Finite Group Databasebeta

← #32268

Subgroups — #32268

Order 256.

358 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 111 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#2563C211yesmaximal, characteristic
2#2487C211yesmaximal, characteristic
3#2077C212yesmaximal, complemented
4#2075C212yesmaximal, complemented
5#2057C212yesmaximal, complemented
6#1920C212yesmaximal, complemented
7#1870C211yesmaximal, characteristic
8#1852C212yesmaximal, complemented
9#1809C212yesmaximal, complemented
10C22×M16C2211yescharacteristic
11C2×C4⋊Q8C2211yescharacteristic
12C2×C4×D4C2211yescharacteristic
13C22×C4⋊C4C2211yescharacteristic
14C2×#107C2212yes—
15C2×#107C2212yes—
16C2×#106C2212yes—
17C2×C22⋊C8C2212yes—
18C2×C22⋊C8C2211yescharacteristic
19#400—21no—
20#399—21no—
21#393—21no—
22#393—21no—
23#390—21no—
24#389C2212yes—
25#386—21no—
26#385—21no—
27#382—21no—
28#380C2212yes—
29#380C2212yes—
30#380C2212yes—
31#380C2211yescharacteristic
32C4×C4⋊C4C2211yescharacteristic
33#340C2212yes—
34#340C2212yes—
35#340C2212yes—
36#340C2212yes—
37#340C2212yes—
38#336C2212yes—
39C22×Q8—21no—
40C22×D4D411yescharacteristic
41C23×C4—21no—
42C23×C4C2311yescharacteristic
43C23×C4C2311yescharacteristic
44C2×M16—24no—
45C2×M16—41no—
46C22×C8C2312yes—
47C22×C8C2312yes—
48C4⋊Q8—41no—
49C4⋊Q8—41no—
50C4×D4—41no—
51C4×D4—41no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—41no—
56C2×C4⋊C4—41no—
57C2×C4⋊C4D411yescharacteristic
58C2×C4⋊C4C2312yes—
59C2×C4⋊C4C2312yes—
60C2×C4⋊C4C2312yes—
61C2×C4⋊C4C2312yes—
62C2×C22⋊C4—21no—
63C2×C22⋊C4—22no—
64C2×C22⋊C4—22no—
65C2×C22⋊C4—22no—
66C2×C22⋊C4D411yescharacteristic
67C2×C22⋊C4D411yescharacteristic
68C2×C42—21no—
69C2×C42C2311yescharacteristic
70#107—42no—
71#107—42no—
72#106—24no—
73C22⋊C8—24no—
74C22⋊C8—41no—
75#95—21no—
76#95—22no—
77#95—22no—
78#95—22no—
79#95—21no—
80#95—22no—
81C24C2×D411yescharacteristic
82C24C2×D411yescharacteristic
83C2×Q8—42no—
84C2×Q8—82no—
85C2×D4—22no—
86C2×D4—22no—
87C2×D4—41no—
88C22×C4—22no—
89C22×C4—22no—
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—22no—
95C22×C4—22no—
96C22×C4—22no—
97C22×C4—22no—
98C22×C4—22no—
99C22×C4—22no—
100C22×C4—42no—
101C22×C4—82no—
102C22×C4C2×D411yescharacteristic
103C22×C4C2×D411yescharacteristic
104C22×C4C2×D411yescharacteristic
105C22×C4C2×D411yescharacteristic
106C22×C4C2411yescharacteristic
107M16—44no—
108C2×C8—22no—
109C2×C8—22no—
110C2×C8—22no—
111C2×C8—22no—
112C2×C8—22no—
113C2×C8#4114yes—
114C4⋊C4—22no—
115C4⋊C4—22no—
116C4⋊C4—22no—
117C4⋊C4—22no—
118C4⋊C4—22no—
119C4⋊C4—22no—
120C4⋊C4—22no—
121C4⋊C4—22no—
122C4⋊C4—41no—
123C4⋊C4—42no—
124C4⋊C4—42no—
125C4⋊C4—82no—
126C22⋊C4—41no—
127C22⋊C4—41no—
128C22⋊C4—42no—
129C22⋊C4—82no—
130C22⋊C4—82no—
131C22⋊C4—82no—
132C42—21no—
133C42—21no—
134C42—42no—
135C23—21no—
136C23—22no—
137C23—21no—
138C23—22no—
139C23—21no—
140C23—22no—
141C23—21no—
142C23—21no—
143C23—22no—
144C23#13712yes—
145C23C22×D411yescharacteristic
146Q8—84no—
147D4—44no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—42no—
157C2×C4—42no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—84no—
175C2×C4#13712yes—
176C2×C4C2×#4112yes—
177C2×C4H(5,𝔽2)11yescharacteristic
178C2×C4H(5,𝔽2)11yescharacteristic
179C2×C42−1+412yes—
180C8—24no—
181C8—42no—
182C22—24no—
183C22—22no—
184C22—42no—
185C22—41no—
186C22—41no—
187C22—42no—
188C22—42no—
189C22—41no—
190C22—41no—
191C22—41no—
192C22#53411yescharacteristic
193C22#54712yes—
194C22C2×#13711yescharacteristic
195C22C2×#13711yescharacteristic
196C22#57612yes—
197C4—22no—
198C4—41no—
199C4—41no—
200C4—41no—
201C4—82no—
202C4—82no—
203C4—82no—
204C4—82no—
205C4#54212yes—
206C4#55111yescharacteristic
207C4#56311yescharacteristic
208C2—42no—
209C2—81no—
210C2#271711yescharacteristic
211C2#319511yescharacteristic
212C2#319911yescharacteristic
213C2#334612yes—
214C2#335712yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #2077C2semidirect
4. #2075C2semidirect
5. #2057C2semidirect
6. #1920C2semidirect
8. #1852C2semidirect
9. #1809C2semidirect

Maximal subgroups

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

subgroupindex
#25632
#24872
#20772
#20752
#20572
#19202
#18702
#18522
#18092