Finite Group Databasebeta

← #30053

Subgroups — #30053

Order 256.

421 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 421 conjugacy classes of subgroups — too many to draw. Only its 107 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#2641C211yesmaximal, characteristic, complemented
2#2326C211yesmaximal, characteristic, complemented
3#2071C212yesmaximal, complemented
4#2026C211yesmaximal, characteristic, complemented
5#2025C211yesmaximal, characteristic, complemented
6#2014C211yesmaximal, characteristic, complemented
7#2012C212yesmaximal, complemented
8#2011C211yesmaximal, characteristic
9#1880C212yesmaximal, complemented
10#1879C212yesmaximal, complemented
11#1840C211yesmaximal, characteristic, complemented
12C22×SD16C2212yes—
13C22×D8C2212yes—
14C2×#127C2211yescharacteristic
15C2×#125C2211yescharacteristic
16C2×#124C2211yescharacteristic
17C2×C4×Q8C2211yescharacteristic
18C2×C4×D4C2211yescharacteristic
19C2×#103C2212yes—
20C2×#102C2212yes—
21C2×#102C2212yes—
22C2×#102C2212yes—
23#394—22no—
24#394—22no—
25#391C2211yescharacteristic
26#390C2211yescharacteristic
27#390C2212yes—
28#390C2212yes—
29#388—22no—
30#385—22no—
31#384C2211yescharacteristic
32#376C2211yescharacteristic
33#340C2212yes—
34#340C2212yes—
35#336C2212yes—
36#336C2212yes—
37#336C2211yescharacteristic
38#336C2211yescharacteristic
39C22×Q8C2311yescharacteristic
40C22×D4—22no—
41C22×D4C2312yes—
42C22×D4C2311yescharacteristic
43C23×C4—21no—
44C2×SD16—24no—
45C2×SD16—42no—
46C2×SD16—42no—
47C2×D8—24no—
48C2×D8—42no—
49C2×D8—42no—
50C22×C8C2312yes—
51C22×C8C2312yes—
52#127—42no—
53#125—42no—
54#124—22no—
55#124—22no—
56C4×Q8—42no—
57C4×D4—42no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4C2311yescharacteristic
62C2×C4⋊C4C2311yescharacteristic
63C2×C4⋊C4C2312yes—
64C2×C22⋊C4—22no—
65C2×C22⋊C4—21no—
66C2×C22⋊C4—22no—
67C2×C22⋊C4—22no—
68C2×C22⋊C4—22no—
69C2×C22⋊C4—22no—
70C2×C22⋊C4—22no—
71C2×C42—21no—
72C2×C42C2311yescharacteristic
73C2×C42C2311yescharacteristic
74C2×C42C2311yescharacteristic
75#103—42no—
76#102—42no—
77#102—42no—
78#102—42no—
79#95—22no—
80#95D412yes—
81#95D412yes—
82C24—21no—
83C24—22no—
84C2×Q8—21no—
85C2×Q8—22no—
86C2×Q8—22no—
87C2×Q8#4112yes—
88C2×D4—21no—
89C2×D4—22no—
90C2×D4—22no—
91C2×D4—22no—
92C2×D4—22no—
93C2×D4—22no—
94C2×D4—22no—
95C2×D4—22no—
96C2×D4—22no—
97C2×D4—44no—
98C2×D4—82no—
99C2×D4—82no—
100C2×D4#4112yes—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—22no—
105C22×C4—42no—
106C22×C4—82no—
107C22×C4C2×D412yes—
108C22×C4C2×D411yescharacteristic
109C22×C4C2×D412yes—
110C22×C4C2×D411yescharacteristic
111C22×C4C2411yescharacteristic
112SD16—48no—
113D8—48no—
114C2×C8—22no—
115C2×C8—22no—
116C2×C8—22no—
117C2×C8—22no—
118C2×C8—22no—
119C2×C8—22no—
120C4⋊C4—22no—
121C4⋊C4—22no—
122C4⋊C4—22no—
123C4⋊C4—22no—
124C4⋊C4—42no—
125C4⋊C4—44no—
126C4⋊C4—82no—
127C22⋊C4—28no—
128C22⋊C4—42no—
129C22⋊C4—82no—
130C22⋊C4—82no—
131C22⋊C4—82no—
132C22⋊C4—82no—
133C22⋊C4—82no—
134C42—21no—
135C42—21no—
136C42—22no—
137C42—22no—
138C42—42no—
139C23—22no—
140C23—41no—
141C23—41no—
142C23—42no—
143C23—42no—
144C23—42no—
145C23—42no—
146C23—42no—
147C23—42no—
148C23—42no—
149C23—42no—
150C23—42no—
151C23C22×D411yescharacteristic
152Q8—24no—
153Q8—42no—
154D4—24no—
155D4—42no—
156D4—42no—
157D4—42no—
158D4—42no—
159D4—42no—
160D4—88no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—84no—
182C2×C4C2×#4111yescharacteristic
183C2×C4C2×#4111yescharacteristic
184C2×C4H(5,𝔽2)12yes—
185C2×C4H(5,𝔽2)12yes—
186C8—42no—
187C8—42no—
188C22—44no—
189C22—42no—
190C22—81no—
191C22—81no—
192C22—82no—
193C22—82no—
194C22—82no—
195C22—82no—
196C22—82no—
197C22—82no—
198C22—82no—
199C22—82no—
200C22#52511yescharacteristic
201C22#53511yescharacteristic
202C22#53511yescharacteristic
203C22#57612yes—
204C22#57712yes—
205C4—42no—
206C4—42no—
207C4—41no—
208C4—41no—
209C4—42no—
210C4—81no—
211C4—81no—
212C4—82no—
213C4#53812yes—
214C4#53812yes—
215C2—82no—
216C2—162no—
217C2#263111yescharacteristic
218C2#311912yes—
219C2#322011yescharacteristic
220C2#322111yescharacteristic
221C2#322412yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2011C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2641C2semidirect
2. #2326C2semidirect
3. #2071C2semidirect
4. #2026C2semidirect
5. #2025C2semidirect
6. #2014C2semidirect
7. #2012C2semidirect
9. #1880C2semidirect
10. #1879C2semidirect
11. #1840C2semidirect

Maximal subgroups

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

subgroupindex
#26412
#23262
#20712
#20262
#20252
#20142
#20122
#20112
#18802
#18792
#18402