Finite Group Databasebeta

← #33119

Subgroups — #33119

Order 256.

415 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 415 conjugacy classes of subgroups — too many to draw. Only its 131 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
1C2×#484C211yesmaximal, characteristic
2C2×#483C211yesmaximal, characteristic
3#2621C211yesmaximal, characteristic
4#2274C211yesmaximal, characteristic
5#1981C212yesmaximal, complemented
6#1949C212yesmaximal, complemented
7#1928C212yesmaximal, complemented
8#1927C212yesmaximal, complemented
9#1873C211yesmaximal, characteristic
10#1873C211yesmaximal, characteristic
11#1798C211yesmaximal, characteristic
12C2×C4⋊Q8C2211yescharacteristic
13C2×C22⋊Q8C2211yescharacteristic
14C2×C22⋊Q8C2211yescharacteristic
15C2×#117C2211yescharacteristic
16C2×#117C2211yescharacteristic
17C22×C4⋊C4C2211yescharacteristic
18#484—24no—
19#483—24no—
20C2×#107C2212yes—
21C2×#106C2212yes—
22C2×C4⋊C8C2212yes—
23C2×#103C2212yes—
24C2×#103C2212yes—
25C2×#103C2212yes—
26C2×#103C2212yes—
27C2×C22⋊C8C2211yescharacteristic
28C2×C22⋊C8C2211yescharacteristic
29#398—22no—
30#397—22no—
31#391C2211yescharacteristic
32#391C2211yescharacteristic
33#382C2211yescharacteristic
34C4×C4⋊C4—22no—
35C4×C4⋊C4C2212yes—
36C4×C4⋊C4C2212yes—
37C4×C22⋊C4—22no—
38#340C2212yes—
39#340C2212yes—
40#336C2212yes—
41C22×Q8C2311yescharacteristic
42C22×Q8C2311yescharacteristic
43C23×C4—21no—
44C23×C4C2311yescharacteristic
45C22×C8C2312yes—
46C22×C8C2312yes—
47C4⋊Q8—24no—
48C22⋊Q8—22no—
49C22⋊Q8—22no—
50C22⋊Q8—22no—
51C22⋊Q8—22no—
52#117—24no—
53#117—24no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4D412yes—
60C2×C4⋊C4D414yes—
61C2×C4⋊C4C2311yescharacteristic
62C2×C4⋊C4C2311yescharacteristic
63C2×C4⋊C4C2312yes—
64C2×C4⋊C4C2312yes—
65C2×C22⋊C4—21no—
66C2×C22⋊C4—21no—
67C2×C22⋊C4—21no—
68C2×C22⋊C4D412yes—
69C2×C42—21no—
70C2×C42—22no—
71C2×C42—22no—
72C2×C42C2311yescharacteristic
73C2×C42C2311yescharacteristic
74#107—24no—
75#106—24no—
76C4⋊C8—24no—
77#103—24no—
78#103—24no—
79#103—24no—
80#103—24no—
81C22⋊C8—22no—
82C22⋊C8—22no—
83#95—22no—
84#95—22no—
85#95—22no—
86#95—22no—
87#95—22no—
88C24C2×D411yescharacteristic
89C2×Q8—21no—
90C2×Q8—21no—
91C2×Q8—22no—
92C2×Q8—22no—
93C2×Q8—21no—
94C2×Q8—21no—
95C2×Q8—22no—
96C2×Q8—22no—
97C22×C4—22no—
98C22×C4—22no—
99C22×C4—22no—
100C22×C4—22no—
101C22×C4—24no—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—82no—
107C22×C4C2×D412yes—
108C22×C4C2×D411yescharacteristic
109C22×C4C2×D412yes—
110C22×C4C2×D412yes—
111C22×C4#4112yes—
112C22×C4C2411yescharacteristic
113C2×C8—22no—
114C2×C8—22no—
115C2×C8—22no—
116C2×C8—22no—
117C2×C8—22no—
118C2×C8—22no—
119C4⋊C4—24no—
120C4⋊C4—24no—
121C4⋊C4—22no—
122C4⋊C4—22no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—42no—
128C4⋊C4—42no—
129C4⋊C4—44no—
130C4⋊C4C2×D414yes—
131C4⋊C4#4114yes—
132C22⋊C4—24no—
133C22⋊C4—24no—
134C22⋊C4—42no—
135C22⋊C4—42no—
136C42—24no—
137C42—22no—
138C42—82no—
139C42—82no—
140C42#4114yes—
141C23—21no—
142C23—21no—
143C23—22no—
144C23—22no—
145C23#12112yes—
146C23C22×D411yescharacteristic
147Q8—42no—
148Q8—42no—
149Q8—42no—
150Q8—42no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—24no—
156C2×C4—24no—
157C2×C4—24no—
158C2×C4—22no—
159C2×C4—24no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—84no—
177C2×C4#12112yes—
178C2×C4#12314yes—
179C2×C4C22×D411yescharacteristic
180C2×C4C2×#4111yescharacteristic
181C2×C4C2×#4112yes—
182C2×C4C2×#4112yes—
183C8—42no—
184C8—42no—
185C22—22no—
186C22—22no—
187C22—22no—
188C22—42no—
189C22—42no—
190C22C2×#12111yescharacteristic
191C22C2×#12311yescharacteristic
192C22#52511yescharacteristic
193C22#57712yes—
194C22#57812yes—
195C4—24no—
196C4—22no—
197C4—42no—
198C4—44no—
199C4—44no—
200C4—81no—
201C4—81no—
202C4—81no—
203C4—81no—
204C4#52912yes—
205C4#53212yes—
206C2—42no—
207C2#285411yescharacteristic
208C2#304211yescharacteristic
209C2#304411yescharacteristic
210C2#309712yes—
211C2#312012yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #1981C2semidirect
6. #1949C2semidirect
7. #1928C2semidirect
8. #1927C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4842
C2×#4832
#26212
#22742
#19812
#19492
#19282
#19272
#18732
#17982