Finite Group Databasebeta

← #9978

Subgroups — #9978

Order 256.

259 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 259 conjugacy classes of subgroups — too many to draw. Only its 51 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
2#3095C211yesmaximal, characteristic, complemented
3#3094C211yesmaximal, characteristic
4#1818C211yesmaximal, characteristic, complemented
5#1472C211yesmaximal, characteristic, complemented
6#1471C211yesmaximal, characteristic
7#1293C211yesmaximal, characteristic, complemented
8C2×C22⋊Q8C2211yescharacteristic
9C2×#121C2211yescharacteristic
10C2×#117C2211yescharacteristic
11#485—22no—
12#484—21no—
13#483—21no—
14#483—21no—
15#482—22no—
16#481—21no—
17#481—21no—
18#480—21no—
19#465—22no—
20#450—21no—
21#448—21no—
22#428—21no—
23#428—21no—
24#419—21no—
25#418—21no—
26#407C2211yescharacteristic
27#407C2211yescharacteristic
28C2×C22⋊C8C2211yescharacteristic
29C2×C22⋊C8C2211yescharacteristic
30#336—22no—
31#324—22no—
32#323—22no—
33C22×Q8D411yescharacteristic
34C22×D4—21no—
35C22×D4D411yescharacteristic
36C23×C4C2311yescharacteristic
37C2×SD16—22no—
38C2×M16—21no—
39C2×M16—21no—
40C22×C8—21no—
41C22×C8D412yes—
42C22⋊Q8—21no—
43C22⋊Q8—21no—
44C22⋊Q8—21no—
45C22⋊Q8—21no—
46#121—21no—
47#121—21no—
48#121—21no—
49#121—21no—
50#117—21no—
51#117—21no—
52#117—21no—
53#117—21no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4D411yescharacteristic
56C2×C4⋊C4D411yescharacteristic
57C2×C4⋊C4D411yescharacteristic
58C2×C4⋊C4D411yescharacteristic
59C2×C22⋊C4—21no—
60C2×C22⋊C4—21no—
61C2×C22⋊C4—21no—
62C2×C42—21no—
63#107—22no—
64#107—22no—
65#106—22no—
66#106—22no—
67#106—22no—
68#103—22no—
69#103—22no—
70#103—22no—
71#102—22no—
72#102—22no—
73#102—22no—
74C22⋊C8—21no—
75C22⋊C8—21no—
76C22⋊C8—21no—
77C22⋊C8—21no—
78C22⋊C8—21no—
79C22⋊C8—21no—
80C22⋊C8D412yes—
81C22⋊C8D412yes—
82C24—21no—
83C24#4111yescharacteristic
84C2×Q8—21no—
85C2×Q8—21no—
86C2×Q8—42no—
87C2×D4—21no—
88C2×D4—21no—
89C2×D4—42no—
90C2×D4—41no—
91C2×D4—41no—
92C2×D4—82no—
93C22×C4—21no—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—42no—
99C22×C4C2×D411yescharacteristic
100C22×C4C2×D411yescharacteristic
101C22×C4C2×D411yescharacteristic
102C22×C4C2×D411yescharacteristic
103C22×C4C2×D411yescharacteristic
104C22×C4C2×D411yescharacteristic
105SD16—82no—
106M16—42no—
107M16—42no—
108C2×C8—22no—
109C2×C8—22no—
110C2×C8—22no—
111C2×C8—42no—
112C2×C8—42no—
113C2×C8—41no—
114C4⋊C4—21no—
115C4⋊C4—21no—
116C4⋊C4—21no—
117C4⋊C4—21no—
118C4⋊C4—21no—
119C4⋊C4—21no—
120C4⋊C4—21no—
121C4⋊C4—21no—
122C4⋊C4—41no—
123C4⋊C4—41no—
124C22⋊C4—41no—
125C22⋊C4—41no—
126C22⋊C4—41no—
127C22⋊C4—41no—
128C22⋊C4—41no—
129C22⋊C4—41no—
130C42—41no—
131C42—41no—
132C23—22no—
133C23—41no—
134C23—41no—
135C23—42no—
136C23—41no—
137C23—81no—
138C23—81no—
139C23#12111yescharacteristic
140C23#12111yescharacteristic
141C23#12111yescharacteristic
142Q8—42no—
143Q8—42no—
144D4—42no—
145D4—42no—
146D4—82no—
147D4—82no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—22no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—42no—
155C2×C4—42no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—42no—
162C2×C4—81no—
163C2×C4—81no—
164C2×C4#12011yescharacteristic
165C2×C4#12011yescharacteristic
166C2×C4#12011yescharacteristic
167C2×C4#12711yescharacteristic
168C8—42no—
169C8—42no—
170C8—42no—
171C8—81no—
172C22—21no—
173C22—21no—
174C22—21no—
175C22—21no—
176C22—21no—
177C22—21no—
178C22—21no—
179C22—21no—
180C22—82no—
181C22—81no—
182C22—81no—
183C22—81no—
184C22—162no—
185C22#39211yescharacteristic
186C22#47112yes—
187C4—21no—
188C4—21no—
189C4—21no—
190C4—81no—
191C4—81no—
192C4—81no—
193C4—81no—
194C4—81no—
195C4#45812yes—
196C2—22no—
197C2—41no—
198C2—41no—
199C2—161no—
200C2#201711yescharacteristic
201C2#201911yescharacteristic
202C2#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. #3095C2semidirect
4. #1818C2semidirect
5. #1472C2semidirect
7. #1293C2semidirect

Maximal subgroups

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

subgroupindex
#31872
#30952
#30942
#18182
#14722
#14712
#12932