Finite Group Databasebeta

← #29561

Subgroups — #29561

Order 256.

422 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 117 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#2858C211yesmaximal, characteristic
2#2295C211yesmaximal, characteristic, complemented
3#2057C211yesmaximal, characteristic, complemented
4#2041C211yesmaximal, characteristic
5#2024C211yesmaximal, characteristic, complemented
6#2014C211yesmaximal, characteristic
7#1973C212yesmaximal, complemented
8#1936C212yesmaximal, complemented
9#1925C212yesmaximal, complemented
10#1880C212yesmaximal, complemented
11#1745C211yesmaximal, characteristic, complemented
12C22×SD16C2212yes—
13C2×C4⋊Q8C2211yescharacteristic
14C2×#127C2211yescharacteristic
15C2×#124—21no—
16C2×#124—21no—
17C2×C4×Q8C2211yescharacteristic
18C2×#107C2212yes—
19C2×#103C2212yes—
20C2×#102C2212yes—
21C2×#102C2212yes—
22C2×#102C2212yes—
23C2×#97C2212yes—
24#391C2211yescharacteristic
25#391C2211yescharacteristic
26#390—21no—
27#390—21no—
28#390C2212yes—
29#390C2212yes—
30#389C2211yescharacteristic
31#389C2211yescharacteristic
32#387—21no—
33#387—21no—
34C4×C4⋊C4—21no—
35C4×C4⋊C4C2211yescharacteristic
36#376—21no—
37C43C2211yescharacteristic
38#340C2212yes—
39#340C2212yes—
40#336C2211yescharacteristic
41#336C2212yes—
42#336C2211yescharacteristic
43C22×Q8—21no—
44C22×Q8C2311yescharacteristic
45C22×D4—21no—
46C22×D4—21no—
47C22×D4C2312yes—
48C2×SD16—24no—
49C2×SD16—24no—
50C2×SD16—42no—
51C22×C8C2312yes—
52C22×C8C2312yes—
53C4⋊Q8—42no—
54#127—42no—
55#124—44no—
56#124—44no—
57C4×Q8—22no—
58C4×Q8—22no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4C2312yes—
64C2×C4⋊C4C2311yescharacteristic
65C2×C4⋊C4C2311yescharacteristic
66C2×C4⋊C4C2311yescharacteristic
67C2×C22⋊C4—22no—
68C2×C22⋊C4—22no—
69C2×C22⋊C4—22no—
70C2×C22⋊C4—22no—
71C2×C42—21no—
72C2×C42—21no—
73C2×C42D411yescharacteristic
74C2×C42D411yescharacteristic
75C2×C42D411yescharacteristic
76C2×C42D411yescharacteristic
77C2×C42C2311yescharacteristic
78C2×C42C2311yescharacteristic
79C2×C42C2311yescharacteristic
80#107—42no—
81#103—42no—
82#102—24no—
83#102—24no—
84#102—42no—
85#97—24no—
86#95—21no—
87#95—21no—
88#95—21no—
89#95—21no—
90#95—21no—
91#95—21no—
92#95—21no—
93C24—22no—
94C2×Q8—21no—
95C2×Q8—22no—
96C2×Q8—22no—
97C2×Q8—42no—
98C2×Q8—42no—
99C2×Q8—42no—
100C2×Q8#4112yes—
101C2×D4—22no—
102C2×D4—22no—
103C2×D4—22no—
104C2×D4—22no—
105C2×D4—22no—
106C2×D4—22no—
107C2×D4—42no—
108C2×D4—42no—
109C2×D4—42no—
110C2×D4—42no—
111C2×D4—42no—
112C2×D4—42no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—22no—
118C22×C4C2×D411yescharacteristic
119C22×C4C2×D411yescharacteristic
120C22×C4C2×D411yescharacteristic
121C22×C4C2×D411yescharacteristic
122C22×C4C2×D411yescharacteristic
123C22×C4C2×D411yescharacteristic
124C22×C4C2411yescharacteristic
125SD16—48no—
126C2×C8—22no—
127C2×C8—22no—
128C2×C8—22no—
129C2×C8—22no—
130C2×C8—22no—
131C2×C8#4114yes—
132C4⋊C4—24no—
133C4⋊C4—21no—
134C4⋊C4—22no—
135C4⋊C4—22no—
136C4⋊C4—22no—
137C4⋊C4—22no—
138C4⋊C4—42no—
139C4⋊C4—42no—
140C4⋊C4—42no—
141C4⋊C4#4112yes—
142C22⋊C4—44no—
143C22⋊C4—44no—
144C22⋊C4—44no—
145C22⋊C4—44no—
146C42—24no—
147C42—22no—
148C42—22no—
149C42—22no—
150C42—22no—
151C42—21no—
152C42—21no—
153C42—22no—
154C42—41no—
155C42—42no—
156C23—42no—
157C23—42no—
158C23—42no—
159C23—42no—
160C23—42no—
161C23—42no—
162C23—42no—
163C23C22×D411yescharacteristic
164Q8—24no—
165Q8—42no—
166Q8—84no—
167D4—42no—
168D4—42no—
169D4—42no—
170D4—42no—
171D4—84no—
172D4—84no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—42no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4AGL(1,ℤ/8)12yes—
204C2×C4AGL(1,ℤ/8)12yes—
205C2×C4#13712yes—
206C2×C4C2×#4111yescharacteristic
207C2×C4C2×#4112yes—
208C2×C4C2×#4111yescharacteristic
209C2×C4H(5,𝔽2)12yes—
210C8—24no—
211C8—42no—
212C22—82no—
213C22—82no—
214C22—82no—
215C22—82no—
216C22—82no—
217C22—82no—
218C22—82no—
219C22#52511yescharacteristic
220C22#53211yescharacteristic
221C22#53511yescharacteristic
222C22C2×AGL(1,ℤ/8)11yescharacteristic
223C22C2×AGL(1,ℤ/8)11yescharacteristic
224C22C2×#13711yescharacteristic
225C22#57511yescharacteristic
226C4—22no—
227C4—22no—
228C4—22no—
229C4—41no—
230C4—42no—
231C4—42no—
232C4—41no—
233C4—41no—
234C4—81no—
235C4—81no—
236C4—82no—
237C4#55112yes—
238C4#55512yes—
239C2—162no—
240C2#261011yescharacteristic
241C2#311211yescharacteristic
242C2#311511yescharacteristic
243C2#317111yescharacteristic
244C2#318211yescharacteristic
245C2#321511yescharacteristic
246C2#321711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2295C2semidirect
3. #2057C2semidirect
5. #2024C2semidirect
7. #1973C2semidirect
8. #1936C2semidirect
9. #1925C2semidirect
10. #1880C2semidirect
11. #1745C2semidirect

Maximal subgroups

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

subgroupindex
#28582
#22952
#20572
#20412
#20242
#20142
#19732
#19362
#19252
#18802
#17452