Finite Group Databasebeta

← #18292

Subgroups — #18292

Order 256.

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

normal subgroups only

This group has 641 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
1C2×#537C211yesmaximal, characteristic, complemented
2#3186C212yesmaximal, complemented
3#2618C211yesmaximal, characteristic, complemented
4#1610C214yesmaximal, complemented
5#1598C214yesmaximal, complemented
6#1519C212yesmaximal, complemented
7C2×#324C211yesmaximal, characteristic, complemented
8C22×#41—21no—
9#537—24no—
10#537—22no—
11#537C2214yes—
12C2×#123—21no—
13C2×C22⋊Q8—22no—
14C2×#121—22no—
15C2×#121—21no—
16C2×#121C2212yes—
17C2×#121C2211yescharacteristic
18C22×C4⋊C4C2211yescharacteristic
19#480—22no—
20#480C2214yescomplemented
21#466—24no—
22#447—22no—
23#447C2214yescomplemented
24C2×C22⋊C8C2212yes—
25#398C2212yescomplemented
26#396—22no—
27#394C2212yescomplemented
28#386—21no—
29#380C2211yescharacteristic
30#327C2218yescomplemented
31#324C2214yescomplemented
32C2×#41—24no—
33C2×#41—24no—
34C2×#41—44no—
35C2×#41—44no—
36C22×Q8—21no—
37C22×D4—21no—
38C22×D4—22no—
39C22×D4—21no—
40C22×D4—21no—
41C22×D4D412yes—
42C22×D4C2311yescharacteristic
43C23×C4—22no—
44C23×C4D411yescharacteristic
45C23×C4D411yescharacteristic
46C23×C4C2311yescharacteristic
47C2×D8—24no—
48C22×C8—22no—
49#123—44no—
50C22⋊Q8—28no—
51C22⋊Q8—28no—
52#121—24no—
53#121—28no—
54#121—28no—
55#121—24no—
56#121—22no—
57#121—22no—
58#121—22no—
59#121—22no—
60#121D414yes—
61#121C2314yes—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4D412yes—
68C2×C4⋊C4D412yes—
69C2×C4⋊C4C2311yescharacteristic
70C2×C4⋊C4C2312yes—
71C2×C22⋊C4—22no—
72C2×C22⋊C4—22no—
73C2×C22⋊C4—21no—
74C2×C22⋊C4—21no—
75C2×C22⋊C4—22no—
76C2×C22⋊C4—22no—
77#107—24no—
78#102—24no—
79#102—24no—
80C22⋊C8—22no—
81C22⋊C8C2314yes—
82#95—22no—
83#95C2312yes—
84C24—22no—
85C24—21no—
86C24C2×D411yescharacteristic
87#41—48no—
88#41—416no—
89#41—48no—
90C2×Q8—24no—
91C2×Q8—44no—
92C2×D4—24no—
93C2×D4—24no—
94C2×D4—24no—
95C2×D4—24no—
96C2×D4—22no—
97C2×D4—22no—
98C2×D4—44no—
99C2×D4—44no—
100C2×D4—44no—
101C2×D4—41no—
102C2×D4—41no—
103C2×D4—42no—
104C2×D4—42no—
105C2×D4—81no—
106C2×D4—82no—
107C2×D4—82no—
108C2×D4C2×D414yes—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—22no—
112C22×C4—28no—
113C22×C4—21no—
114C22×C4—22no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—22no—
118C22×C4—48no—
119C22×C4—81no—
120C22×C4—81no—
121C22×C4D814yescomplemented
122C22×C4C2×D412yes—
123C22×C4C2×D412yes—
124C22×C4C2×D411yescharacteristic
125C22×C4C2×D412yes—
126C22×C4C2×D412yes—
127C22×C4C2411yescharacteristic
128D8—84no—
129C2×C8—24no—
130C2×C8—82no—
131C4⋊C4—24no—
132C4⋊C4—24no—
133C4⋊C4—22no—
134C4⋊C4—24no—
135C4⋊C4—22no—
136C4⋊C4—44no—
137C4⋊C4—81no—
138C4⋊C4C2×D414yes—
139C22⋊C4—28no—
140C22⋊C4—24no—
141C22⋊C4—24no—
142C22⋊C4—42no—
143C22⋊C4—42no—
144C22⋊C4—44no—
145C22⋊C4—82no—
146C23—24no—
147C23—41no—
148C23—41no—
149C23—44no—
150C23—44no—
151C23—42no—
152C23—42no—
153C23—44no—
154C23—41no—
155C23—41no—
156C23—41no—
157C23#12011yescharacteristic
158C23#12011yescharacteristic
159C23C2×D812yes—
160C23AGL(1,ℤ/8)12yes—
161C23C22×D411yescharacteristic
162Q8—48no—
163D4—44no—
164D4—48no—
165D4—44no—
166D4—44no—
167D4—44no—
168D4—48no—
169D4—82no—
170D4—84no—
171D4—82no—
172D4—82no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—24no—
176C2×C4—24no—
177C2×C4—22no—
178C2×C4—24no—
179C2×C4—41no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—41no—
183C2×C4—416no—
184C2×C4—42no—
185C2×C4—44no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—41no—
190C2×C4—42no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—81no—
194C2×C4#12012yes—
195C2×C4C2×D814yes—
196C2×C4C22×D412yes—
197C8—44no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—21no—
204C22—42no—
205C22—42no—
206C22—44no—
207C22—44no—
208C22—44no—
209C22—88no—
210C22—82no—
211C22—82no—
212C22—81no—
213C22—81no—
214C22—81no—
215C22#44712yes—
216C22#44712yes—
217C22C2×#12011yescharacteristic
218C22C22×D811yescharacteristic
219C22C2×AGL(1,ℤ/8)11yescharacteristic
220C4—24no—
221C4—22no—
222C4—22no—
223C4—44no—
224C4—42no—
225C4—82no—
226C4—81no—
227C2—21no—
228C2—22no—
229C2—21no—
230C2—22no—
231C2—84no—
232C2—161no—
233C2C2×#44711yescharacteristic
234C2#302011yescharacteristic
235C2#302611yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#537C2semidirect
2. #3186C2semidirect
3. #2618C2semidirect
4. #1610C2semidirect
5. #1598C2semidirect
6. #1519C2semidirect
7. C2×#324C2semidirect
20. #480C22semidirect
23. #447C22semidirect
25. #398C22semidirect
27. #394C22semidirect
30. #327C22semidirect
31. #324C22semidirect
121. C22×C4D8semidirect

Maximal subgroups

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

subgroupindex
C2×#5372
#31862
#26182
#16102
#15982
#15192
C2×#3242