Finite Group Databasebeta

← #32834

Subgroups — #32834

Order 256.

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

normal subgroups only

This group has 360 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#2868C211yesmaximal, characteristic, complemented
2#2860C211yesmaximal, characteristic
3#2099C212yesmaximal, complemented
4#2093C212yesmaximal
5#2083C212yesmaximal, complemented
6#2071C212yesmaximal
7#2057C212yesmaximal, complemented
8#2023C212yesmaximal
9#1839C211yesmaximal, characteristic, complemented
10C2×C4⋊Q8—21no—
11C2×C4⋊Q8—21no—
12C2×C4⋊Q8C2212yes—
13C2×#126—21no—
14C2×#107C2214yes—
15C2×#102C2214yes—
16#390C2211yescharacteristic
17#390C2211yescharacteristic
18#390C2211yescharacteristic
19#389C2211yescharacteristic
20#389C2211yescharacteristic
21#389C2211yescharacteristic
22#388—21no—
23#388—21no—
24#388—21no—
25#384—21no—
26#384—21no—
27#384C2212yes—
28#384C2212yes—
29C43C2211yescharacteristic
30#340C2214yes—
31#340C2214yes—
32#336C2212yes—
33#336C2212yes—
34#336C2212yes—
35C22×Q8—22no—
36C22×D4C2311yescharacteristic
37C22×C8C2314yes—
38C4⋊Q8—44no—
39C4⋊Q8—44no—
40C4⋊Q8—44no—
41#126—44no—
42C2×C4⋊C4—21no—
43C2×C4⋊C4—21no—
44C2×C4⋊C4—21no—
45C2×C4⋊C4—22no—
46C2×C4⋊C4—22no—
47C2×C4⋊C4—22no—
48C2×C4⋊C4—22no—
49C2×C4⋊C4—22no—
50C2×C4⋊C4C2314yes—
51C2×C4⋊C4C2311yescharacteristic
52C2×C4⋊C4C2311yescharacteristic
53C2×C4⋊C4C2311yescharacteristic
54C2×C22⋊C4—21no—
55C2×C22⋊C4—21no—
56C2×C22⋊C4—21no—
57C2×C22⋊C4—21no—
58C2×C22⋊C4—21no—
59C2×C22⋊C4—21no—
60C2×C42D411yescharacteristic
61C2×C42D411yescharacteristic
62C2×C42D411yescharacteristic
63C2×C42D411yescharacteristic
64C2×C42C2311yescharacteristic
65C2×C42C2311yescharacteristic
66C2×C42C2311yescharacteristic
67#107—44no—
68#102—44no—
69#95—21no—
70#95—21no—
71#95—21no—
72C24—21no—
73C2×Q8—44no—
74C2×Q8—44no—
75C2×Q8—44no—
76C2×D4—22no—
77C2×D4—22no—
78C2×D4—22no—
79C22×C4—21no—
80C22×C4—21no—
81C22×C4—21no—
82C22×C4—24no—
83C22×C4C2×D411yescharacteristic
84C22×C4C2×D411yescharacteristic
85C22×C4C2×D411yescharacteristic
86C22×C4C2×D411yescharacteristic
87C22×C4C2×D411yescharacteristic
88C22×C4C2×D411yescharacteristic
89C22×C4C2411yescharacteristic
90C2×C8—24no—
91C2×C8—24no—
92C2×C8—24no—
93C4⋊C4—24no—
94C4⋊C4—22no—
95C4⋊C4—24no—
96C4⋊C4—22no—
97C4⋊C4—22no—
98C4⋊C4—44no—
99C4⋊C4—44no—
100C4⋊C4—42no—
101C4⋊C4—42no—
102C4⋊C4—42no—
103C4⋊C4—44no—
104C4⋊C4—44no—
105C4⋊C4—44no—
106C22⋊C4—42no—
107C22⋊C4—42no—
108C22⋊C4—42no—
109C22⋊C4—42no—
110C22⋊C4—42no—
111C22⋊C4—42no—
112C42—22no—
113C42—22no—
114C42—22no—
115C42—22no—
116C42—22no—
117C42—22no—
118C42D814yes—
119C23—41no—
120C23—41no—
121C23—41no—
122C23—41no—
123C23—41no—
124C23—41no—
125C23—41no—
126C23C22×D411yescharacteristic
127Q8—88no—
128D4—44no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—44no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—44no—
149C2×C4—41no—
150C2×C4—44no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4C2×D812yes—
154C2×C4C2×D812yes—
155C2×C4C2×D812yes—
156C2×C4AGL(1,ℤ/8)12yes—
157C2×C4#13712yes—
158C2×C4#13712yes—
159C2×C4H(5,𝔽2)12yes—
160C2×C42−1+412yes—
161C2×C42−1+412yes—
162C8—44no—
163C22—81no—
164C22—81no—
165C22—81no—
166C22—81no—
167C22—81no—
168C22—81no—
169C22—81no—
170C22#53511yescharacteristic
171C22#53611yescharacteristic
172C22#53611yescharacteristic
173C22C22×D811yescharacteristic
174C22C2×AGL(1,ℤ/8)11yescharacteristic
175C22C2×#13711yescharacteristic
176C22C2×#13711yescharacteristic
177C4—22no—
178C4—22no—
179C4—22no—
180C4—22no—
181C4—22no—
182C4—22no—
183C4—81no—
184C4—84no—
185C4—81no—
186C4—81no—
187C4#56314yes—
188C2—161no—
189C2#282111yescharacteristic
190C2#320511yescharacteristic
191C2#321811yescharacteristic
192C2#322611yescharacteristic
193C2#322611yescharacteristic
194C2#323811yescharacteristic
195C2#323911yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2868C2semidirect
3. #2099C2semidirect
5. #2083C2semidirect
7. #2057C2semidirect
9. #1839C2semidirect

Maximal subgroups

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

subgroupindex
#28682
#28602
#20992
#20932
#20832
#20712
#20572
#20232
#18392