Finite Group Databasebeta

← #30175

Subgroups — #30175

Order 256.

373 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 373 conjugacy classes of subgroups — too many to draw. Only its 115 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
2C4×C22⋊Q8C211yesmaximal, characteristic
3#2083C211yesmaximal, characteristic, complemented
4#2077C212yesmaximal
5#2072C211yesmaximal, characteristic, complemented
6#2058C211yesmaximal, characteristic, complemented
7#2036C212yesmaximal
8#2024C211yesmaximal, characteristic, complemented
9#1918C212yesmaximal, complemented
10#1865C212yesmaximal, complemented
11#1817C211yesmaximal, characteristic
12C2×C4⋊Q8C2211yescharacteristic
13C2×C22⋊Q8—21no—
14C2×C22⋊Q8C2211yescharacteristic
15C2×C4×Q8C2211yescharacteristic
16C22×C42C2211yescharacteristic
17C2×#107C2212yes—
18C2×#106C2212yes—
19C2×#103C2212yes—
20C2×#103C2212yes—
21C2×C22⋊C8C2212yes—
22#391—21no—
23#391C2211yescharacteristic
24#389—21no—
25#389C2211yescharacteristic
26#389C2212yes—
27#387—21no—
28#385—21no—
29#384C2211yescharacteristic
30#380—21no—
31#380C2211yescharacteristic
32#380C2212yes—
33C4×C4⋊C4—21no—
34C4×C4⋊C4C2211yescharacteristic
35C4×C22⋊C4—21no—
36#340C2212yes—
37#340C2212yes—
38#340C2212yes—
39#340C2212yes—
40#336C2211yescharacteristic
41#336C2211yescharacteristic
42#336C2212yes—
43C22×Q8—21no—
44C22×Q8C2311yescharacteristic
45C23×C4D411yescharacteristic
46C23×C4D411yescharacteristic
47C23×C4C2311yescharacteristic
48C22×C8C2312yes—
49C22×C8C2312yes—
50C4⋊Q8—42no—
51C22⋊Q8—22no—
52C22⋊Q8—22no—
53C22⋊Q8—44no—
54C4×Q8—42no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4C2311yescharacteristic
63C2×C4⋊C4C2312yes—
64C2×C4⋊C4C2311yescharacteristic
65C2×C4⋊C4C2312yes—
66C2×C4⋊C4C2311yescharacteristic
67C2×C22⋊C4—21no—
68C2×C22⋊C4—22no—
69C2×C22⋊C4—21no—
70C2×C42—21no—
71C2×C42—21no—
72C2×C42—42no—
73C2×C42D411yescharacteristic
74C2×C42D411yescharacteristic
75C2×C42C2311yescharacteristic
76C2×C42C2311yescharacteristic
77#107—24no—
78#106—42no—
79#103—24no—
80#103—42no—
81C22⋊C8—42no—
82#95—21no—
83#95—21no—
84#95—21no—
85#95—21no—
86#95—21no—
87#95—22no—
88#95—22no—
89C24C2×D411yescharacteristic
90C2×Q8—21no—
91C2×Q8—22no—
92C2×Q8—22no—
93C2×Q8—42no—
94C2×Q8—42no—
95C2×Q8—42no—
96C2×Q8#4112yes—
97C22×C4—22no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—22no—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—41no—
112C22×C4—41no—
113C22×C4C2×D411yescharacteristic
114C22×C4C2×D411yescharacteristic
115C22×C4C2×D411yescharacteristic
116C22×C4C2×D411yescharacteristic
117C22×C4C2×D411yescharacteristic
118C22×C4C2411yescharacteristic
119C2×C8—22no—
120C2×C8—22no—
121C2×C8—22no—
122C2×C8—22no—
123C2×C8—22no—
124C2×C8—22no—
125C4⋊C4—24no—
126C4⋊C4—21no—
127C4⋊C4—22no—
128C4⋊C4—22no—
129C4⋊C4—22no—
130C4⋊C4—22no—
131C4⋊C4—22no—
132C4⋊C4—22no—
133C4⋊C4—42no—
134C4⋊C4—42no—
135C4⋊C4—42no—
136C4⋊C4—42no—
137C4⋊C4—42no—
138C4⋊C4—44no—
139C4⋊C4#4112yes—
140C22⋊C4—24no—
141C22⋊C4—44no—
142C22⋊C4—42no—
143C42—22no—
144C42—22no—
145C42—41no—
146C42—41no—
147C42—42no—
148C42—42no—
149C23—21no—
150C23—21no—
151C23—21no—
152C23—21no—
153C23—21no—
154C23—21no—
155C23#13712yes—
156C23C22×D411yescharacteristic
157Q8—24no—
158Q8—42no—
159Q8—84no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—44no—
175C2×C4—44no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—41no—
184C2×C4—42no—
185C2×C4—41no—
186C2×C4—42no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—42no—
194C2×C4#13512yes—
195C2×C4AGL(1,ℤ/8)12yes—
196C2×C4#13712yes—
197C2×C4C2×#4111yescharacteristic
198C2×C4C2×#4111yescharacteristic
199C2×C4H(5,𝔽2)12yes—
200C2×C42−1+412yes—
201C8—42no—
202C8—42no—
203C22—22no—
204C22—22no—
205C22—22no—
206C22—41no—
207C22—41no—
208C22—41no—
209C22—41no—
210C22#52511yescharacteristic
211C22#53411yescharacteristic
212C22#53611yescharacteristic
213C22C2×#13511yescharacteristic
214C22C2×AGL(1,ℤ/8)11yescharacteristic
215C22C2×#13711yescharacteristic
216C22C2×#13711yescharacteristic
217C4—22no—
218C4—22no—
219C4—22no—
220C4—42no—
221C4—41no—
222C4—42no—
223C4—41no—
224C4—82no—
225C4—81no—
226C4—82no—
227C4—81no—
228C4#53912yes—
229C4#54212yes—
230C2—42no—
231C2#263811yescharacteristic
232C2#310611yescharacteristic
233C2#311311yescharacteristic
234C2#319011yescharacteristic
235C2#319311yescharacteristic
236C2#323111yescharacteristic
237C2#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. #2083C2semidirect
5. #2072C2semidirect
6. #2058C2semidirect
8. #2024C2semidirect
9. #1918C2semidirect
10. #1865C2semidirect

Maximal subgroups

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

subgroupindex
#28572
C4×C22⋊Q82
#20832
#20772
#20722
#20582
#20362
#20242
#19182
#18652
#18172