Finite Group Databasebeta

← #18218

Subgroups — #18218

Order 256.

611 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 611 conjugacy classes of subgroups — too many to draw. Only its 127 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#3497C211yesmaximal, characteristic, complemented
2#3100C212yesmaximal, complemented
3#2536C211yesmaximal, characteristic, complemented
4#1610C214yesmaximal, complemented
5#1598C214yesmaximal, complemented
6#1517C212yesmaximal, complemented
7#1461C211yesmaximal, characteristic, complemented
8C2×2−1+4—21no—
9#550—22no—
10D4×Q8—22no—
11#548—24no—
12#537—21no—
13#537C2214yes—
14#532—22no—
15#532C2212yes—
16C2×C22⋊Q8—21no—
17C2×C4×Q8C2211yescharacteristic
18C2×C4×D4C2211yescharacteristic
19#480C2214yes—
20#474—22no—
21#459—22no—
22#447C2214yes—
23C4×D8—24no—
24#431C2212yes—
25#398C2212yes—
26#394C2212yes—
27#388—22no—
28#382C2211yescharacteristic
29#380—21no—
30#327C2218yes—
31#324C2214yes—
322−1+4—44no—
332−1+4—44no—
34C2×#41—24no—
35C2×#41—24no—
36C2×#41—22no—
37C22×Q8—24no—
38C22×Q8D411yescharacteristic
39C22×D4C2311yescharacteristic
40C23×C4—21no—
41C2×D8—24no—
42C4⋊Q8—22no—
43C4⋊Q8D412yes—
44#127—22no—
45#127D412yes—
46#124—24no—
47#124—22no—
48C22⋊Q8—24no—
49C22⋊Q8—28no—
50C22⋊Q8—44no—
51#121—24no—
52#121—28no—
53#121—22no—
54#121—24no—
55#121C2314yes—
56C4×Q8—22no—
57C4×Q8—22no—
58C4×D4—24no—
59C4×D4—24no—
60C4×D4—22no—
61C4×D4—22no—
62C4×D4—22no—
63C4×D4D414yes—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4D411yescharacteristic
67C2×C4⋊C4C2311yescharacteristic
68C2×C4⋊C4C2312yes—
69C2×C22⋊C4—21no—
70C2×C22⋊C4—22no—
71C2×C22⋊C4—22no—
72C2×C42D412yes—
73C2×C42C2311yescharacteristic
74#107—24no—
75C4⋊C8—22no—
76#102—24no—
77#102—24no—
78C22⋊C8C2314yes—
79C4×C8—22no—
80#95—22no—
81#95—21no—
82#95C2312yes—
83C24—21no—
84#41—48no—
85#41—416no—
86#41—48no—
87#41—44no—
88#41—44no—
89C2×Q8—24no—
90C2×Q8—22no—
91C2×Q8—22no—
92C2×Q8—22no—
93C2×Q8—416no—
94C2×Q8D814yescomplemented
95C2×D4—24no—
96C2×D4—24no—
97C2×D4—24no—
98C2×D4—22no—
99C2×D4—22no—
100C2×D4—22no—
101C2×D4—42no—
102C2×D4C2×D414yes—
103C22×C4—28no—
104C22×C4—24no—
105C22×C4—22no—
106C22×C4—21no—
107C22×C4—42no—
108C22×C4—42no—
109C22×C4—42no—
110C22×C4C2×D412yes—
111C22×C4C2×D412yes—
112C22×C4C2×D411yescharacteristic
113C22×C4C2×D411yescharacteristic
114C22×C4C2411yescharacteristic
115D8—48no—
116C2×C8—24no—
117C4⋊C4—24no—
118C4⋊C4—22no—
119C4⋊C4—22no—
120C4⋊C4—24no—
121C4⋊C4—22no—
122C4⋊C4—42no—
123C4⋊C4—44no—
124C4⋊C4C2×D414yes—
125C22⋊C4—28no—
126C22⋊C4—24no—
127C22⋊C4—24no—
128C22⋊C4—44no—
129C22⋊C4—41no—
130C22⋊C4—41no—
131C22⋊C4—44no—
132C42—22no—
133C42—22no—
134C42C2×D412yes—
135C42C2×D412yes—
136C23—22no—
137C23—24no—
138C23—41no—
139C23—41no—
140C23—41no—
141C23—41no—
142C23—41no—
143C23—41no—
144C23C22×D411yescharacteristic
145Q8—28no—
146Q8—48no—
147Q8—44no—
148D4—24no—
149D4—44no—
150D4—44no—
151D4—42no—
152D4—48no—
153D4—44no—
154D4—44no—
155D4—44no—
156D4—84no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—24no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—416no—
164C2×C4—48no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—41no—
169C2×C4—42no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—84no—
173C2×C4#12011yescharacteristic
174C2×C4#12011yescharacteristic
175C2×C4#12012yes—
176C2×C4C2×D814yes—
177C2×C4C2×D812yes—
178C2×C4AGL(1,ℤ/8)12yes—
179C2×C4C22×D412yes—
180C8—44no—
181C22—21no—
182C22—21no—
183C22—42no—
184C22—42no—
185C22—44no—
186C22—44no—
187C22—42no—
188C22—44no—
189C22—81no—
190C22—81no—
191C22—81no—
192C22—81no—
193C22C2×#12011yescharacteristic
194C22C22×D811yescharacteristic
195C22C2×AGL(1,ℤ/8)11yescharacteristic
196C4—24no—
197C4—21no—
198C4—21no—
199C4—22no—
200C4—44no—
201C4—42no—
202C4—82no—
203C4—81no—
204C4#44714yes—
205C2—21no—
206C2—21no—
207C2—84no—
208C2—82no—
209C2C2×#44711yescharacteristic
210C2#302011yescharacteristic
211C2#302511yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3497C2semidirect
2. #3100C2semidirect
3. #2536C2semidirect
4. #1610C2semidirect
5. #1598C2semidirect
6. #1517C2semidirect
7. #1461C2semidirect
94. C2×Q8D8semidirect

Maximal subgroups

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

subgroupindex
#34972
#31002
#25362
#16102
#15982
#15172
#14612