Finite Group Databasebeta

← #25922

Subgroups — #25922

Order 256.

536 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 127 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, complemented
2#3140C212yesmaximal, complemented
3#2544C211yesmaximal, characteristic
4#2028C212yesmaximal, complemented
5#2027C212yesmaximal, complemented
6#1986C212yesmaximal, complemented
7#1887C212yesmaximal
8#1885C212yesmaximal, complemented
9C4×#100C211yesmaximal, characteristic, complemented
10#554—22no—
11#554—24no—
12D4×Q8—22no—
13#544C2212yes—
14#543—22no—
15#533C2212yes—
16C2×#127C2211yescharacteristic
17C2×#124C2211yescharacteristic
18C2×#124C2211yescharacteristic
19C2×C22⋊Q8—22no—
20#518C2212yes—
21C2×C4×D4C2211yescharacteristic
22#489—22no—
23#489—22no—
24#488—24no—
25#419C2214yes—
26#418C2214yes—
27C2×#100C2212yes—
28C4×M16C2212yes—
29#394—22no—
30#391C2211yescharacteristic
31#390C2211yescharacteristic
32#386—22no—
33C4×C4⋊C4C2211yescharacteristic
34#343C2212yes—
35#339C2214yes—
36#339C2214yes—
37C22×Q8—22no—
38C22×Q8C2311yescharacteristic
39C22×D4—22no—
40C22×D4C2311yescharacteristic
41C22×D4C2311yescharacteristic
42C23×C4D412yes—
43C2×M16C2314yes—
44C4⋊Q8—24no—
45C4⋊Q8D412yescomplemented
46C4⋊Q8D412yescomplemented
47#127—24no—
48#125—24no—
49#125D414yescomplemented
50#124—24no—
51#124—22no—
52#124—22no—
53C22⋊Q8—24no—
54C22⋊Q8—24no—
55C22⋊Q8—28no—
56C22⋊Q8—28no—
57C22⋊Q8—22no—
58C22⋊Q8—24no—
59C22⋊Q8—22no—
60C4×Q8—22no—
61C4×Q8—22no—
62C4×D4—24no—
63C4×D4—24no—
64C4×D4—22no—
65C4×D4—22no—
66#117C2314yes—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—24no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4C2311yescharacteristic
71C2×C22⋊C4—24no—
72C2×C22⋊C4—22no—
73C2×C22⋊C4—21no—
74C2×C22⋊C4—21no—
75C2×C22⋊C4—22no—
76C2×C22⋊C4D412yes—
77C2×C42C2311yescharacteristic
78C2×C42C2311yescharacteristic
79C2×C42C2311yescharacteristic
80#103—24no—
81#103—24no—
82#102—24no—
83#102—24no—
84#100—24no—
85#97—22no—
86C4×C8—22no—
87#95—21no—
88#95—22no—
89C24—21no—
90C24C2×D412yes—
91C2×Q8—24no—
92C2×Q8—22no—
93C2×Q8—22no—
94C2×Q8—22no—
95C2×Q8—22no—
96C2×Q8—22no—
97C2×Q8—84no—
98C2×D4—22no—
99C2×D4—22no—
100C2×D4—22no—
101C2×D4—22no—
102C2×D4—22no—
103C2×D4—44no—
104C2×D4—44no—
105C2×D4—82no—
106C2×D4#4114yes—
107C22×C4—21no—
108C22×C4—24no—
109C22×C4—24no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—82no—
115C22×C4C2×D412yes—
116C22×C4C2×D411yescharacteristic
117C22×C4C2×D411yescharacteristic
118C22×C4C2411yescharacteristic
119M16—24no—
120M16—24no—
121C2×C8—24no—
122C4⋊C4—24no—
123C4⋊C4—28no—
124C4⋊C4—24no—
125C4⋊C4—22no—
126C4⋊C4—24no—
127C4⋊C4—22no—
128C4⋊C4—22no—
129C4⋊C4—42no—
130C4⋊C4C2×D414yes—
131C4⋊C4C2×D414yes—
132C22⋊C4—28no—
133C22⋊C4—24no—
134C22⋊C4—24no—
135C22⋊C4—44no—
136C22⋊C4—42no—
137C22⋊C4—41no—
138C22⋊C4—41no—
139C22⋊C4—82no—
140C42—24no—
141C42—22no—
142C42—22no—
143C42C2×D412yes—
144C42C2×D412yes—
145C23—22no—
146C23—22no—
147C23—42no—
148C23—42no—
149C23—42no—
150C23—42no—
151C23—41no—
152C23—41no—
153C23—41no—
154C23#12414yes—
155C23C22×D411yescharacteristic
156Q8—44no—
157Q8—44no—
158Q8—82no—
159Q8—84no—
160D4—28no—
161D4—44no—
162D4—88no—
163C2×C4—22no—
164C2×C4—24no—
165C2×C4—24no—
166C2×C4—22no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—44no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—42no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—82no—
180C2×C4—84no—
181C2×C4—84no—
182C2×C4#12012yes—
183C2×C4#12011yescharacteristic
184C2×C4#12011yescharacteristic
185C2×C4C22×D412yes—
186C2×C4C2×#4112yes—
187C2×C42−1+412yes—
188C8—44no—
189C22—21no—
190C22—21no—
191C22—24no—
192C22—44no—
193C22—42no—
194C22—42no—
195C22—82no—
196C22—82no—
197C22—82no—
198C22—81no—
199C22—81no—
200C22—81no—
201C22C2×#12011yescharacteristic
202C22C2×#12411yescharacteristic
203C22#53611yescharacteristic
204C4—22no—
205C4—21no—
206C4—21no—
207C4—42no—
208C4—44no—
209C4—42no—
210C4—44no—
211C4—81no—
212C4—81no—
213C4#54814yes—
214C2—21no—
215C2—21no—
216C2—44no—
217C2—161no—
218C2#243611yescharacteristic
219C2#301811yescharacteristic
220C2#301911yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#127C22
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
1. #3501C2semidirect
2. #3140C2semidirect
4. #2028C2semidirect
5. #2027C2semidirect
6. #1986C2semidirect
8. #1885C2semidirect
9. C4×#100C2semidirect
45. C4⋊Q8D4semidirect
46. C4⋊Q8D4semidirect
49. #125D4semidirect

Maximal subgroups

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

subgroupindex
#35012
#31402
#25442
#20282
#20272
#19862
#18872
#18852
C4×#1002