Finite Group Databasebeta

← #9065

Subgroups — #9065

Order 256.

267 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 267 conjugacy classes of subgroups — too many to draw. Only its 87 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×#487C211yesmaximal, characteristic
2#2365C211yesmaximal, characteristic
3#1987C211yesmaximal, characteristic
4#1987C211yesmaximal, characteristic
5C2×#332C211yesmaximal, characteristic
6#1279C211yesmaximal, characteristic
7#1279C211yesmaximal, characteristic
8C2×C4⋊Q8C2211yescharacteristic
9C2×C4⋊Q8C2211yescharacteristic
10C2×C4×Q8—21no—
11#487—22no—
12#487C414yescomplemented
13C2×C4⋊C8C2211yescharacteristic
14C2×C4⋊C8C2211yescharacteristic
15C2×#103—22no—
16C2×C4×C8C2211yescharacteristic
17#391—21no—
18#389—21no—
19#387—22no—
20C4×C4⋊C4—21no—
21C4×C4⋊C4C2211yescharacteristic
22C4×C4⋊C4C2211yescharacteristic
23#340—21no—
24#340—21no—
25#340—21no—
26#340—21no—
27#332—22no—
28C22×Q8—21no—
29C22×Q8—21no—
30C22×C8—21no—
31C22×C8—21no—
32C22×C8—21no—
33C4⋊Q8—21no—
34C4⋊Q8—21no—
35C4⋊Q8—21no—
36C4⋊Q8—21no—
37C4⋊Q8—21no—
38C4⋊Q8C2×C412yes—
39C4⋊Q8C2×C412yes—
40C4⋊Q8D412yes—
41C4×Q8—42no—
42C4×Q8—42no—
43C2×C4⋊C4—21no—
44C2×C4⋊C4—21no—
45C2×C4⋊C4—21no—
46C2×C4⋊C4—22no—
47C2×C4⋊C4D411yescharacteristic
48C2×C4⋊C4D411yescharacteristic
49C2×C4⋊C4D412yes—
50C2×C4⋊C4D411yescharacteristic
51C2×C4⋊C4D411yescharacteristic
52C2×C42—22no—
53C2×C42—21no—
54C2×C42—21no—
55C2×C42C2311yescharacteristic
56C4⋊C8—21no—
57C4⋊C8—21no—
58C4⋊C8—21no—
59C4⋊C8—21no—
60#103—28no—
61C4×C8—21no—
62C4×C8C2×C412yes—
63#95—22no—
64#95—22no—
65#95—21no—
66#95—21no—
67C2×Q8—22no—
68C2×Q8—22no—
69C2×Q8—22no—
70C2×Q8—42no—
71C2×Q8—42no—
72C2×Q8—41no—
73C2×Q8—41no—
74C2×Q8—41no—
75C2×Q8—41no—
76C2×Q8—41no—
77C22×C4—21no—
78C22×C4—22no—
79C22×C4—21no—
80C22×C4—21no—
81C22×C4—21no—
82C22×C4—41no—
83C22×C4C2×D411yescharacteristic
84C22×C4C2×D411yescharacteristic
85C22×C4C2×D411yescharacteristic
86C2×C8—22no—
87C2×C8—41no—
88C2×C8—41no—
89C2×C8—41no—
90C2×C8—41no—
91C2×C8—41no—
92C2×C8—41no—
93C2×C8—41no—
94C2×C8—41no—
95C4⋊C4—22no—
96C4⋊C4—22no—
97C4⋊C4—22no—
98C4⋊C4—21no—
99C4⋊C4—21no—
100C4⋊C4—22no—
101C4⋊C4—22no—
102C4⋊C4—21no—
103C4⋊C4—21no—
104C4⋊C4—21no—
105C4⋊C4—21no—
106C4⋊C4—44no—
107C4⋊C4—41no—
108C4⋊C4—41no—
109C4⋊C4C22⋊C414yes—
110C4⋊C4SD1612yes—
111C4⋊C4Q1612yes—
112C42—44no—
113C42—41no—
114C42—41no—
115C42—41no—
116C42—41no—
117C42C22×C411yescharacteristic
118C42C2×D411yescharacteristic
119C42#4111yescharacteristic
120C42#4111yescharacteristic
121C23#12011yescharacteristic
122Q8—44no—
123Q8—42no—
124Q8—42no—
125Q8—42no—
126Q8—42no—
127Q8—82no—
128C2×C4—21no—
129C2×C4—21no—
130C2×C4—24no—
131C2×C4—22no—
132C2×C4—22no—
133C2×C4—41no—
134C2×C4—42no—
135C2×C4—41no—
136C2×C4—42no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—42no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—81no—
147C2×C4—81no—
148C2×C4C2×C22⋊C411yescharacteristic
149C2×C4C4×D411yescharacteristic
150C2×C4C4×D411yescharacteristic
151C2×C4#12111yescharacteristic
152C2×C4#12111yescharacteristic
153C2×C4#12311yescharacteristic
154C2×C4C2×SD1611yescharacteristic
155C2×C4C2×Q1611yescharacteristic
156C2×C4#13511yescharacteristic
157C2×C4#13511yescharacteristic
158C2×C4AGL(1,ℤ/8)11yescharacteristic
159C2×C4#13711yescharacteristic
160C2×C4#13711yescharacteristic
161C2×C4#13711yescharacteristic
162C8—42no—
163C8—81no—
164C8—81no—
165C22#38611yescharacteristic
166C22#45011yescharacteristic
167C22#45111yescharacteristic
168C22#45211yescharacteristic
169C22#45211yescharacteristic
170C22#45411yescharacteristic
171C22#45411yescharacteristic
172C4—21no—
173C4—21no—
174C4—44no—
175C4—42no—
176C4—42no—
177C4—82no—
178C4—81no—
179C4—81no—
180C4C4×SD1611yescharacteristic
181C4C4×Q1611yescharacteristic
182C4#44011yescharacteristic
183C4#44111yescharacteristic
184C4#46111yescharacteristic
185C4#46211yescharacteristic
186C4#46311yescharacteristic
187C4#46411yescharacteristic
188C2#163511yescharacteristic
189C2#163611yescharacteristic
190C2#163711yescharacteristic
191C2#163811yescharacteristic
192C2#187711yescharacteristic
193C2#187811yescharacteristic
194C2#188811yescharacteristic
subgroupG/H
center, socle, prime radicalC23#120
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
12. #487C4semidirect

Maximal subgroups

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

subgroupindex
C2×#4872
#23652
#19872
C2×#3322
#12792