Finite Group Databasebeta

← #32789

Subgroups — #32789

Order 256.

336 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 109 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#2857C211yesmaximal, characteristic
2#2851C211yesmaximal, characteristic
3#2101C212yesmaximal, complemented
4#2099C211yesmaximal, characteristic, complemented
5#2099C211yesmaximal, characteristic, complemented
6#2077C212yesmaximal
7#2076C212yesmaximal
8#2059C212yesmaximal, complemented
9#2058C211yesmaximal, characteristic, complemented
10#2057C211yesmaximal, characteristic, complemented
11#1817C211yesmaximal, characteristic
12C2×C4⋊Q8C2211yescharacteristic
13C2×#125C2211yescharacteristic
14C2×C22⋊Q8—21no—
15C22×C42C2211yescharacteristic
16C2×#107C2212yes—
17C2×#106C2212yes—
18C2×C22⋊C8C2212yes—
19#391—21no—
20#389—21no—
21#389C2211yescharacteristic
22#389C2211yescharacteristic
23#389C2212yes—
24#387—21no—
25#387—21no—
26#385—21no—
27#385—21no—
28#384C2211yescharacteristic
29#382C2211yescharacteristic
30#380—21no—
31#380C2211yescharacteristic
32#380C2212yes—
33#380C2211yescharacteristic
34#340C2212yes—
35#340C2212yes—
36#340C2212yes—
37#340C2212yes—
38#340C2212yes—
39#340C2212yes—
40#336C2211yescharacteristic
41#336C2211yescharacteristic
42#336C2212yes—
43C22×Q8—21no—
44C23×C4D411yescharacteristic
45C23×C4D411yescharacteristic
46C23×C4C2311yescharacteristic
47C22×C8C2312yes—
48C22×C8C2312yes—
49C4⋊Q8—42no—
50#125—42no—
51C22⋊Q8—44no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—22no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4C2311yescharacteristic
61C2×C4⋊C4C2311yescharacteristic
62C2×C4⋊C4C2312yes—
63C2×C4⋊C4C2311yescharacteristic
64C2×C4⋊C4C2311yescharacteristic
65C2×C4⋊C4C2312yes—
66C2×C22⋊C4—22no—
67C2×C22⋊C4—21no—
68C2×C22⋊C4—21no—
69C2×C42—42no—
70C2×C42D411yescharacteristic
71C2×C42D411yescharacteristic
72C2×C42C2311yescharacteristic
73C2×C42C2311yescharacteristic
74#107—42no—
75#106—42no—
76C22⋊C8—42no—
77#95—22no—
78#95—21no—
79#95—21no—
80#95—22no—
81#95—21no—
82#95—21no—
83#95—21no—
84#95—21no—
85C24C2×D411yescharacteristic
86C2×Q8—42no—
87C2×Q8—42no—
88C2×Q8—42no—
89C22×C4—22no—
90C22×C4—22no—
91C22×C4—22no—
92C22×C4—22no—
93C22×C4—22no—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—22no—
101C22×C4—22no—
102C22×C4—41no—
103C22×C4—41no—
104C22×C4—41no—
105C22×C4C2×D411yescharacteristic
106C22×C4C2×D411yescharacteristic
107C22×C4C2×D411yescharacteristic
108C22×C4C2×D411yescharacteristic
109C22×C4C2×D411yescharacteristic
110C22×C4C2411yescharacteristic
111C2×C8—22no—
112C2×C8—22no—
113C2×C8—22no—
114C2×C8—22no—
115C2×C8—22no—
116C2×C8—22no—
117C4⋊C4—22no—
118C4⋊C4—22no—
119C4⋊C4—22no—
120C4⋊C4—22no—
121C4⋊C4—21no—
122C4⋊C4—21no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—24no—
126C4⋊C4—44no—
127C4⋊C4—42no—
128C4⋊C4—42no—
129C4⋊C4—42no—
130C4⋊C4—42no—
131C4⋊C4—42no—
132C4⋊C4—42no—
133C4⋊C4—42no—
134C22⋊C4—42no—
135C22⋊C4—44no—
136C22⋊C4—42no—
137C42—22no—
138C42—22no—
139C42—41no—
140C42—41no—
141C23—21no—
142C23—21no—
143C23—21no—
144C23—21no—
145C23—21no—
146C23—21no—
147C23#13712yes—
148C23C22×D411yescharacteristic
149Q8—84no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—44no—
164C2×C4—44no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—42no—
175C2×C4—41no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—42no—
185C2×C4#13512yes—
186C2×C4AGL(1,ℤ/8)12yes—
187C2×C4H(5,𝔽2)12yes—
188C2×C42−1+412yes—
189C2×C42−1+411yescharacteristic
190C2×C42−1+411yescharacteristic
191C8—42no—
192C8—42no—
193C22—22no—
194C22—22no—
195C22—22no—
196C22—41no—
197C22—41no—
198C22—41no—
199C22—41no—
200C22#53411yescharacteristic
201C22#53611yescharacteristic
202C22#53611yescharacteristic
203C22C2×#13511yescharacteristic
204C22C2×AGL(1,ℤ/8)11yescharacteristic
205C22C2×#13711yescharacteristic
206C22#57511yescharacteristic
207C4—22no—
208C4—22no—
209C4—41no—
210C4—41no—
211C4—41no—
212C4—81no—
213C4—81no—
214C4—82no—
215C4—82no—
216C4—81no—
217C4—81no—
218C4#56312yes—
219C4#56312yes—
220C2—42no—
221C2#281711yescharacteristic
222C2#319011yescharacteristic
223C2#319411yescharacteristic
224C2#322811yescharacteristic
225C2#323311yescharacteristic
226C2#323611yescharacteristic
227C2#323911yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #2101C2semidirect
4. #2099C2semidirect
5. #2099C2semidirect
8. #2059C2semidirect
9. #2058C2semidirect
10. #2057C2semidirect

Maximal subgroups

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

subgroupindex
#28572
#28512
#21012
#20992
#20772
#20762
#20592
#20582
#20572
#18172