Finite Group Databasebeta

← #10419

Subgroups — #10419

Order 256.

244 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 75 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#2546C211yesmaximal, characteristic
2#1990C211yesmaximal, characteristic
3#1947C211yesmaximal, characteristic
4#1803C211yesmaximal, characteristic
5C2×#332C211yesmaximal, characteristic
6C2×Q8⋊C8C211yesmaximal, characteristic
7#1279C211yesmaximal, characteristic
8C2×C4⋊Q8—21no—
9C2×C4⋊Q8C2211yescharacteristic
10C2×C4×Q8C2211yescharacteristic
11C2×#107—21no—
12C2×C4⋊C8C2211yescharacteristic
13C2×C4⋊C8C2211yescharacteristic
14C2×#103—21no—
15C2×C4×C8C2211yescharacteristic
16#398—22no—
17#391—21no—
18#389—21no—
19#389—21no—
20#384C2211yescharacteristic
21C4×C4⋊C4C2211yescharacteristic
22#340—21no—
23#340—21no—
24#340—22no—
25#332—22no—
26Q8⋊C8—22no—
27C22×Q8—21no—
28C22×Q8D411yescharacteristic
29C22×C8—21no—
30C22×C8—21no—
31C22×C8—21no—
32C4⋊Q8—21no—
33C4⋊Q8—21no—
34C4⋊Q8—21no—
35C4⋊Q8—21no—
36C4⋊Q8—42no—
37C4⋊Q8—42no—
38C4×Q8—21no—
39C4×Q8—21no—
40C4×Q8—21no—
41C4×Q8—21no—
42C2×C4⋊C4—22no—
43C2×C4⋊C4—22no—
44C2×C4⋊C4—21no—
45C2×C4⋊C4—21no—
46C2×C4⋊C4—21no—
47C2×C4⋊C4—21no—
48C2×C4⋊C4—21no—
49C2×C4⋊C4D411yescharacteristic
50C2×C4⋊C4D412yes—
51C2×C4⋊C4D411yescharacteristic
52C2×C4⋊C4D411yescharacteristic
53C2×C42—21no—
54C2×C42—21no—
55C2×C42C2311yescharacteristic
56#107—42no—
57C4⋊C8—21no—
58C4⋊C8—21no—
59C4⋊C8—21no—
60C4⋊C8D412yes—
61#103—42no—
62C4×C8—21no—
63C4×C8—21no—
64#95—22no—
65#95—22no—
66#95—21no—
67C2×Q8—21no—
68C2×Q8—21no—
69C2×Q8—21no—
70C2×Q8—21no—
71C2×Q8—21no—
72C2×Q8—41no—
73C2×Q8—41no—
74C2×Q8—41no—
75C2×Q8—41no—
76C2×Q8—41no—
77C2×Q8—41no—
78C2×Q8D812yes—
79C22×C4—21no—
80C22×C4—21no—
81C22×C4—21no—
82C22×C4—21no—
83C22×C4—22no—
84C22×C4—41no—
85C22×C4C2×D411yescharacteristic
86C22×C4C2×D411yescharacteristic
87C22×C4C2×D411yescharacteristic
88C2×C8—22no—
89C2×C8—22no—
90C2×C8—41no—
91C2×C8—41no—
92C2×C8—41no—
93C2×C8—41no—
94C2×C8—41no—
95C2×C8—41no—
96C2×C8—41no—
97C4⋊C4—21no—
98C4⋊C4—21no—
99C4⋊C4—21no—
100C4⋊C4—22no—
101C4⋊C4—21no—
102C4⋊C4—22no—
103C4⋊C4—44no—
104C4⋊C4—44no—
105C4⋊C4—42no—
106C4⋊C4—41no—
107C4⋊C4—41no—
108C4⋊C4—41no—
109C4⋊C4—41no—
110C4⋊C4—41no—
111C4⋊C4—41no—
112C4⋊C4—41no—
113C4⋊C4—41no—
114C4⋊C4SD1612yes—
115C4⋊C4Q1612yes—
116C42—41no—
117C42—41no—
118C42—41no—
119C42—41no—
120C42C2×D411yescharacteristic
121C42#4111yescharacteristic
122C42#4111yescharacteristic
123C42#4111yescharacteristic
124C23#12011yescharacteristic
125Q8—22no—
126Q8—22no—
127Q8—41no—
128Q8—41no—
129Q8—81no—
130Q8—81no—
131Q8—81no—
132Q8—81no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—22no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—42no—
146C2×C4—42no—
147C2×C4—42no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—81no—
151C2×C4—81no—
152C2×C4—81no—
153C2×C4#12111yescharacteristic
154C2×C4#12111yescharacteristic
155C2×C4#12111yescharacteristic
156C2×C4#12311yescharacteristic
157C2×C4#12311yescharacteristic
158C2×C4#12411yescharacteristic
159C2×C4C2×D811yescharacteristic
160C2×C4C2×SD1611yescharacteristic
161C2×C4C2×Q1611yescharacteristic
162C2×C4#13511yescharacteristic
163C2×C4AGL(1,ℤ/8)11yescharacteristic
164C2×C4#13711yescharacteristic
165C2×C4#13711yescharacteristic
166C2×C4#13711yescharacteristic
167C8—42no—
168C8—42no—
169C8—81no—
170C22#39411yescharacteristic
171C22#44711yescharacteristic
172C22#44811yescharacteristic
173C22#45111yescharacteristic
174C22#45211yescharacteristic
175C22#45411yescharacteristic
176C22#45611yescharacteristic
177C4—21no—
178C4—21no—
179C4—42no—
180C4—42no—
181C4—42no—
182C4—82no—
183C4—81no—
184C4—161no—
185C4#46111yescharacteristic
186C4#46211yescharacteristic
187C4#46611yescharacteristic
188C4#46811yescharacteristic
189C4#48011yescharacteristic
190C4#48211yescharacteristic
191C4#48711yescharacteristic
192C4#49111yescharacteristic
193C2#162711yescharacteristic
194C2#163611yescharacteristic
195C2#164911yescharacteristic
196C2#165311yescharacteristic
197C2#203311yescharacteristic
198C2#203811yescharacteristic
199C2#204311yescharacteristic
subgroupG/H
center, socle, prime radicalC23#120
commutator, Frattini, cosocleC2×C42C23
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#25462
#19902
#19472
#18032
C2×#3322
C2×Q8⋊C82
#12792