Finite Group Databasebeta

← #32923

Subgroups — #32923

Order 256.

667 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 667 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C4×2−1+4C211yesmaximal, characteristic
2#3039C212yesmaximal, complemented
3C2×#456C212yesmaximal
4#2546C211yesmaximal, characteristic, complemented
5#1886C212yesmaximal, complemented
6#1882C214yesmaximal, complemented
7#1806C212yesmaximal, complemented
8#1801C211yesmaximal, characteristic, complemented
9C2×2−1+4C2211yescharacteristic
10C2×#137C2214yes—
11C2×C4⋊Q8C2212yes—
12C2×C4⋊Q8C2211yescharacteristic
13#520—22no—
14#520—22no—
15#520C2212yes—
16#519—22no—
17#519C2211yescharacteristic
18C4×#41—22no—
19C4×#41—22no—
20C4×#41C2212yes—
21C2×C4×Q8—22no—
22C2×C4×Q8C2211yescharacteristic
23#462—22no—
24#462—22no—
25#461—22no—
26#461—22no—
27#456—28no—
28#423C2212yes—
29C2×#104C2214yes—
30#419C2214yes—
31C2×#101C2212yes—
32#398—22no—
33#391C2212yes—
34#389—22no—
35C4×C4⋊C4C2211yescharacteristic
36#343C2212yes—
37#339C2214yes—
382−1+4—24no—
392−1+4—22no—
402−1+4D414yes—
41C2×#41—22no—
42C2×#41—22no—
43C2×#41C2312yes—
44C22×Q8—22no—
45C22×Q8C2312yes—
46C22×Q8C2311yescharacteristic
47#137—28no—
48#137—28no—
49C2×Q16—24no—
50C2×SD16—24no—
51C2×M16C2314yes—
52C4⋊Q8—24no—
53C4⋊Q8—24no—
54C4⋊Q8—22no—
55C4⋊Q8—22no—
56C4×Q8—22no—
57C4×Q8—24no—
58C4×Q8—22no—
59C4×Q8—22no—
60C4×Q8—22no—
61C4×Q8—22no—
62C4×Q8—22no—
63C4×Q8—22no—
64C4×Q8D412yes—
65C4×Q8D412yes—
66C4×D4—22no—
67C4×D4—24no—
68C4×D4—22no—
69C4×D4—22no—
70C4×D4—22no—
71C4×D4—24no—
72C4×D4—22no—
73C4×D4D412yes—
74C4×D4D412yes—
75#117—24no—
76#117—24no—
77#117—22no—
78#117—22no—
79#117—22no—
80#117C2312yes—
81C2×C4⋊C4—24no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—24no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4D412yes—
88C2×C4⋊C4C2311yescharacteristic
89C2×C42—24no—
90C2×C42—22no—
91C2×C42D412yes—
92C2×C42C2312yes—
93C2×C42C2311yescharacteristic
94C4⋊C8—22no—
95C4⋊C8—22no—
96#104—28no—
97#103—24no—
98#103—24no—
99#101—24no—
100#95—21no—
101#95—24no—
102#41—28no—
103#41—28no—
104#41—28no—
105#41—28no—
106#41—22no—
107#41—22no—
108#41C2×D414yes—
109#41#4114yes—
110C2×Q8—22no—
111C2×Q8—28no—
112C2×Q8—28no—
113C2×Q8—22no—
114C2×Q8—22no—
115C2×Q8—24no—
116C2×Q8—22no—
117C2×Q8—22no—
118C2×Q8—24no—
119C2×Q8C2×D412yes—
120C2×Q8C2×D412yes—
121C2×Q8#4112yes—
122C2×D4—22no—
123C2×D4—22no—
124C2×D4C2×D412yes—
125C22×C4—24no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—24no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4C2×D411yescharacteristic
133C22×C4C2×D412yes—
134C22×C4C2×D412yes—
135C22×C4C2×D411yescharacteristic
136C22×C4C2411yescharacteristic
137Q16—48no—
138SD16—48no—
139M16—24no—
140M16—24no—
141C2×C8—24no—
142C4⋊C4—24no—
143C4⋊C4—24no—
144C4⋊C4—22no—
145C4⋊C4—22no—
146C4⋊C4—22no—
147C4⋊C4—24no—
148C4⋊C4—22no—
149C4⋊C4—22no—
150C4⋊C4—41no—
151C4⋊C4—42no—
152C4⋊C4—41no—
153C4⋊C4—48no—
154C4⋊C4C2×D412yes—
155C4⋊C4C2×D412yes—
156C22⋊C4—24no—
157C22⋊C4—22no—
158C22⋊C4—22no—
159C22⋊C4—22no—
160C42—24no—
161C42—24no—
162C42—22no—
163C42—22no—
164C42—22no—
165C42—22no—
166C42—22no—
167C42—22no—
168C42—22no—
169C42C2×D412yes—
170C42C2×D412yes—
171C23—22no—
172C23C22×D411yescharacteristic
173Q8—28no—
174Q8—24no—
175Q8—22no—
176Q8—24no—
177Q8—44no—
178Q8—44no—
179Q8#12114yes—
180D4—28no—
181D4—24no—
182D4—24no—
183D4—22no—
184D4#12114yes—
185C2×C4—216no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—24no—
190C2×C4—24no—
191C2×C4—22no—
192C2×C4—24no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—44no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—82no—
205C2×C4—84no—
206C2×C4#12011yescharacteristic
207C2×C4#12012yes—
208C2×C4#12011yescharacteristic
209C2×C4#12114yes—
210C2×C4C22×D411yescharacteristic
211C2×C4C22×D412yes—
212C2×C4C2×#4112yes—
213C2×C4C2×#4111yescharacteristic
214C8—44no—
215C22—21no—
216C22—21no—
217C22—24no—
218C22—42no—
219C22—42no—
220C22C2×#12011yescharacteristic
221C22C2×#12111yescharacteristic
222C22#52511yescharacteristic
223C4—24no—
224C4—24no—
225C4—21no—
226C4—22no—
227C4—44no—
228C4—42no—
229C4—42no—
230C4—42no—
231C4—82no—
232C4#45612yescomplemented
233C4C2×#12112yes—
234C4#52512yes—
235C2—21no—
236C2—21no—
237C2—44no—
238C2#284711yescharacteristic
239C2C2×#45611yescharacteristic
240C2#302211yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×2−1+4C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3039C2semidirect
4. #2546C2semidirect
5. #1886C2semidirect
6. #1882C2semidirect
7. #1806C2semidirect
8. #1801C2semidirect
232. C4#456semidirect

Maximal subgroups

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

subgroupindex
C4×2−1+42
#30392
C2×#4562
#25462
#18862
#18822
#18062
#18012