Finite Group Databasebeta

← #11757

Subgroups — #11757

Order 256.

350 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 53 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#2539C211yesmaximal, characteristic
2C2×#342C211yesmaximal, characteristic, complemented
3#1462C211yesmaximal, characteristic, complemented
4#1352C212yesmaximal, complemented
5#1343C212yesmaximal, complemented
6C2×#123—21no—
7C2×#121C411yescharacteristic, complemented
8C2×#120—21no—
9C22×C22⋊C4—21no—
10C22×C22⋊C4C2211yescharacteristic
11#407—21no—
12#399C411yescharacteristic, complemented
13#399C2211yescharacteristic
14#392C411yescharacteristic, complemented
15#392C2211yescharacteristic
16#390—21no—
17#390—21no—
18C4×C22⋊C4C411yescharacteristic, complemented
19#342—21no—
20#342C2212yes—
21#323—21no—
22#323C2212yes—
23C25D411yescharacteristic
24C22×D4—21no—
25C22×D4—21no—
26C22×D4—21no—
27C22×D4C2×C411yescharacteristic, complemented
28C23×C4—41no—
29C23×C4C2×C411yescharacteristic
30C23×C4D411yescharacteristic
31C2×M16—41no—
32#123—44no—
33#121—41no—
34#121—41no—
35#120—44no—
36C2×C4⋊C4—21no—
37C2×C4⋊C4—21no—
38C2×C4⋊C4C2×C411yescharacteristic, complemented
39C2×C22⋊C4—21no—
40C2×C22⋊C4—24no—
41C2×C22⋊C4—22no—
42C2×C22⋊C4—21no—
43C2×C22⋊C4—21no—
44C2×C22⋊C4—21no—
45C2×C22⋊C4—21no—
46C2×C22⋊C4—21no—
47C2×C22⋊C4—41no—
48C2×C22⋊C4—41no—
49C2×C22⋊C4—41no—
50C2×C22⋊C4—42no—
51C2×C22⋊C4C2×C411yescharacteristic
52C2×C22⋊C4D411yescharacteristic
53C2×C22⋊C4D411yescharacteristic
54C2×C22⋊C4C2311yescharacteristic
55C2×C42—21no—
56C22⋊C8—22no—
57C22⋊C8—41no—
58#95C2×C411yescharacteristic, complemented
59#95C2×C411yescharacteristic, complemented
60C24—21no—
61C24—21no—
62C24—22no—
63C24—21no—
64C24—21no—
65C24—21no—
66C24—42no—
67C24—81no—
68C24C22⋊C411yescharacteristic
69C24C22⋊C411yescharacteristic
70C24C2×D411yescharacteristic
71C2×D4—41no—
72C2×D4—42no—
73C2×D4—42no—
74C2×D4—42no—
75C2×D4—42no—
76C2×D4—41no—
77C2×D4—41no—
78C2×D4—42no—
79C2×D4—41no—
80C2×D4—41no—
81C2×D4—81no—
82C2×D4—81no—
83C2×D4—81no—
84C22×C4—21no—
85C22×C4—21no—
86C22×C4—21no—
87C22×C4—21no—
88C22×C4—21no—
89C22×C4—21no—
90C22×C4—21no—
91C22×C4—42no—
92C22×C4—44no—
93C22×C4—42no—
94C22×C4—81no—
95C22×C4—82no—
96C22×C4—82no—
97C22×C4C22⋊C411yescharacteristic
98C22×C4C22⋊C411yescharacteristic
99C22×C4C22×C411yescharacteristic
100C22×C4C2×D411yescharacteristic
101M16—82no—
102C2×C8—42no—
103C4⋊C4—41no—
104C4⋊C4—42no—
105C4⋊C4—42no—
106C22⋊C4—24no—
107C22⋊C4—24no—
108C22⋊C4—41no—
109C22⋊C4—41no—
110C22⋊C4—41no—
111C22⋊C4—41no—
112C22⋊C4—42no—
113C22⋊C4—42no—
114C22⋊C4—42no—
115C22⋊C4—42no—
116C22⋊C4—42no—
117C22⋊C4—44no—
118C22⋊C4—42no—
119C42—81no—
120C23—22no—
121C23—22no—
122C23—22no—
123C23—22no—
124C23—22no—
125C23—21no—
126C23—21no—
127C23—21no—
128C23—21no—
129C23—21no—
130C23—21no—
131C23—41no—
132C23—42no—
133C23—41no—
134C23—41no—
135C23—41no—
136C23—41no—
137C23—41no—
138C23—41no—
139C23—41no—
140C23—41no—
141C23—42no—
142C23—41no—
143C23—41no—
144C23—41no—
145C23—81no—
146C23—81no—
147C23—84no—
148C23—82no—
149C23—82no—
150C23#9911yescharacteristic
151C23#9911yescharacteristic
152C23#10412yes—
153C23C2×C22⋊C411yescharacteristic
154C23AGL(1,ℤ/8)11yescharacteristic
155C23#13711yescharacteristic
156D4—84no—
157D4—82no—
158D4—82no—
159D4—82no—
160D4—82no—
161D4—82no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—42no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—44no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—81no—
182C2×C4—81no—
183C2×C4—81no—
184C2×C4—88no—
185C2×C4—82no—
186C2×C4—82no—
187C8—82no—
188C22—21no—
189C22—21no—
190C22—21no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—21no—
195C22—21no—
196C22—22no—
197C22—21no—
198C22—42no—
199C22—42no—
200C22—42no—
201C22—41no—
202C22—41no—
203C22—44no—
204C22—44no—
205C22—41no—
206C22—41no—
207C22—41no—
208C22—41no—
209C22—81no—
210C22—81no—
211C22—81no—
212C22—81no—
213C22—81no—
214C22—81no—
215C22—82no—
216C22—82no—
217C22—81no—
218C22—81no—
219C22—81no—
220C22#35312yes—
221C22C2×#9911yescharacteristic
222C22#41711yescharacteristic
223C22C2×#10411yescharacteristic
224C4—42no—
225C4—42no—
226C4—41no—
227C4—41no—
228C4—41no—
229C4—41no—
230C4—84no—
231C4—81no—
232C2—21no—
233C2—21no—
234C2—22no—
235C2—41no—
236C2—41no—
237C2—42no—
238C2—41no—
239C2—161no—
240C2#150411yescharacteristic
241C2#212111yescharacteristic
242C2C2×#35311yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C22⋊C4C23
prime radical#2539C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#342C2semidirect
3. #1462C2semidirect
4. #1352C2semidirect
5. #1343C2semidirect
7. C2×#121C4semidirect
12. #399C4semidirect
14. #392C4semidirect
18. C4×C22⋊C4C4semidirect
27. C22×D4C2×C4semidirect
38. C2×C4⋊C4C2×C4semidirect
58. #95C2×C4semidirect
59. #95C2×C4semidirect

Maximal subgroups

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

subgroupindex
#25392
C2×#3422
#14622
#13522
#13432