Finite Group Databasebeta

← #28258

Subgroups — #28258

Order 256.

463 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 463 conjugacy classes of subgroups — too many to draw. Only its 135 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
1C2×#464C212yesmaximal
2C2×#462C211yesmaximal, characteristic, complemented
3C2×#461C211yesmaximal, characteristic
4#2327C211yesmaximal, characteristic, complemented
5#2326C211yesmaximal, characteristic
6#2056C212yesmaximal, complemented
7#2055C212yesmaximal
8#1843C211yesmaximal, characteristic, complemented
9#1805C211yesmaximal, characteristic, complemented
10#1804C211yesmaximal, characteristic
11#1803C212yesmaximal, complemented
12C22×Q16C2212yes—
13C22×SD16C2212yes—
14C2×C4⋊Q8C2211yescharacteristic
15C2×#124C2211yescharacteristic
16C2×C4×Q8C2212yes—
17C2×C4×Q8C2211yescharacteristic
18C2×C4×D4C2211yescharacteristic
19#464—28no—
20#462—24no—
21#461—24no—
22C2×C4⋊C8C2212yes—
23C2×C4⋊C8C2211yescharacteristic
24C2×C4⋊C8C2211yescharacteristic
25C2×C4⋊C8C2212yes—
26C2×#103C2212yes—
27C2×#103C2212yes—
28C2×#103C2212yes—
29C2×#102C2212yes—
30#391C2212yes—
31#391C2211yescharacteristic
32#390C2211yescharacteristic
33#388—22no—
34#387—22no—
35#387—22no—
36#385—22no—
37#384C2211yescharacteristic
38#382C2211yescharacteristic
39#376C2211yescharacteristic
40#340C2212yes—
41#340C2212yes—
42C22×Q8C2311yescharacteristic
43C22×Q8C2312yes—
44C22×D4C2311yescharacteristic
45C23×C4—21no—
46C2×Q16—24no—
47C2×Q16—24no—
48C2×Q16—42no—
49C2×SD16—24no—
50C2×SD16—24no—
51C2×SD16—42no—
52C22×C8C2312yes—
53C22×C8C2312yes—
54C4⋊Q8—22no—
55C4⋊Q8D414yes—
56#124—22no—
57#124D414yes—
58C4×Q8—22no—
59C4×Q8—24no—
60C4×Q8—22no—
61C4×Q8—24no—
62C4×D4—22no—
63C4×D4—22no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4C2311yescharacteristic
70C2×C4⋊C4C2312yes—
71C2×C4⋊C4C2311yescharacteristic
72C2×C22⋊C4—22no—
73C2×C22⋊C4—21no—
74C2×C22⋊C4—22no—
75C2×C42—22no—
76C2×C42—21no—
77C2×C42C2311yescharacteristic
78C2×C42C2311yescharacteristic
79C2×C42C2311yescharacteristic
80C4⋊C8—22no—
81C4⋊C8—22no—
82C4⋊C8—24no—
83C4⋊C8—42no—
84#103—24no—
85#103—24no—
86#103—24no—
87#102—24no—
88#95—21no—
89#95—21no—
90#95—22no—
91#95—24no—
92#95D412yes—
93#95D412yes—
94C24—21no—
95C2×Q8—21no—
96C2×Q8—22no—
97C2×Q8—22no—
98C2×Q8—21no—
99C2×Q8—21no—
100C2×Q8—22no—
101C2×Q8—22no—
102C2×Q8—21no—
103C2×Q8—21no—
104C2×Q8—22no—
105C2×Q8C2×D412yes—
106C2×Q8C2×D414yes—
107C2×D4—21no—
108C2×D4—21no—
109C2×D4—21no—
110C2×D4—21no—
111C2×D4—21no—
112C2×D4C2×D412yes—
113C22×C4—21no—
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—42no—
119C22×C4—82no—
120C22×C4C2×D412yes—
121C22×C4C2×D411yescharacteristic
122C22×C4C2×D411yescharacteristic
123C22×C4C2×D412yes—
124C22×C4C2411yescharacteristic
125Q16—48no—
126SD16—48no—
127C2×C8—22no—
128C2×C8—22no—
129C2×C8—22no—
130C2×C8—22no—
131C2×C8—22no—
132C2×C8—22no—
133C4⋊C4—24no—
134C4⋊C4—24no—
135C4⋊C4—22no—
136C4⋊C4—21no—
137C4⋊C4—21no—
138C4⋊C4—21no—
139C4⋊C4—22no—
140C4⋊C4—21no—
141C4⋊C4—44no—
142C4⋊C4—42no—
143C4⋊C4—82no—
144C22⋊C4—28no—
145C22⋊C4—42no—
146C22⋊C4—82no—
147C42—21no—
148C42—21no—
149C42—44no—
150C42—42no—
151C42C2×D412yes—
152C42C2×D412yes—
153C42#4112yes—
154C42#4112yes—
155C23—22no—
156C23—41no—
157C23—41no—
158C23—42no—
159C23—42no—
160C23C22×D411yescharacteristic
161Q8—24no—
162Q8—22no—
163Q8—22no—
164Q8—24no—
165Q8—42no—
166Q8—41no—
167Q8—41no—
168Q8—42no—
169D4—22no—
170D4—22no—
171D4—41no—
172D4—41no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—24no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—41no—
183C2×C4—42no—
184C2×C4—41no—
185C2×C4—42no—
186C2×C4—41no—
187C2×C4—42no—
188C2×C4—41no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—84no—
194C2×C4#12114yes—
195C2×C4#12112yes—
196C2×C4#12112yes—
197C2×C4C22×D411yescharacteristic
198C2×C4C22×D411yescharacteristic
199C2×C4C2×#4111yescharacteristic
200C2×C4C2×#4111yescharacteristic
201C2×C4H(5,𝔽2)11yescharacteristic
202C2×C42−1+411yescharacteristic
203C8—42no—
204C8—42no—
205C22—44no—
206C22—42no—
207C22—82no—
208C22—81no—
209C22—81no—
210C22C2×#12111yescharacteristic
211C22C2×#12111yescharacteristic
212C22#53711yescharacteristic
213C22#57712yes—
214C22#57812yes—
215C4—22no—
216C4—22no—
217C4—24no—
218C4—42no—
219C4—42no—
220C4—44no—
221C4—81no—
222C4—81no—
223C4—82no—
224C4D4211yescharacteristic
225C4#54711yescharacteristic
226C4#54711yescharacteristic
227C4#55511yescharacteristic
228C2—82no—
229C2#253711yescharacteristic
230C2#304311yescharacteristic
231C2#304411yescharacteristic
232C2#304612yes—
233C2#325812yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#462C2semidirect
4. #2327C2semidirect
6. #2056C2semidirect
8. #1843C2semidirect
9. #1805C2semidirect
11. #1803C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4642
C2×#4622
C2×#4612
#23272
#23262
#20562
#20552
#18432
#18052
#18042
#18032