Finite Group Databasebeta

← #30193

Subgroups — #30193

Order 256.

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

normal subgroups only

This group has 353 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#2628C211yesmaximal, characteristic
2#2621C211yesmaximal, characteristic
3#2101C212yesmaximal, complemented
4#2074C212yesmaximal, complemented
5#2043C212yesmaximal, complemented
6#2024C212yesmaximal, complemented
7#1894C212yesmaximal
8#1892C212yesmaximal
9#1729C211yesmaximal, characteristic
10C22×M16C2211yescharacteristic
11C23×C8C2211yescharacteristic
12C2×C4⋊Q8C2211yescharacteristic
13C2×C4⋊Q8C2211yescharacteristic
14C2×C22⋊Q8C2212yes—
15C2×#117C2211yescharacteristic
16C2×#103C2212yes—
17C2×#103C2212yes—
18C2×#103C2212yes—
19C2×#103C2212yes—
20#400—21no—
21#399—21no—
22#398—21no—
23#398—21no—
24#397—21no—
25#397—21no—
26#395—21no—
27#393—21no—
28#391C2212yes—
29#389C2212yes—
30#382C2211yescharacteristic
31#382C2211yescharacteristic
32#380C2212yes—
33#340C2212yes—
34#340C2212yes—
35#340C2212yes—
36#340C2212yes—
37#336C2212yes—
38#336C2212yes—
39C22×Q8—21no—
40C22×Q8C2312yes—
41C23×C4C2311yescharacteristic
42C2×M16—22no—
43C2×M16—41no—
44C2×M16—41no—
45C22×C8—22no—
46C22×C8—41no—
47C22×C8—41no—
48C22×C8C2312yes—
49C22×C8C2312yes—
50C4⋊Q8—41no—
51C4⋊Q8—41no—
52C4⋊Q8—41no—
53C4⋊Q8—41no—
54C22⋊Q8—24no—
55C22⋊Q8—24no—
56#117—41no—
57#117—41no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4D411yescharacteristic
64C2×C4⋊C4D411yescharacteristic
65C2×C4⋊C4C2312yes—
66C2×C4⋊C4C2312yes—
67C2×C4⋊C4C2312yes—
68C2×C22⋊C4—22no—
69C2×C22⋊C4—22no—
70C2×C22⋊C4D411yescharacteristic
71C2×C22⋊C4D411yescharacteristic
72C2×C42C2311yescharacteristic
73C2×C42C2311yescharacteristic
74#103—42no—
75#103—42no—
76#103—42no—
77#103—42no—
78#95—22no—
79#95—21no—
80#95—21no—
81#95—21no—
82#95—21no—
83#95—22no—
84#95—22no—
85#95—22no—
86C24C2×D411yescharacteristic
87C2×Q8—22no—
88C2×Q8—22no—
89C2×Q8—22no—
90C2×Q8—22no—
91C2×Q8—22no—
92C2×Q8—22no—
93C2×Q8—42no—
94C2×Q8—82no—
95C22×C4—21no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—22no—
100C22×C4—22no—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4C2×D411yescharacteristic
104C22×C4C2×D411yescharacteristic
105C22×C4C2×D411yescharacteristic
106C22×C4C2×D411yescharacteristic
107C22×C4C2×D411yescharacteristic
108C22×C4#4112yes—
109C22×C4#4112yes—
110C22×C4C2411yescharacteristic
111M16—44no—
112C2×C8—22no—
113C2×C8—22no—
114C2×C8—22no—
115C2×C8—22no—
116C2×C8—22no—
117C2×C8—22no—
118C2×C8—44no—
119C4⋊C4—22no—
120C4⋊C4—22no—
121C4⋊C4—22no—
122C4⋊C4—22no—
123C4⋊C4—22no—
124C4⋊C4—22no—
125C4⋊C4—41no—
126C4⋊C4—41no—
127C4⋊C4—42no—
128C4⋊C4—42no—
129C4⋊C4—42no—
130C4⋊C4—44no—
131C4⋊C4—82no—
132C22⋊C4—41no—
133C22⋊C4—41no—
134C22⋊C4—44no—
135C22⋊C4—82no—
136C42—21no—
137C42—21no—
138C42—21no—
139C42—21no—
140C23—21no—
141C23—21no—
142C23—22no—
143C23—22no—
144C23#12412yes—
145C23C22×D411yescharacteristic
146Q8—42no—
147Q8—42no—
148Q8—42no—
149Q8—42no—
150Q8—84no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—22no—
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—42no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4#12412yes—
178C2×C4C2×#4111yescharacteristic
179C2×C4C2×#4111yescharacteristic
180C2×C4H(5,𝔽2)11yescharacteristic
181C2×C4H(5,𝔽2)11yescharacteristic
182C2×C4H(5,𝔽2)11yescharacteristic
183C2×C42−1+411yescharacteristic
184C8—42no—
185C8—42no—
186C22—24no—
187C22—22no—
188C22—41no—
189C22—41no—
190C22—42no—
191C22C2×#12411yescharacteristic
192C22#53411yescharacteristic
193C22#53711yescharacteristic
194C22#57712yes—
195C22#57812yes—
196C4—22no—
197C4—41no—
198C4—41no—
199C4—41no—
200C4—41no—
201C4—82no—
202C4—82no—
203C4—82no—
204C4—82no—
205C4#53811yescharacteristic
206C4#53811yescharacteristic
207C4#53911yescharacteristic
208C4#53911yescharacteristic
209C2—42no—
210C2#263911yescharacteristic
211C2#313712yes—
212C2#320011yescharacteristic
213C2#320211yescharacteristic
214C2#325812yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #2101C2semidirect
4. #2074C2semidirect
5. #2043C2semidirect
6. #2024C2semidirect

Maximal subgroups

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

subgroupindex
#26282
#26212
#21012
#20742
#20432
#20242
#18942
#18922
#17292