Finite Group Databasebeta

← #30050

Subgroups — #30050

Order 256.

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

normal subgroups only

This group has 376 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#2640C211yesmaximal, characteristic
2#2327C211yesmaximal, characteristic, complemented
3#2074C211yesmaximal, characteristic, complemented
4#2073C211yesmaximal, characteristic
5#2024C212yesmaximal, complemented
6#2016C211yesmaximal, characteristic, complemented
7#2015C212yesmaximal
8#2013C211yesmaximal, characteristic
9#1880C212yesmaximal
10#1878C212yesmaximal, complemented
11#1840C211yesmaximal, characteristic, complemented
12C22×Q16C2212yes—
13C22×SD16C2212yes—
14C2×C4⋊Q8C2211yescharacteristic
15C2×C4⋊Q8C2211yescharacteristic
16C2×#124C2211yescharacteristic
17C2×C4×Q8C2212yes—
18C2×#103C2212yes—
19C2×#103C2212yes—
20C2×#103C2212yes—
21C2×#102C2212yes—
22#397—22no—
23#393—22no—
24#391C2212yes—
25#391C2211yescharacteristic
26#391C2211yescharacteristic
27#390C2211yescharacteristic
28#390C2211yescharacteristic
29#387—22no—
30#387—22no—
31#382C2211yescharacteristic
32#376C2211yescharacteristic
33#340C2212yes—
34#340C2212yes—
35#336C2211yescharacteristic
36#336C2211yescharacteristic
37#336C2212yes—
38#336C2212yes—
39C22×Q8—21no—
40C22×Q8C2311yescharacteristic
41C22×Q8C2312yes—
42C22×D4C2311yescharacteristic
43C2×Q16—24no—
44C2×Q16—42no—
45C2×Q16—42no—
46C2×SD16—24no—
47C2×SD16—42no—
48C2×SD16—42no—
49C22×C8C2312yes—
50C22×C8C2312yes—
51C4⋊Q8—24no—
52C4⋊Q8—42no—
53#124—42no—
54C4×Q8—42no—
55C4×Q8—42no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4C2312yes—
60C2×C4⋊C4C2311yescharacteristic
61C2×C4⋊C4C2311yescharacteristic
62C2×C22⋊C4—21no—
63C2×C22⋊C4—21no—
64C2×C22⋊C4—22no—
65C2×C22⋊C4—22no—
66C2×C42—22no—
67C2×C42C2311yescharacteristic
68C2×C42C2311yescharacteristic
69C2×C42C2311yescharacteristic
70#103—42no—
71#103—42no—
72#103—42no—
73#102—42no—
74#95—22no—
75#95—22no—
76#95—22no—
77#95—22no—
78#95—22no—
79#95D412yes—
80#95D412yes—
81C24—21no—
82C2×Q8—22no—
83C2×Q8—22no—
84C2×Q8—22no—
85C2×Q8—22no—
86C2×Q8—22no—
87C2×Q8—22no—
88C2×Q8—21no—
89C2×Q8—22no—
90C2×Q8—21no—
91C2×Q8—42no—
92C2×Q8—82no—
93C2×Q8#4114yes—
94C2×D4—22no—
95C2×D4—21no—
96C2×D4—22no—
97C2×D4—21no—
98C22×C4—22no—
99C22×C4—22no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4C2×D412yes—
104C22×C4C2×D411yescharacteristic
105C22×C4C2×D411yescharacteristic
106C22×C4C2×D412yes—
107C22×C4C2411yescharacteristic
108Q16—48no—
109SD16—48no—
110C2×C8—22no—
111C2×C8—22no—
112C2×C8—22no—
113C2×C8—22no—
114C2×C8—22no—
115C2×C8—22no—
116C4⋊C4—28no—
117C4⋊C4—22no—
118C4⋊C4—22no—
119C4⋊C4—22no—
120C4⋊C4—22no—
121C4⋊C4—44no—
122C4⋊C4—42no—
123C22⋊C4—42no—
124C22⋊C4—42no—
125C22⋊C4—82no—
126C22⋊C4—82no—
127C42—22no—
128C42—21no—
129C42—21no—
130C42—22no—
131C42—44no—
132C23—41no—
133C23—41no—
134C23—41no—
135C23—42no—
136C23—42no—
137C23C22×D411yescharacteristic
138Q8—24no—
139Q8—24no—
140Q8—42no—
141Q8—42no—
142Q8—42no—
143Q8—42no—
144Q8—84no—
145D4—42no—
146D4—42no—
147C2×C4—22no—
148C2×C4—24no—
149C2×C4—22no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—41no—
161C2×C4—42no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4C2×#4112yes—
171C2×C4H(5,𝔽2)12yes—
172C2×C4H(5,𝔽2)11yescharacteristic
173C2×C4H(5,𝔽2)11yescharacteristic
174C8—42no—
175C8—42no—
176C22—82no—
177C22—81no—
178C22—81no—
179C22—81no—
180C22—82no—
181C22#52511yescharacteristic
182C22#53511yescharacteristic
183C22#53511yescharacteristic
184C22#57712yes—
185C22#57812yes—
186C4—42no—
187C4—44no—
188C4—41no—
189C4—41no—
190C4—42no—
191C4—81no—
192C4—81no—
193C4—81no—
194C4—82no—
195C4#53812yes—
196C4#53812yes—
197C2—161no—
198C2#263111yescharacteristic
199C2#312112yes—
200C2#322211yescharacteristic
201C2#322311yescharacteristic
202C2#322512yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
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
2. #2327C2semidirect
3. #2074C2semidirect
5. #2024C2semidirect
6. #2016C2semidirect
10. #1878C2semidirect
11. #1840C2semidirect

Maximal subgroups

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

subgroupindex
#26402
#23272
#20742
#20732
#20242
#20162
#20152
#20132
#18802
#18782
#18402