Finite Group Databasebeta

← #9960

Subgroups — #9960

Order 256.

325 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 325 conjugacy classes of subgroups — too many to draw. Only its 63 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#3187C211yesmaximal, characteristic
2C2×#483C211yesmaximal, characteristic, complemented
3#2501C211yesmaximal, characteristic
4#1996C211yesmaximal, characteristic, complemented
5#1823C211yesmaximal, characteristic, complemented
6#1519C211yesmaximal, characteristic
7#1282C211yesmaximal, characteristic, complemented
8C2×#123—22no—
9C2×C22⋊Q8C2211yescharacteristic
10C2×#121C2211yescharacteristic
11C22×C4⋊C4C2211yescharacteristic
12#483—22no—
13#483—21no—
14#483—22no—
15#481—21no—
16#465—22no—
17#450—21no—
18#448—21no—
19C2×#106—21no—
20C2×#103—21no—
21C2×C22⋊C8C2211yescharacteristic
22C2×C22⋊C8C2211yescharacteristic
23#399—22no—
24#390—21no—
25#384—21no—
26#380C2211yescharacteristic
27C4×C22⋊C4C2211yescharacteristic
28#342—22no—
29#340—22no—
30#327—22no—
31C22×Q8D411yescharacteristic
32C22×D4—21no—
33C22×D4D411yescharacteristic
34C23×C4—21no—
35C23×C4C2311yescharacteristic
36C2×SD16—22no—
37C22×C8—21no—
38C22×C8D412yes—
39#123—44no—
40#123—44no—
41C22⋊Q8—21no—
42C22⋊Q8—21no—
43C22⋊Q8—21no—
44C22⋊Q8—21no—
45#121—21no—
46#121—21no—
47#121—21no—
48#121—21no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4—21no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—24no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4D412yes—
58C2×C4⋊C4D411yescharacteristic
59C2×C4⋊C4D411yescharacteristic
60C2×C22⋊C4—24no—
61C2×C22⋊C4—21no—
62C2×C22⋊C4—21no—
63C2×C22⋊C4—21no—
64C2×C22⋊C4D412yes—
65C2×C42—21no—
66#106—24no—
67#106—22no—
68#103—24no—
69#103—22no—
70#102—22no—
71C22⋊C8—21no—
72C22⋊C8—21no—
73C22⋊C8—21no—
74C22⋊C8D412yes—
75#95—21no—
76#95—21no—
77C24—21no—
78C24C2×D411yescharacteristic
79C2×Q8—21no—
80C2×Q8—21no—
81C2×Q8—42no—
82C2×D4—21no—
83C2×D4—21no—
84C2×D4—42no—
85C2×D4—41no—
86C2×D4—41no—
87C2×D4—42no—
88C2×D4—42no—
89C22×C4—22no—
90C22×C4—22no—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—42no—
95C22×C4—41no—
96C22×C4—41no—
97C22×C4—42no—
98C22×C4—42no—
99C22×C4—41no—
100C22×C4C2×D411yescharacteristic
101C22×C4C2×D411yescharacteristic
102C22×C4C2×D411yescharacteristic
103C22×C4C2×D411yescharacteristic
104C22×C4C2×D411yescharacteristic
105C22×C4#4111yescharacteristic
106SD16—82no—
107C2×C8—22no—
108C2×C8—22no—
109C2×C8—42no—
110C2×C8—42no—
111C4⋊C4—21no—
112C4⋊C4—22no—
113C4⋊C4—21no—
114C4⋊C4—21no—
115C4⋊C4—22no—
116C4⋊C4—21no—
117C4⋊C4—48no—
118C4⋊C4—42no—
119C4⋊C4—41no—
120C4⋊C4—41no—
121C4⋊C4—41no—
122C4⋊C4—41no—
123C22⋊C4—22no—
124C22⋊C4—48no—
125C22⋊C4—41no—
126C22⋊C4—41no—
127C22⋊C4—41no—
128C22⋊C4—41no—
129C22⋊C4—42no—
130C22⋊C4SD1614yescomplemented
131C42—42no—
132C23—41no—
133C23—41no—
134C23—42no—
135C23—42no—
136C23—41no—
137C23—42no—
138C23#12011yescharacteristic
139C23#12111yescharacteristic
140C23#12711yescharacteristic
141C23C2×SD1612yes—
142C23AGL(1,ℤ/8)12yes—
143Q8—42no—
144Q8—42no—
145D4—42no—
146D4—81no—
147D4—82no—
148D4—81no—
149D4—81no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—24no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—42no—
158C2×C4—42no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—42no—
163C2×C4—44no—
164C2×C4—42no—
165C2×C4—82no—
166C2×C4—81no—
167C2×C4—81no—
168C2×C4—81no—
169C2×C4—81no—
170C2×C4—81no—
171C2×C4—81no—
172C2×C4—81no—
173C2×C4#12011yescharacteristic
174C2×C4#12011yescharacteristic
175C2×C4#12111yescharacteristic
176C2×C4#12111yescharacteristic
177C8—42no—
178C8—42no—
179C22—21no—
180C22—21no—
181C22—21no—
182C22—21no—
183C22—21no—
184C22—21no—
185C22—42no—
186C22—42no—
187C22—84no—
188C22—81no—
189C22—81no—
190C22—81no—
191C22#39211yescharacteristic
192C22#45012yes—
193C22#46012yes—
194C22#49211yescharacteristic
195C22#49611yescharacteristic
196C4—21no—
197C4—21no—
198C4—21no—
199C4—44no—
200C4—81no—
201C4—82no—
202C4—81no—
203C4—81no—
204C4—161no—
205C4#45812yes—
206C2—22no—
207C2—21no—
208C2—22no—
209C2—21no—
210C2—161no—
211C2#200911yescharacteristic
212C2#201411yescharacteristic
213C2#202211yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC23×C4C23
prime radicalC2×#121C22
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#483C2semidirect
4. #1996C2semidirect
5. #1823C2semidirect
7. #1282C2semidirect
130. C22⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
#31872
C2×#4832
#25012
#19962
#18232
#15192
#12822