611yescharacteristic
183C22#57612yes—
184C22#57812yes—
185C4—22no—
186C4—44no—
187C4—41no—
188C4—41no—
189C4—41no—
190C4—41no—
191C4—82no—
192C4—82no—
193C4—82no—
194C4#53911yescharacteristic
195C4#53911yescharacteristic
196C4#54411yescharacteristic
197C4#54411yescharacteristic
198C2—42no—
199C2#262711yescharacteristic
200C2#308312yes—
201C2#319911yescharacteristic
202C2#320211yescharacteristic
203C2#324312yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #2101C2semidirect
4. #2087C2semidirect
5. #2086C2semidirect
6. #2057C2semidirect
49. #117Q8semidirect

Maximal subgroups

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

subgroupindex
#26282
#23172
#21012
#20872
#20862
#20572
#18522
#18252
#17982