Finite Group Databasebeta

← #29782

Subgroups — #29782

Order 256.

463 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 463 conjugacy classes of subgroups — too many to draw. Only its 123 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
1C2×#469C211yesmaximal, characteristic
2C2×#423C211yesmaximal, characteristic, complemented
3#2854C212yesmaximal, complemented
4#2088C212yesmaximal, complemented
5#2067C212yesmaximal, complemented
6#2065C212yesmaximal, complemented
7#2054C212yesmaximal, complemented
8#1895C212yesmaximal, complemented
9#1809C211yesmaximal, characteristic, complemented
10C22×D8C2211yescharacteristic
11C22×M16C2211yescharacteristic
12C22×M16C2211yescharacteristic
13C2×#123—22no—
14C2×#121C2212yes—
15C22×C42C2211yescharacteristic
16#469—24no—
17C2×#107C2211yescharacteristic
18C2×#106C2211yescharacteristic
19C2×#106C2211yescharacteristic
20#423—24no—
21C2×C4⋊C8C2212yes—
22C2×C4⋊C8C2212yes—
23C2×#102C2212yes—
24C2×#102C2212yes—
25C2×#102C2212yes—
26#390C2212yes—
27#390C2212yes—
28#389C2212yes—
29#389C2212yes—
30#388—22no—
31#388—22no—
32#386—22no—
33#380C2212yes—
34#340C2212yes—
35#340C2212yes—
36#340C2212yes—
37C22×D4—22no—
38C22×D4C2312yes—
39C23×C4D412yes—
40C23×C4C2311yescharacteristic
41C2×D8—22no—
42C2×D8—41no—
43C2×D8—41no—
44C2×M16—22no—
45C2×M16—41no—
46C2×M16—41no—
47C2×M16—41no—
48C2×M16—41no—
49C2×M16D414yes—
50C22×C8C2312yes—
51C22×C8C2311yescharacteristic
52C22×C8C2311yescharacteristic
53#123—48no—
54#121—24no—
55#121—24no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4C2312yes—
59C2×C4⋊C4C2312yes—
60C2×C4⋊C4C2312yes—
61C2×C22⋊C4—22no—
62C2×C22⋊C4—22no—
63C2×C22⋊C4—22no—
64C2×C22⋊C4—22no—
65C2×C22⋊C4—22no—
66C2×C22⋊C4—22no—
67C2×C42—22no—
68C2×C42—22no—
69C2×C42D411yescharacteristic
70C2×C42D411yescharacteristic
71C2×C42C2312yes—
72#107—41no—
73#106—22no—
74#106—41no—
75C4⋊C8—24no—
76C4⋊C8—24no—
77#102—24no—
78#102—42no—
79#102—42no—
80#95—22no—
81#95—22no—
82#95—22no—
83C24—22no—
84C24C2×D411yescharacteristic
85C2×D4—22no—
86C2×D4—22no—
87C2×D4—22no—
88C2×D4—22no—
89C2×D4—22no—
90C2×D4—22no—
91C2×D4—44no—
92C2×D4—44no—
93C2×D4—44no—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—24no—
99C22×C4—24no—
100C22×C4—22no—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4—42no—
104C22×C4C2×D412yes—
105C22×C4C2×D411yescharacteristic
106C22×C4C2×D412yes—
107C22×C4C2×D412yes—
108C22×C4#4112yes—
109C22×C4C2411yescharacteristic
110D8—44no—
111M16—44no—
112M16—44no—
113C2×C8—21no—
114C2×C8—21no—
115C2×C8—21no—
116C2×C8—21no—
117C2×C8—22no—
118C2×C8—22no—
119C2×C8—22no—
120C2×C8C2×D412yes—
121C2×C8C2×D412yes—
122C4⋊C4—22no—
123C4⋊C4—22no—
124C4⋊C4—22no—
125C4⋊C4—22no—
126C4⋊C4—22no—
127C4⋊C4—22no—
128C4⋊C4—44no—
129C4⋊C4—44no—
130C22⋊C4—44no—
131C22⋊C4—44no—
132C22⋊C4—44no—
133C22⋊C4—44no—
134C22⋊C4—44no—
135C22⋊C4—44no—
136C42—22no—
137C42—22no—
138C42—22no—
139C42—22no—
140C23—21no—
141C23—21no—
142C23—22no—
143C23—22no—
144C23—42no—
145C23—42no—
146C23—42no—
147C23—42no—
148C23—42no—
149C23—42no—
150C23—42no—
151C23#12112yes—
152C23C22×D411yescharacteristic
153D4—42no—
154D4—42no—
155D4—42no—
156D4—42no—
157D4—88no—
158C2×C4—22no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—48no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4#12112yes—
180C2×C4AGL(1,ℤ/8)14yes—
181C2×C4C22×D411yescharacteristic
182C2×C4C2×#4111yescharacteristic
183C2×C4H(5,𝔽2)11yescharacteristic
184C2×C4H(5,𝔽2)11yescharacteristic
185C2×C42−1+411yescharacteristic
186C2×C42−1+411yescharacteristic
187C8—22no—
188C8—22no—
189C8—42no—
190C22—22no—
191C22—24no—
192C22—42no—
193C22—41no—
194C22—41no—
195C22—82no—
196C22—82no—
197C22—82no—
198C22—82no—
199C22—82no—
200C22—82no—
201C22—82no—
202C22C2×#12111yescharacteristic
203C22#53711yescharacteristic
204C22#53711yescharacteristic
205C22C2×AGL(1,ℤ/8)12yes—
206C22#57512yes—
207C4—22no—
208C4—24no—
209C4—42no—
210C4—82no—
211C4—82no—
212C4—82no—
213C4#53411yescharacteristic
214C4#53611yescharacteristic
215C4#53911yescharacteristic
216C4#53911yescharacteristic
217C2—42no—
218C2—162no—
219C2#261811yescharacteristic
220C2#305512yes—
221C2#325112yes—
222C2#325311yescharacteristic
223C2#325611yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#469C2
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
2. C2×#423C2semidirect
3. #2854C2semidirect
4. #2088C2semidirect
5. #2067C2semidirect
6. #2065C2semidirect
7. #2054C2semidirect
8. #1895C2semidirect
9. #1809C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4692
C2×#4232
#28542
#20882
#20672
#20652
#20542
#18952
#18092