Finite Group Databasebeta

← #24318

Subgroups — #24318

Order 256.

507 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 507 conjugacy classes of subgroups — too many to draw. Only its 135 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#3078C212yesmaximal, complemented
3#2326C211yesmaximal, characteristic
4#2060C212yesmaximal, complemented
5#2028C212yesmaximal, complemented
6#1995C212yesmaximal, complemented
7#1885C212yesmaximal
8#1856C212yesmaximal
9#1800C211yesmaximal, characteristic
10#554—22no—
11#554—24no—
12D4×Q8—22no—
13#544C2212yes—
14#543—22no—
15#533C2212yes—
16C2×#124C2211yescharacteristic
17C2×C22⋊Q8—22no—
18#518C2212yes—
19C2×C4×Q8C2211yescharacteristic
20C2×C4×D4C2211yescharacteristic
21#479—24no—
22#477—22no—
23#475—22no—
24#428C2214yes—
25#423C2212yes—
26#419C2214yes—
27C2×#100C2212yes—
28#391C2211yescharacteristic
29#390C2211yescharacteristic
30#388—22no—
31#385—22no—
32#384C2211yescharacteristic
33#376C2211yescharacteristic
34#343C2212yes—
35#339C2214yes—
36#339C2214yes—
37C22×Q8—22no—
38C22×Q8C2311yescharacteristic
39C22×D4C2311yescharacteristic
40C23×C4D412yes—
41C2×M16C2314yes—
42C4⋊Q8—24no—
43C4⋊Q8D412yes—
44C4⋊Q8D412yes—
45#125—24no—
46#125D414yes—
47#124—22no—
48#124Q814yescomplemented
49C22⋊Q8—24no—
50C22⋊Q8—24no—
51C22⋊Q8—28no—
52C22⋊Q8—28no—
53C22⋊Q8—22no—
54C22⋊Q8—24no—
55C22⋊Q8—22no—
56C4×Q8—22no—
57C4×Q8—22no—
58C4×Q8—22no—
59C4×Q8—22no—
60C4×D4—24no—
61C4×D4—24no—
62C4×D4—22no—
63C4×D4—22no—
64#117C2314yes—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—24no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4C2311yescharacteristic
70C2×C4⋊C4C2311yescharacteristic
71C2×C22⋊C4—24no—
72C2×C22⋊C4—22no—
73C2×C22⋊C4—22no—
74C2×C22⋊C4D412yes—
75C2×C42—21no—
76C2×C42C2311yescharacteristic
77C2×C42C2311yescharacteristic
78C2×C42C2311yescharacteristic
79#107—24no—
80#106—24no—
81C4⋊C8—22no—
82C4⋊C8—22no—
83#103—24no—
84#103—24no—
85#100—24no—
86#95—21no—
87#95—21no—
88#95—22no—
89C24C2×D412yes—
90C2×Q8—24no—
91C2×Q8—22no—
92C2×Q8—22no—
93C2×Q8—21no—
94C2×Q8—22no—
95C2×Q8—22no—
96C2×Q8—84no—
97C2×Q8C2×Q812yescomplemented
98C2×D4—22no—
99C2×D4—22no—
100C2×D4C2×Q812yes—
101C2×D4#4112yes—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—24no—
105C22×C4—24no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—82no—
111C22×C4C2×D412yes—
112C22×C4C2×D411yescharacteristic
113C22×C4C2×D411yescharacteristic
114C22×C4C2411yescharacteristic
115M16—24no—
116M16—24no—
117C2×C8—24no—
118C4⋊C4—24no—
119C4⋊C4—28no—
120C4⋊C4—24no—
121C4⋊C4—22no—
122C4⋊C4—24no—
123C4⋊C4—22no—
124C4⋊C4—22no—
125C4⋊C4—22no—
126C4⋊C4—41no—
127C4⋊C4—41no—
128C4⋊C4—82no—
129C4⋊C4C2×D414yes—
130C4⋊C4C2×D414yes—
131C22⋊C4—28no—
132C22⋊C4—28no—
133C22⋊C4—24no—
134C22⋊C4—24no—
135C22⋊C4—82no—
136C42—24no—
137C42—21no—
138C42—22no—
139C42—42no—
140C42C2×D412yes—
141C42C2×D412yes—
142C42C2×Q812yescomplemented
143C23—22no—
144C23—22no—
145C23—42no—
146C23—42no—
147C23C22⋊Q814yes—
148C23C22×D411yescharacteristic
149Q8—24no—
150Q8—44no—
151Q8—42no—
152Q8—82no—
153Q8—84no—
154D4—24no—
155D4—24no—
156C2×C4—24no—
157C2×C4—24no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—44no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—42no—
169C2×C4—41no—
170C2×C4—42no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—82no—
175C2×C4—84no—
176C2×C4—84no—
177C2×C4#12012yes—
178C2×C4#12011yescharacteristic
179C2×C4#12011yescharacteristic
180C2×C4C22×D412yes—
181C2×C4C22×Q811yescharacteristic
182C2×C4C2×#4111yescharacteristic
183C2×C42−1+412yes—
184C8—44no—
185C22—21no—
186C22—21no—
187C22—24no—
188C22—44no—
189C22—42no—
190C22—42no—
191C22—82no—
192C22C2×#12011yescharacteristic
193C22C2×C22⋊Q811yescharacteristic
194C22#53611yescharacteristic
195C4—21no—
196C4—21no—
197C4—22no—
198C4—42no—
199C4—42no—
200C4—42no—
201C4—44no—
202C4—44no—
203C4—81no—
204C4—81no—
205C4#54812yes—
206C4D4×Q812yes—
207C2—21no—
208C2—21no—
209C2—44no—
210C2#240911yescharacteristic
211C2#301911yescharacteristic
212C2#302211yescharacteristic
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. #3078C2semidirect
4. #2060C2semidirect
5. #2028C2semidirect
6. #1995C2semidirect
48. #124Q8semidirect
97. C2×Q8C2×Q8semidirect
142. C42C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#35012
#30782
#23262
#20602
#20282
#19952
#18852
#18562
#18002