Finite Group Databasebeta

← #21174

Subgroups — #21174

Order 256.

495 conjugacy classes of subgroups; only the 247 normal subgroups are shown.

normal subgroups only

This group has 495 conjugacy classes of subgroups — too many to draw. Only its 247 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×#428C211yesmaximal, characteristic
2C22×#106C211yesmaximal, characteristic
3C2×#424C211yesmaximal, characteristic
4#2317C211yesmaximal, characteristic
5#2274C211yesmaximal, characteristic
6#1949C214yesmaximal, complemented
7#1948C214yesmaximal, complemented
8#1736C211yesmaximal, characteristic
9#1735C211yesmaximal, characteristic
10C22×M16C2211yescharacteristic
11C23×C8C2211yescharacteristic
12C2×#117C2211yescharacteristic
13C2×#117C2211yescharacteristic
14C2×#117C2211yescharacteristic
15C22×C4⋊C4C2211yescharacteristic
16C22×C4⋊C4C2211yescharacteristic
17#428—24no—
18C2×#107C2212yes—
19C2×#106—24no—
20C2×#106C2214yes—
21C2×#106C2212yes—
22#424C418yescomplemented
23C2×C4⋊C8C2214yes—
24#389—22no—
25#385—22no—
26#384C2212yes—
27#382C2212yes—
28C4×C4⋊C4—22no—
29C4×C4⋊C4C2214yes—
30C4×C22⋊C4—22no—
31#340C2212yes—
32#340C2214yes—
33#340C2212yes—
34C23×C4—21no—
35C23×C4—21no—
36C23×C4C2311yescharacteristic
37C2×M16—22no—
38C2×M16—22no—
39C2×M16C2×C414yes—
40C22×C8—22no—
41C22×C8—22no—
42C22×C8C2×C414yes—
43C22×C8C2312yes—
44C22×C8C2312yes—
45#117—22no—
46#117—22no—
47#117—22no—
48#117C2×C414yes—
49#117Q814yes—
50#117Q814yes—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—22no—
53C2×C4⋊C4—22no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—24no—
57C2×C4⋊C4D414yes—
58C2×C4⋊C4D412yes—
59C2×C4⋊C4C2312yes—
60C2×C4⋊C4C2311yescharacteristic
61C2×C4⋊C4C2311yescharacteristic
62C2×C4⋊C4C2312yes—
63C2×C4⋊C4C2312yes—
64C2×C22⋊C4—21no—
65C2×C22⋊C4—21no—
66C2×C22⋊C4D412yes—
67C2×C42—21no—
68C2×C42—21no—
69C2×C42—24no—
70C2×C42C2312yes—
71#107—24no—
72#106—28no—
73#106—24no—
74C4⋊C8C2×C4116yescomplemented
75#95—24no—
76#95—24no—
77C24C2×D411yescharacteristic
78C22×C4—22no—
79C22×C4—22no—
80C22×C4—24no—
81C22×C4—21no—
82C22×C4—22no—
83C22×C4—21no—
84C22×C4—42no—
85C22×C4—82no—
86C22×C4—82no—
87C22×C4C22×C412yes—
88C22×C4C2×D414yes—
89C22×C4C2×D411yescharacteristic
90C22×C4C2×D412yes—
91C22×C4C2×Q812yes—
92C22×C4C2×Q812yes—
93C22×C4#4112yes—
94C22×C4#4112yes—
95C22×C4C2411yescharacteristic
96M16—28no—
97C2×C8—28no—
98C2×C8—22no—
99C2×C8—22no—
100C2×C8—22no—
101C2×C8—22no—
102C2×C8C22×C414yes—
103C2×C8C22×C414yes—
104C4⋊C4—24no—
105C4⋊C4—22no—
106C4⋊C4—22no—
107C4⋊C4—22no—
108C4⋊C4—21no—
109C4⋊C4—21no—
110C4⋊C4—22no—
111C4⋊C4—21no—
112C4⋊C4—21no—
113C4⋊C4—42no—
114C4⋊C4—84no—
115C4⋊C4C4⋊C418yes—
116C4⋊C4C2×D414yes—
117C4⋊C4C2×Q814yes—
118C22⋊C4—24no—
119C22⋊C4—42no—
120C22⋊C4C4⋊C418yes—
121C42—24no—
122C42—42no—
123C42—84no—
124C42C22×C414yes—
125C42C2×Q814yes—
126C23—22no—
127C23—22no—
128C23C2×C4⋊C412yes—
129C23#12112yes—
130C23C22⋊Q812yes—
131C23C22×D411yescharacteristic
132C2×C4—24no—
133C2×C4—24no—
134C2×C4—24no—
135C2×C4—24no—
136C2×C4—42no—
137C2×C4—44no—
138C2×C4—42no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—42no—
142C2×C4—42no—
143C2×C4—44no—
144C2×C4—42no—
145C2×C4—84no—
146C2×C4—84no—
147C2×C4C2×C4⋊C418yes—
148C2×C4C2×C4⋊C412yes—
149C2×C4C4×D414yes—
150C2×C4C4×Q814yes—
151C2×C4#12112yes—
152C2×C4C22⋊Q812yes—
153C2×C4#12514yes—
154C2×C4C4⋊Q814yes—
155C2×C4C23×C411yescharacteristic
156C2×C4C22×D411yescharacteristic
157C2×C4C22×Q811yescharacteristic
158C2×C4C22×Q811yescharacteristic
159C2×C4C2×#4111yescharacteristic
160C2×C4C2×#4111yescharacteristic
161C8—24no—
162C8—24no—
163C22—24no—
164C22—24no—
165C22—22no—
166C22—22no—
167C22#38914yes—
168C22C22×C4⋊C411yescharacteristic
169C22C2×#12111yescharacteristic
170C22C2×C22⋊Q811yescharacteristic
171C22#57712yes—
172C22#57712yes—
173C4—28no—
174C4—44no—
175C4—44no—
176C4—81no—
177C4—81no—
178C4—82no—
179C4#38914yes—
180C4C2×C4×D411yescharacteristic
181C4C2×C4×Q811yescharacteristic
182C4C2×#12511yescharacteristic
183C4C2×C4⋊Q811yescharacteristic
184C2—24no—
185C2C2×#38911yescharacteristic
186C2#291212yes—
187C2#304211yescharacteristic
188C2#304311yescharacteristic
189C2#308412yes—
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C2×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
6. #1949C2semidirect
7. #1948C2semidirect
22. #424C4semidirect
74. C4⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#4282
C22×#1062
C2×#4242
#23172
#22742
#19492
#19482
#17362
#17352