Finite Group Databasebeta

← #32825

Subgroups — #32825

Order 256.

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

normal subgroups only

This group has 331 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#2861C211yesmaximal, characteristic
2#2859C211yesmaximal, characteristic, complemented
3#2096C211yesmaximal, characteristic, complemented
4#2093C211yesmaximal, characteristic
5#2072C211yesmaximal, characteristic, complemented
6#2071C211yesmaximal, characteristic
7#2068C212yesmaximal, complemented
8#2065C212yesmaximal
9#2059C212yesmaximal, complemented
10#2042C212yesmaximal
11#1831C211yesmaximal, characteristic, complemented
12C2×#125—21no—
13C2×#125C2211yescharacteristic
14C2×#124C2211yescharacteristic
15C2×#107C2212yes—
16C2×#106C2212yes—
17C2×C4⋊C8C2212yes—
18C2×#103C2212yes—
19C2×#103C2212yes—
20C2×#102C2212yes—
21C2×#102C2212yes—
22#391C2211yescharacteristic
23#391C2211yescharacteristic
24#390C2211yescharacteristic
25#390C2211yescharacteristic
26#389—21no—
27#389—21no—
28#389C2212yes—
29#389C2212yes—
30#388—21no—
31#388—21no—
32#388—21no—
33#388—21no—
34#384C2211yescharacteristic
35#384C2211yescharacteristic
36#382—21no—
37C43C2211yescharacteristic
38#340C2212yes—
39#340C2212yes—
40#336C2212yes—
41#336C2211yescharacteristic
42#336C2211yescharacteristic
43C22×Q8C2311yescharacteristic
44C22×D4C2311yescharacteristic
45C22×C8C2312yes—
46C22×C8C2312yes—
47#125—42no—
48#125—44no—
49#124—42no—
50C2×C4⋊C4—21no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—22no—
53C2×C4⋊C4—22no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4C2311yescharacteristic
59C2×C4⋊C4C2311yescharacteristic
60C2×C4⋊C4C2312yes—
61C2×C4⋊C4C2312yes—
62C2×C22⋊C4—21no—
63C2×C22⋊C4—21no—
64C2×C22⋊C4—21no—
65C2×C22⋊C4—21no—
66C2×C22⋊C4—21no—
67C2×C22⋊C4—21no—
68C2×C42D411yescharacteristic
69C2×C42D411yescharacteristic
70C2×C42D411yescharacteristic
71C2×C42D411yescharacteristic
72C2×C42C2311yescharacteristic
73C2×C42C2311yescharacteristic
74C2×C42C2311yescharacteristic
75#107—42no—
76#106—42no—
77C4⋊C8—42no—
78#103—42no—
79#103—42no—
80#102—42no—
81#102—42no—
82#95—22no—
83#95—22no—
84#95—21no—
85#95—21no—
86#95—21no—
87#95—21no—
88C24—21no—
89C2×Q8—21no—
90C2×Q8—22no—
91C2×Q8—22no—
92C2×Q8—21no—
93C2×D4—21no—
94C2×D4—22no—
95C2×D4—22no—
96C2×D4—21no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—22no—
101C22×C4—22no—
102C22×C4C2×D411yescharacteristic
103C22×C4C2×D411yescharacteristic
104C22×C4C2×D411yescharacteristic
105C22×C4C2×D411yescharacteristic
106C22×C4C2×D411yescharacteristic
107C22×C4C2×D411yescharacteristic
108C22×C4C2411yescharacteristic
109C2×C8—22no—
110C2×C8—22no—
111C2×C8—22no—
112C2×C8—22no—
113C2×C8—22no—
114C2×C8—22no—
115C4⋊C4—22no—
116C4⋊C4—22no—
117C4⋊C4—22no—
118C4⋊C4—22no—
119C4⋊C4—22no—
120C4⋊C4—22no—
121C4⋊C4—42no—
122C4⋊C4—42no—
123C4⋊C4—42no—
124C4⋊C4—44no—
125C4⋊C4—44no—
126C4⋊C4—42no—
127C4⋊C4—42no—
128C4⋊C4—42no—
129C22⋊C4—42no—
130C22⋊C4—42no—
131C22⋊C4—42no—
132C22⋊C4—42no—
133C22⋊C4—42no—
134C22⋊C4—42no—
135C42—22no—
136C42—22no—
137C42—22no—
138C42—21no—
139C42—22no—
140C42—22no—
141C42—21no—
142C42—41no—
143C23—41no—
144C23—41no—
145C23—41no—
146C23—41no—
147C23—41no—
148C23—41no—
149C23—41no—
150C23C22×D411yescharacteristic
151Q8—42no—
152Q8—42no—
153D4—42no—
154D4—42no—
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—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—42no—
174C2×C4—41no—
175C2×C4—42no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—42no—
179C2×C4—41no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—42no—
185C2×C4#13512yes—
186C2×C4AGL(1,ℤ/8)12yes—
187C2×C4#13712yes—
188C2×C4H(5,𝔽2)11yescharacteristic
189C2×C4H(5,𝔽2)12yes—
190C2×C42−1+412yes—
191C2×C42−1+411yescharacteristic
192C8—42no—
193C8—42no—
194C22—81no—
195C22—81no—
196C22—81no—
197C22—81no—
198C22—81no—
199C22—81no—
200C22—81no—
201C22#53511yescharacteristic
202C22#53611yescharacteristic
203C22#53711yescharacteristic
204C22C2×#13511yescharacteristic
205C22C2×AGL(1,ℤ/8)11yescharacteristic
206C22C2×#13711yescharacteristic
207C22#57511yescharacteristic
208C4—22no—
209C4—22no—
210C4—22no—
211C4—41no—
212C4—41no—
213C4—41no—
214C4—81no—
215C4—81no—
216C4—82no—
217C4—81no—
218C4—82no—
219C4#56212yes—
220C4#56312yes—
221C2—161no—
222C2#282011yescharacteristic
223C2#320711yescharacteristic
224C2#321611yescharacteristic
225C2#323211yescharacteristic
226C2#323811yescharacteristic
227C2#324811yescharacteristic
228C2#325111yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
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. #2859C2semidirect
3. #2096C2semidirect
5. #2072C2semidirect
7. #2068C2semidirect
9. #2059C2semidirect
11. #1831C2semidirect

Maximal subgroups

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

subgroupindex
#28612
#28592
#20962
#20932
#20722
#20712
#20682
#20652
#20592
#20422
#18312