Finite Group Databasebeta

← #25828

Subgroups — #25828

Order 256.

497 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 497 conjugacy classes of subgroups — too many to draw. Only its 143 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#3501C211yesmaximal, characteristic
2#3166C212yesmaximal, complemented
3#2566C211yesmaximal, characteristic
4#2061C212yesmaximal, complemented
5#2060C212yesmaximal, complemented
6#1986C212yesmaximal, complemented
7#1856C212yesmaximal
8#1856C212yesmaximal
9C4×#100C211yesmaximal, characteristic
10#554—22no—
11#554—24no—
12D4×Q8—22no—
13#544C2212yes—
14#543—22no—
15#533C2212yes—
16C2×#125C2211yescharacteristic
17C2×C22⋊Q8—22no—
18#518C2212yes—
19C2×C4×D4C2211yescharacteristic
20#501—22no—
21#501—22no—
22#499—24no—
23#428C2214yes—
24#428C2214yes—
25C2×#100C2212yes—
26C4×M16C2212yes—
27#396—22no—
28#390C2211yescharacteristic
29#390C2211yescharacteristic
30#385—22no—
31#384C2211yescharacteristic
32#382C2211yescharacteristic
33C4×C4⋊C4C2211yescharacteristic
34#343C2212yes—
35#339C2214yes—
36#339C2214yes—
37C22×Q8—22no—
38C22×D4C2311yescharacteristic
39C23×C4D412yes—
40C2×M16C2314yes—
41C4⋊Q8—24no—
42C4⋊Q8D412yes—
43C4⋊Q8D412yes—
44#125—24no—
45#125—22no—
46#125—22no—
47#125D414yes—
48C22⋊Q8—24no—
49C22⋊Q8—24no—
50C22⋊Q8—28no—
51C22⋊Q8—28no—
52C22⋊Q8—22no—
53C22⋊Q8—24no—
54C22⋊Q8—22no—
55C4×Q8—22no—
56C4×Q8—22no—
57C4×D4—24no—
58C4×D4—24no—
59C4×D4—22no—
60C4×D4—22no—
61#117C2314yes—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—24no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4C2311yescharacteristic
69C2×C4⋊C4C2311yescharacteristic
70C2×C4⋊C4C2311yescharacteristic
71C2×C22⋊C4—24no—
72C2×C22⋊C4—22no—
73C2×C22⋊C4—22no—
74C2×C22⋊C4D412yes—
75C2×C42C2311yescharacteristic
76C2×C42C2311yescharacteristic
77C2×C42C2311yescharacteristic
78#107—24no—
79#107—24no—
80#106—24no—
81#106—24no—
82#100Q818yescomplemented
83#97—22no—
84C4×C8—22no—
85#95—21no—
86#95—22no—
87C24C2×D412yes—
88C2×Q8—24no—
89C2×Q8—22no—
90C2×Q8—22no—
91C2×Q8—84no—
92C2×D4—22no—
93C2×D4—22no—
94C2×D4C2×Q814yes—
95C22×C4—21no—
96C22×C4—24no—
97C22×C4—24no—
98C22×C4—22no—
99C22×C4—22no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—82no—
104C22×C4C2×D412yes—
105C22×C4C2×D411yescharacteristic
106C22×C4C2×D411yescharacteristic
107C22×C4C2411yescharacteristic
108M16—24no—
109M16C2×Q818yescomplemented
110C2×C8—24no—
111C4⋊C4—24no—
112C4⋊C4—28no—
113C4⋊C4—24no—
114C4⋊C4—22no—
115C4⋊C4—24no—
116C4⋊C4—22no—
117C4⋊C4—22no—
118C4⋊C4—22no—
119C4⋊C4—22no—
120C4⋊C4—42no—
121C4⋊C4—42no—
122C4⋊C4—41no—
123C4⋊C4—41no—
124C4⋊C4—82no—
125C4⋊C4C2×D414yes—
126C4⋊C4C2×D414yes—
127C22⋊C4—28no—
128C22⋊C4—24no—
129C22⋊C4—24no—
130C22⋊C4—44no—
131C22⋊C4—82no—
132C42—24no—
133C42—22no—
134C42—22no—
135C42C2×D412yes—
136C42C2×D412yes—
137C23—22no—
138C23—22no—
139C23—42no—
140C23—42no—
141C23C4⋊Q814yes—
142C23C22×D411yescharacteristic
143Q8—44no—
144Q8—82no—
145Q8—84no—
146D4—28no—
147C2×C4—22no—
148C2×C4—24no—
149C2×C4—24no—
150C2×C4—22no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—44no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—42no—
157C2×C4—42no—
158C2×C4—41no—
159C2×C4—42no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—82no—
165C2×C4—84no—
166C2×C4—84no—
167C2×C4#12012yes—
168C2×C4#12011yescharacteristic
169C2×C4#12011yescharacteristic
170C2×C4C22×D412yes—
171C2×C4C22×Q812yes—
172C2×C42−1+412yes—
173C8—28no—
174C22—21no—
175C22—21no—
176C22—24no—
177C22—44no—
178C22—42no—
179C22—42no—
180C22—82no—
181C22C2×#12011yescharacteristic
182C22C2×C4⋊Q811yescharacteristic
183C22#53611yescharacteristic
184C4—22no—
185C4—21no—
186C4—21no—
187C4—42no—
188C4—44no—
189C4—42no—
190C4—44no—
191C4—81no—
192C4—81no—
193C4—81no—
194C4D4×Q814yes—
195C2—21no—
196C2—21no—
197C2—44no—
198C2#243411yescharacteristic
199C2#301811yescharacteristic
200C2#301911yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3166C2semidirect
4. #2061C2semidirect
5. #2060C2semidirect
6. #1986C2semidirect
82. #100Q8semidirect
109. M16C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#35012
#31662
#25662
#20612
#20602
#19862
#18562
C4×#1002