Finite Group Databasebeta

← #12795

Subgroups — #12795

Order 256.

300 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 65 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#2321C211yesmaximal, characteristic
2C2×AΣL(1,𝔽16)C212yesmaximal
3#1800C211yesmaximal, characteristic, complemented
4#1799C211yesmaximal, characteristic, complemented
5#1390C212yesmaximal, complemented
6C2×#120C2211yescharacteristic
7C2×C4×D4C2211yescharacteristic
8C22×C22⋊C4C411yescharacteristic, complemented
9#423—21no—
10C2×#100C2212yes—
11C2×#99C2212yes—
12#390—21no—
13#388—21no—
14#388—21no—
15#386—21no—
16#386C2211yescharacteristic
17#385C411yescharacteristic, complemented
18#380—21no—
19#379C411yescharacteristic, complemented
20C4×C22⋊C4C411yescharacteristic, complemented
21AΣL(1,𝔽16)—24no—
22#343—21no—
23#342—21no—
24C25C2×C411yescharacteristic
25C22×D4—21no—
26C22×D4C2311yescharacteristic
27C23×C4C2×C411yescharacteristic
28C23×C4C2×C411yescharacteristic
29C23×C4D411yescharacteristic
30C2×M16—22no—
31#120—21no—
32#120—21no—
33#120D412yes—
34#120Q812yes—
35C4×D4—41no—
36C4×D4—41no—
37C2×C4⋊C4—21no—
38C2×C4⋊C4—41no—
39C2×C4⋊C4D411yescharacteristic
40C2×C22⋊C4—21no—
41C2×C22⋊C4—21no—
42C2×C22⋊C4—21no—
43C2×C22⋊C4—22no—
44C2×C22⋊C4—21no—
45C2×C22⋊C4—21no—
46C2×C22⋊C4—21no—
47C2×C22⋊C4—41no—
48C2×C22⋊C4—41no—
49C2×C22⋊C4C2×C411yescharacteristic
50C2×C22⋊C4C2×C411yescharacteristic
51C2×C22⋊C4C2×C411yescharacteristic
52C2×C22⋊C4D411yescharacteristic
53C2×C42—21no—
54C2×C42D411yescharacteristic
55C4⋊C8—41no—
56C4⋊C8—41no—
57#100—22no—
58#100—22no—
59#99—22no—
60#99—22no—
61#95—21no—
62#95—21no—
63#95—21no—
64#95—41no—
65C24—21no—
66C24—21no—
67C24—41no—
68C24—41no—
69C24—81no—
70C24—81no—
71C24C4⋊C412yescomplemented
72C24C22×C411yescharacteristic
73C24C2×D411yescharacteristic
74C2×D4—22no—
75C2×D4—22no—
76C2×D4—41no—
77C2×D4—41no—
78C2×D4—81no—
79C2×D4—81no—
80C2×D4C2×D411yescharacteristic
81C2×D4C2×Q811yescharacteristic
82C2×D4#4111yescharacteristic
83C2×D4#4111yescharacteristic
84C22×C4—21no—
85C22×C4—21no—
86C22×C4—21no—
87C22×C4—21no—
88C22×C4—21no—
89C22×C4—21no—
90C22×C4—21no—
91C22×C4—41no—
92C22×C4—42no—
93C22×C4—81no—
94C22×C4—81no—
95C22×C4—81no—
96C22×C4C22⋊C411yescharacteristic
97C22×C4C22⋊C411yescharacteristic
98C22×C4C22⋊C411yescharacteristic
99C22×C4C22⋊C411yescharacteristic
100C22×C4C2×D411yescharacteristic
101M16—42no—
102M16—42no—
103C2×C8—42no—
104C4⋊C4—41no—
105C4⋊C4—81no—
106C4⋊C4—82no—
107C22⋊C4—21no—
108C22⋊C4—22no—
109C22⋊C4—22no—
110C22⋊C4—41no—
111C22⋊C4—41no—
112C22⋊C4—42no—
113C22⋊C4—41no—
114C22⋊C4—41no—
115C22⋊C4—41no—
116C22⋊C4—42no—
117C22⋊C4—42no—
118C22⋊C4—81no—
119C22⋊C4—81no—
120C22⋊C4—81no—
121C22⋊C4C4⋊C412yescomplemented
122C42—21no—
123C42—21no—
124C42—81no—
125C23—21no—
126C23—21no—
127C23—21no—
128C23—21no—
129C23—22no—
130C23—41no—
131C23—41no—
132C23—41no—
133C23—41no—
134C23—41no—
135C23—41no—
136C23—41no—
137C23—41no—
138C23—42no—
139C23—42no—
140C23—41no—
141C23—41no—
142C23—82no—
143C23—82no—
144C23—82no—
145C23—82no—
146C23—84no—
147C23C2×C22⋊C411yescharacteristic
148C23C2×C4⋊C411yescharacteristic
149C23#11711yescharacteristic
150C23#12111yescharacteristic
151C23C22⋊Q811yescharacteristic
152D4—41no—
153D4—41no—
154D4—41no—
155D4—41no—
156D4—82no—
157D4—82no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—22no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—41no—
165C2×C4—42no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—81no—
178C2×C4—81no—
179C2×C4—81no—
180C2×C4—82no—
181C2×C4—82no—
182C2×C4—82no—
183C2×C4—82no—
184C2×C4#9911yescharacteristic
185C2×C4#9911yescharacteristic
186C2×C4#12111yescharacteristic
187C2×C4C22⋊Q811yescharacteristic
188C8—82no—
189C22—21no—
190C22—21no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—21no—
195C22—41no—
196C22—41no—
197C22—41no—
198C22—41no—
199C22—42no—
200C22—42no—
201C22—42no—
202C22—41no—
203C22—41no—
204C22—41no—
205C22—41no—
206C22—42no—
207C22—81no—
208C22—81no—
209C22—81no—
210C22—84no—
211C22—84no—
212C22—81no—
213C22—81no—
214C22#38011yescharacteristic
215C22C2×#9911yescharacteristic
216C22#41011yescharacteristic
217C4—21no—
218C4—21no—
219C4—21no—
220C4—21no—
221C4—42no—
222C4—41no—
223C4—41no—
224C4—41no—
225C4—81no—
226C4—81no—
227C4—162no—
228C2—21no—
229C2—21no—
230C2—41no—
231C2—41no—
232C2—41no—
233C2—41no—
234C2—82no—
235C2#179911yescharacteristic
236C2#212111yescharacteristic
237C2#212211yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC24C22×C4
Frattini, cosocleC22×D4C23
prime radicalC2×#120C22
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #1800C2semidirect
4. #1799C2semidirect
5. #1390C2semidirect
8. C22×C22⋊C4C4semidirect
17. #385C4semidirect
19. #379C4semidirect
20. C4×C22⋊C4C4semidirect
71. C24C4⋊C4semidirect
121. C22⋊C4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#23212
C2×AΣL(1,𝔽16)2
#18002
#17992
#13902