Finite Group Databasebeta

← #11213

Subgroups — #11213

Order 256.

341 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#428C211yesmaximal, characteristic
2#2316C211yesmaximal, characteristic
3C2×#344C211yesmaximal, characteristic
4#1740C211yesmaximal, characteristic
5#1736C211yesmaximal, characteristic
6#1733C211yesmaximal, characteristic
7C2×H(3,ℤ/4)C211yesmaximal, characteristic
8C22×M16C2211yescharacteristic
9C22×M16C2211yescharacteristic
10C2×#117C2211yescharacteristic
11C2×#117C2211yescharacteristic
12C2×#117C2211yescharacteristic
13C2×#117C2211yescharacteristic
14C22×C4⋊C4C2211yescharacteristic
15#428—22no—
16#428—22no—
17C2×#107C412yescomplemented
18C2×#106C412yescomplemented
19#389—22no—
20#389—22no—
21#380—22no—
22#344—22no—
23#342—22no—
24#340—22no—
25#339—22no—
26H(3,ℤ/4)—22no—
27C23×C4—21no—
28C23×C4C2311yescharacteristic
29C2×M16—21no—
30C2×M16—21no—
31C2×M16—21no—
32C2×M16—21no—
33C2×M16—21no—
34C2×M16—21no—
35C2×M16—21no—
36C2×M16—21no—
37C2×M16—21no—
38C2×M16D412yes—
39C2×M16Q812yes—
40C2×M16Q812yes—
41C22×C8—21no—
42C22×C8C2×C412yes—
43#117—21no—
44#117—21no—
45#117—21no—
46#117—21no—
47#117—21no—
48#117—21no—
49#117—21no—
50#117—21no—
51#117—21no—
52#117—21no—
53#117—21no—
54#117—21no—
55#117—21no—
56#117—21no—
57#117—21no—
58#117Q812yes—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—24no—
66C2×C4⋊C4C2×C411yescharacteristic
67C2×C4⋊C4C2×C411yescharacteristic
68C2×C4⋊C4C2×C412yes—
69C2×C4⋊C4D412yes—
70C2×C4⋊C4D412yes—
71C2×C22⋊C4—21no—
72C2×C22⋊C4—21no—
73C2×C22⋊C4—21no—
74C2×C22⋊C4D412yes—
75C2×C42—21no—
76C2×C42—21no—
77C2×C42—21no—
78C2×C42D412yes—
79#107—24no—
80#106—24no—
81#95—24no—
82C24C2×D411yescharacteristic
83C22×C4—22no—
84C22×C4—22no—
85C22×C4—22no—
86C22×C4—22no—
87C22×C4—22no—
88C22×C4—21no—
89C22×C4—21no—
90C22×C4—41no—
91C22×C4—82no—
92C22×C4C22×C411yescharacteristic
93C22×C4C2×D411yescharacteristic
94C22×C4C2×D411yescharacteristic
95C22×C4C2×D411yescharacteristic
96C22×C4C2×D411yescharacteristic
97C22×C4C2×Q811yescharacteristic
98C22×C4C2×Q811yescharacteristic
99C22×C4C2×Q811yescharacteristic
100C22×C4#4111yescharacteristic
101C22×C4#4111yescharacteristic
102C22×C4#4111yescharacteristic
103C22×C4#4111yescharacteristic
104C22×C4#4111yescharacteristic
105C22×C4#4111yescharacteristic
106M16—22no—
107M16—22no—
108M16—22no—
109M16—22no—
110M16—22no—
111M16—22no—
112M16—41no—
113M16—41no—
114C2×C8—22no—
115C2×C8—22no—
116C2×C8—22no—
117C2×C8—41no—
118C2×C8—41no—
119C2×C8C4⋊C414yescomplemented
120C4⋊C4—22no—
121C4⋊C4—22no—
122C4⋊C4—22no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—22no—
128C4⋊C4—41no—
129C4⋊C4—41no—
130C4⋊C4—41no—
131C4⋊C4—41no—
132C4⋊C4—41no—
133C4⋊C4—84no—
134C4⋊C4C22⋊C414yes—
135C22⋊C4—22no—
136C22⋊C4—22no—
137C22⋊C4—22no—
138C22⋊C4—41no—
139C22⋊C4—41no—
140C22⋊C4—41no—
141C22⋊C4—41no—
142C22⋊C4—41no—
143C42—22no—
144C42—22no—
145C42—22no—
146C42—41no—
147C42—41no—
148C42—41no—
149C42—41no—
150C42—41no—
151C23—41no—
152C23—41no—
153C23C4×D411yescharacteristic
154C23#12011yescharacteristic
155C23#12111yescharacteristic
156C23#12111yescharacteristic
157C23C22⋊Q811yescharacteristic
158C23#12311yescharacteristic
159C23C4⋊Q811yescharacteristic
160C2×C4—24no—
161C2×C4—44no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—82no—
172C2×C4—82no—
173C2×C4—81no—
174C2×C4—81no—
175C2×C4—81no—
176C2×C4—81no—
177C2×C4C2×C22⋊C411yescharacteristic
178C2×C4C2×C4⋊C411yescharacteristic
179C2×C4#11711yescharacteristic
180C2×C4C4×D411yescharacteristic
181C2×C4C4×Q811yescharacteristic
182C2×C4C4×Q811yescharacteristic
183C2×C4#12111yescharacteristic
184C2×C4#12111yescharacteristic
185C2×C4#12111yescharacteristic
186C2×C4#12111yescharacteristic
187C2×C4C22⋊Q811yescharacteristic
188C2×C4C22⋊Q811yescharacteristic
189C2×C4C22⋊Q811yescharacteristic
190C2×C4C22⋊Q811yescharacteristic
191C2×C4C22⋊Q811yescharacteristic
192C2×C4C22⋊Q811yescharacteristic
193C2×C4C22⋊Q811yescharacteristic
194C2×C4#12311yescharacteristic
195C2×C4#12311yescharacteristic
196C2×C4#12411yescharacteristic
197C2×C4#12511yescharacteristic
198C2×C4#12511yescharacteristic
199C2×C4#12511yescharacteristic
200C2×C4#12511yescharacteristic
201C2×C4#12611yescharacteristic
202C2×C4#12611yescharacteristic
203C2×C4C4⋊Q811yescharacteristic
204C2×C4C4⋊Q811yescharacteristic
205C8—24no—
206C8—42no—
207C22—21no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—21no—
212C22—21no—
213C22—41no—
214C22—41no—
215C22—41no—
216C22—41no—
217C22#38611yescharacteristic
218C22#38711yescharacteristic
219C22#38911yescharacteristic
220C22#39411yescharacteristic
221C22#39511yescharacteristic
222C22#39611yescharacteristic
223C22#39811yescharacteristic
224C4—44no—
225C4—82no—
226C4—82no—
227C4—82no—
228C4—81no—
229C4—81no—
230C4#38011yescharacteristic
231C4#38711yescharacteristic
232C4#38911yescharacteristic
233C4#39111yescharacteristic
234C4#39611yescharacteristic
235C4#39811yescharacteristic
236C4#39811yescharacteristic
237C4#40011yescharacteristic
238C2—21no—
239C2—21no—
240C2—21no—
241C2—21no—
242C2—21no—
243C2—21no—
244C2#144111yescharacteristic
245C2#190411yescharacteristic
246C2#190511yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
17. C2×#107C4semidirect
18. C2×#106C4semidirect
119. C2×C8C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#4282
#23162
C2×#3442
#17402
#17362
#17332
C2×H(3,ℤ/4)2