Finite Group Databasebeta

← #10328

Subgroups — #10328

Order 256.

349 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 349 conjugacy classes of subgroups — too many to draw. Only its 53 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#3192C211yesmaximal, characteristic
2#2504C211yesmaximal, characteristic, complemented
3#2501C211yesmaximal, characteristic, complemented
4#1821C211yesmaximal, characteristic, complemented
5#1507C211yesmaximal, characteristic, complemented
6#1507C211yesmaximal, characteristic, complemented
7#1297C211yesmaximal, characteristic, complemented
8C2×#123—22no—
9C2×#121C2211yescharacteristic
10C2×#121C2211yescharacteristic
11C22×C4⋊C4C2211yescharacteristic
12#481—21no—
13#481—21no—
14#469—22no—
15#447—21no—
16#447—21no—
17#407C2211yescharacteristic
18#407C2211yescharacteristic
19#399—22no—
20#397—22no—
21#390—21no—
22#390—21no—
23#386—22no—
24#384—21no—
25#382—21no—
26C4×C22⋊C4C2211yescharacteristic
27C4×C22⋊C4C2211yescharacteristic
28#343—22no—
29#327—22no—
30#327—22no—
31C22×D4—21no—
32C22×D4—21no—
33C22×D4D411yescharacteristic
34C22×D4D411yescharacteristic
35C23×C4D412yes—
36C23×C4C2311yescharacteristic
37C2×D8—22no—
38C2×M16—21no—
39C2×M16D412yes—
40#123—48no—
41#121—21no—
42#121—21no—
43#121—21no—
44#121—21no—
45#121—21no—
46#121—21no—
47#121—21no—
48#121—21no—
49C2×C4⋊C4—24no—
50C2×C4⋊C4—21no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4D411yescharacteristic
57C2×C4⋊C4D411yescharacteristic
58C2×C22⋊C4—21no—
59C2×C22⋊C4—24no—
60C2×C22⋊C4—21no—
61C2×C22⋊C4—21no—
62C2×C22⋊C4—21no—
63C2×C22⋊C4—24no—
64C2×C42—21no—
65C2×C42—21no—
66#106—22no—
67#102—22no—
68#102—22no—
69C22⋊C8—21no—
70C22⋊C8—21no—
71C22⋊C8—21no—
72C22⋊C8—21no—
73#95—24no—
74#95D412yes—
75#95D412yes—
76C24—21no—
77C24—21no—
78C24C2×D411yescharacteristic
79C2×D4—21no—
80C2×D4—21no—
81C2×D4—21no—
82C2×D4—21no—
83C2×D4—42no—
84C2×D4—42no—
85C2×D4—44no—
86C2×D4—44no—
87C2×D4—41no—
88C2×D4—41no—
89C2×D4—41no—
90C2×D4—41no—
91C22×C4—22no—
92C22×C4—21no—
93C22×C4—22no—
94C22×C4—22no—
95C22×C4—22no—
96C22×C4—21no—
97C22×C4—22no—
98C22×C4—42no—
99C22×C4—44no—
100C22×C4C2×D411yescharacteristic
101C22×C4C2×D411yescharacteristic
102C22×C4C2×D411yescharacteristic
103C22×C4C2×D411yescharacteristic
104C22×C4C2×D411yescharacteristic
105C22×C4#4111yescharacteristic
106D8—82no—
107M16—42no—
108M16—42no—
109C2×C8—22no—
110C2×C8—41no—
111C4⋊C4—21no—
112C4⋊C4—21no—
113C4⋊C4—21no—
114C4⋊C4—21no—
115C4⋊C4—48no—
116C4⋊C4—41no—
117C4⋊C4—41no—
118C4⋊C4—41no—
119C4⋊C4—41no—
120C22⋊C4—24no—
121C22⋊C4—48no—
122C22⋊C4—42no—
123C22⋊C4—41no—
124C22⋊C4—41no—
125C22⋊C4—41no—
126C22⋊C4—41no—
127C22⋊C4—84no—
128C42—42no—
129C42—42no—
130C23—22no—
131C23—22no—
132C23—41no—
133C23—42no—
134C23—42no—
135C23—42no—
136C23—42no—
137C23—41no—
138C23—41no—
139C23—42no—
140C23—42no—
141C23#12011yescharacteristic
142C23#12011yescharacteristic
143C23#12111yescharacteristic
144D4—42no—
145D4—42no—
146D4—81no—
147D4—81no—
148D4—82no—
149D4—82no—
150D4—82no—
151D4—82no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—24no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—44no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—44no—
171C2×C4—82no—
172C2×C4#12011yescharacteristic
173C2×C4#12111yescharacteristic
174C2×C4#12111yescharacteristic
175C2×C4#12711yescharacteristic
176C8—42no—
177C8—81no—
178C22—21no—
179C22—21no—
180C22—21no—
181C22—21no—
182C22—21no—
183C22—21no—
184C22—21no—
185C22—44no—
186C22—84no—
187C22—84no—
188C22—81no—
189C22—81no—
190C22—81no—
191C22—81no—
192C22—81no—
193C22—81no—
194C22#39211yescharacteristic
195C22#45312yes—
196C22#45412yes—
197C4—21no—
198C4—21no—
199C4—21no—
200C4—44no—
201C4—82no—
202C4—82no—
203C4—81no—
204C4—81no—
205C4#45812yes—
206C2—22no—
207C2—22no—
208C2—41no—
209C2—161no—
210C2—161no—
211C2#200211yescharacteristic
212C2#200311yescharacteristic
213C2#202211yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC23×C4C23
prime radical#3192C2
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. #2504C2semidirect
3. #2501C2semidirect
4. #1821C2semidirect
5. #1507C2semidirect
6. #1507C2semidirect
7. #1297C2semidirect

Maximal subgroups

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

subgroupindex
#31922
#25042
#25012
#18212
#15072
#12972