Finite Group Databasebeta

← #25623

Subgroups — #25623

Order 256.

529 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 529 conjugacy classes of subgroups — too many to draw. Only its 125 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#3507C211yesmaximal, characteristic
2#3152C211yesmaximal, characteristic
3#3139C211yesmaximal, characteristic, complemented
4#2546C211yesmaximal, characteristic, complemented
5#2005C214yesmaximal, complemented
6#1986C211yesmaximal, characteristic, complemented
7#1980C211yesmaximal, characteristic
8#1882C214yesmaximal, complemented
9C4×#101C211yesmaximal, characteristic, complemented
10C2×#137C2214yes—
11#559—21no—
12#557—21no—
13#556—21no—
14#556—22no—
15#554—21no—
16#552—22no—
17#539—22no—
18#536C2212yes—
19#533C2211yescharacteristic
20#532C2211yescharacteristic
21C2×C4⋊Q8C2212yes—
22C2×C4⋊Q8C2211yescharacteristic
23#529—22no—
24#520C2212yes—
25C2×C4×Q8C2211yescharacteristic
26#497—22no—
27#494—21no—
28#492—21no—
29#491—22no—
30#487—21no—
31#487—21no—
32C2×#104C2214yes—
33#419C2214yes—
34C2×#101C2212yes—
35C4×M16C2211yescharacteristic
36C4×M16C2211yescharacteristic
37#398—22no—
38#391C2212yes—
39#389—22no—
40C4×C4⋊C4C2211yescharacteristic
41#343C2212yes—
42#339C2214yes—
43C2×#41C2312yes—
44C22×Q8C2312yes—
45C22×Q8C2311yescharacteristic
46#137—28no—
47#137—44no—
48C2×Q16—24no—
49C2×SD16—24no—
50C2×M16C2312yes—
51C2×M16C2312yes—
52C4⋊Q8—22no—
53C4⋊Q8—21no—
54C4⋊Q8—21no—
55C4⋊Q8—21no—
56C4⋊Q8—21no—
57C4⋊Q8—44no—
58C4⋊Q8D411yescharacteristic
59C4⋊Q8D411yescharacteristic
60C4⋊Q8D411yescharacteristic
61#127D411yescharacteristic
62#126—24no—
63#126—24no—
64#125—22no—
65#125—24no—
66#125D412yes—
67#124—22no—
68#124—22no—
69#124D412yes—
70#123—24no—
71#123—24no—
72C22⋊Q8—24no—
73C22⋊Q8—21no—
74C22⋊Q8—21no—
75C22⋊Q8—22no—
76C22⋊Q8—22no—
77#121—21no—
78#121—21no—
79C4×Q8—22no—
80C4×Q8—22no—
81C4×Q8—21no—
82C4×Q8—21no—
83C4×Q8—21no—
84C4×Q8—21no—
85C4×Q8—21no—
86C4×Q8—21no—
87C4×D4—22no—
88C4×D4—22no—
89C4×D4—24no—
90C4×D4—21no—
91C4×D4—21no—
92#117—24no—
93#117—22no—
94#117C2312yes—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—24no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4D412yes—
101C2×C4⋊C4C2311yescharacteristic
102C2×C42D412yes—
103C2×C42C2312yes—
104C2×C42C2311yescharacteristic
105#104—44no—
106#103—24no—
107#103—24no—
108#101—24no—
109#97—21no—
110#97—21no—
111C4×C8—21no—
112C4×C8—21no—
113#95—21no—
114#95—24no—
115#41—24no—
116#41—24no—
117C2×Q8—21no—
118C2×Q8—21no—
119C2×Q8—21no—
120C2×Q8—21no—
121C2×Q8—22no—
122C2×Q8—22no—
123C2×Q8—22no—
124C2×Q8—22no—
125C2×Q8—24no—
126C2×Q8—24no—
127C2×Q8C2×D412yes—
128C2×D4—22no—
129C2×D4—21no—
130C2×D4—21no—
131C2×D4C2×D412yes—
132C22×C4—22no—
133C22×C4—24no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4C2×D412yes—
137C22×C4C2×D411yescharacteristic
138C22×C4C2×D412yes—
139C22×C4C2×D411yescharacteristic
140C22×C4C2411yescharacteristic
141Q16—48no—
142SD16—48no—
143M16—22no—
144M16—22no—
145M16—22no—
146M16#4114yes—
147C2×C8—22no—
148C2×C8—22no—
149C4⋊C4—24no—
150C4⋊C4—24no—
151C4⋊C4—24no—
152C4⋊C4—24no—
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—22no—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—22no—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—21no—
163C4⋊C4—48no—
164C4⋊C4—41no—
165C4⋊C4—41no—
166C4⋊C4—41no—
167C4⋊C4—41no—
168C4⋊C4C2×D412yes—
169C4⋊C4C2×D412yes—
170C22⋊C4—24no—
171C22⋊C4—24no—
172C22⋊C4—24no—
173C22⋊C4—22no—
174C22⋊C4—22no—
175C22⋊C4—22no—
176C42—24no—
177C42—22no—
178C42—22no—
179C42—22no—
180C42—24no—
181C42C2×D411yescharacteristic
182C42C2×D411yescharacteristic
183C42C2×D411yescharacteristic
184C42C2×D411yescharacteristic
185C23—22no—
186C23C22×D411yescharacteristic
187Q8—24no—
188Q8—42no—
189Q8—42no—
190Q8—42no—
191Q8—44no—
192Q8—44no—
193Q8—81no—
194D4—24no—
195D4—42no—
196D4—42no—
197D4—42no—
198D4—81no—
199C2×C4—22no—
200C2×C4—24no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—41no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—44no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—84no—
217C2×C4#12011yescharacteristic
218C2×C4#12011yescharacteristic
219C2×C4#12011yescharacteristic
220C2×C4#12011yescharacteristic
221C2×C4C22×D411yescharacteristic
222C2×C4C22×D411yescharacteristic
223C2×C4C2×#4112yes—
224C2×C4H(5,𝔽2)12yes—
225C8—24no—
226C8—42no—
227C22—21no—
228C22—21no—
229C22—42no—
230C22—42no—
231C22—42no—
232C22C2×#12011yescharacteristic
233C22#53211yescharacteristic
234C22#53511yescharacteristic
235C4—21no—
236C4—21no—
237C4—21no—
238C4—44no—
239C4—42no—
240C4—42no—
241C4—42no—
242C4—42no—
243C4—42no—
244C4—82no—
245C4#45612yes—
246C4#54612yes—
247C4#55012yes—
248C2—21no—
249C2—21no—
250C2—82no—
251C2#243111yescharacteristic
252C2C2×#45611yescharacteristic
253C2#302211yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radical#532C22
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
3. #3139C2semidirect
4. #2546C2semidirect
5. #2005C2semidirect
6. #1986C2semidirect
8. #1882C2semidirect
9. C4×#101C2semidirect

Maximal subgroups

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

subgroupindex
#35072
#31522
#31392
#25462
#20052
#19862
#19802
#18822
C4×#1012