Finite Group Databasebeta

← #18210

Subgroups — #18210

Order 256.

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

normal subgroups only

This group has 669 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#3496C211yesmaximal, characteristic, complemented
2#3100C212yesmaximal, complemented
3#2535C211yesmaximal, characteristic, complemented
4#1607C214yesmaximal, complemented
5#1595C214yesmaximal, complemented
6#1517C212yesmaximal, complemented
7#1460C211yesmaximal, characteristic, complemented
8C2×H(5,𝔽2)—21no—
9#547—22no—
10#546—24no—
11D42—22no—
12#537—21no—
13#534C2214yes—
14#532C2212yes—
15C2×#121—22no—
16C2×#121—21no—
17C2×C4×D4C2211yescharacteristic
18C2×C4×D4C2211yescharacteristic, complemented
19#480C2214yescomplemented
20#474—22no—
21#459—22no—
22#447C2214yescomplemented
23C4×D8—24no—
24#431C2212yescomplemented
25#397C2212yescomplemented
26#392C2212yescomplemented
27#386—22no—
28#382C2211yescharacteristic, complemented
29#380—21no—
30#327C2218yescomplemented
31#323C2214yescomplemented
32H(5,𝔽2)—44no—
33H(5,𝔽2)—44no—
34C2×#41—24no—
35C2×#41—22no—
36C22×D4—24no—
37C22×D4—24no—
38C22×D4—22no—
39C22×D4—22no—
40C22×D4D411yescharacteristic
41C22×D4C2311yescharacteristic
42C23×C4—21no—
43C23×C4D412yes—
44C2×D8—24no—
45C4⋊Q8D412yescomplemented
46#127D412yescomplemented
47#124—22no—
48#123—28no—
49C22⋊Q8—24no—
50#121—24no—
51#121—24no—
52#121—24no—
53#121—22no—
54#121—24no—
55#121—44no—
56#121C2314yes—
57#120—28no—
58C4×D4—22no—
59C4×D4—22no—
60C4×D4—22no—
61C4×D4—22no—
62C4×D4—22no—
63C4×D4D414yescomplemented
64C2×C4⋊C4—22no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4D411yescharacteristic
67C2×C4⋊C4C2311yescharacteristic
68C2×C22⋊C4—24no—
69C2×C22⋊C4—22no—
70C2×C22⋊C4—21no—
71C2×C22⋊C4—22no—
72C2×C22⋊C4C2312yes—
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—24no—
84C24—21no—
85C24C2×D412yes—
86#41—416no—
87#41—44no—
88#41—44no—
89C2×Q8—22no—
90C2×D4—28no—
91C2×D4—24no—
92C2×D4—24no—
93C2×D4—22no—
94C2×D4—22no—
95C2×D4—22no—
96C2×D4—22no—
97C2×D4—22no—
98C2×D4—42no—
99C2×D4—416no—
100C2×D4—48no—
101C2×D4—48no—
102C2×D4—44no—
103C2×D4—44no—
104C2×D4—84no—
105C2×D4—82no—
106C2×D4D814yescomplemented
107C2×D4C2×D414yes—
108C22×C4—24no—
109C22×C4—22no—
110C22×C4—24no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—21no—
114C22×C4—42no—
115C22×C4—42no—
116C22×C4—42no—
117C22×C4—82no—
118C22×C4C2×D412yes—
119C22×C4C2×D411yescharacteristic
120C22×C4C2×D411yescharacteristic
121C22×C4C2411yescharacteristic
122D8—48no—
123C2×C8—24no—
124C4⋊C4—24no—
125C4⋊C4—22no—
126C4⋊C4—22no—
127C4⋊C4—22no—
128C4⋊C4—42no—
129C4⋊C4C2×D414yes—
130C22⋊C4—28no—
131C22⋊C4—24no—
132C22⋊C4—24no—
133C22⋊C4—24no—
134C22⋊C4—44no—
135C22⋊C4—41no—
136C22⋊C4—41no—
137C22⋊C4—82no—
138C42C2×D412yes—
139C42C2×D412yes—
140C23—22no—
141C23—22no—
142C23—22no—
143C23—24no—
144C23—42no—
145C23—42no—
146C23—41no—
147C23—41no—
148C23—41no—
149C23—48no—
150C23—41no—
151C23—41no—
152C23—41no—
153C23—84no—
154C23—84no—
155C23C2×D814yes—
156C23C22×D411yescharacteristic
157Q8—44no—
158D4—24no—
159D4—28no—
160D4—44no—
161D4—42no—
162D4—44no—
163D4—416no—
164D4—44no—
165D4—44no—
166D4—44no—
167D4—84no—
168D4—88no—
169C2×C4—22no—
170C2×C4—24no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—44no—
174C2×C4—48no—
175C2×C4—48no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—42no—
182C2×C4—41no—
183C2×C4—82no—
184C2×C4—84no—
185C2×C4#12011yescharacteristic
186C2×C4#12011yescharacteristic
187C2×C4#12012yes—
188C2×C4C2×D812yes—
189C2×C4AGL(1,ℤ/8)12yes—
190C2×C4C22×D412yes—
191C8—44no—
192C22—21no—
193C22—21no—
194C22—24no—
195C22—44no—
196C22—42no—
197C22—42no—
198C22—42no—
199C22—42no—
200C22—44no—
201C22—44no—
202C22—42no—
203C22—44no—
204C22—82no—
205C22—81no—
206C22—81no—
207C22—81no—
208C22—81no—
209C22—168no—
210C22C2×#12011yescharacteristic
211C22C22×D811yescharacteristic
212C22C2×AGL(1,ℤ/8)11yescharacteristic
213C4—21no—
214C4—21no—
215C4—22no—
216C4—44no—
217C4—42no—
218C4—82no—
219C4—81no—
220C4#44714yes—
221C2—21no—
222C2—21no—
223C2—44no—
224C2—82no—
225C2—84no—
226C2C2×#44711yescharacteristic
227C2#301911yescharacteristic
228C2#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. #3496C2semidirect
2. #3100C2semidirect
3. #2535C2semidirect
4. #1607C2semidirect
5. #1595C2semidirect
6. #1517C2semidirect
7. #1460C2semidirect
18. C2×C4×D4C22semidirect
19. #480C22semidirect
22. #447C22semidirect
24. #431C22semidirect
25. #397C22semidirect
26. #392C22semidirect
28. #382C22semidirect
30. #327C22semidirect
31. #323C22semidirect
45. C4⋊Q8D4semidirect
46. #127D4semidirect
63. C4×D4D4semidirect
106. C2×D4D8semidirect

Maximal subgroups

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

subgroupindex
#34962
#31002
#25352
#16072
#15952
#15172
#14602