Finite Group Databasebeta

← #29774

Subgroups — #29774

Order 256.

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

normal subgroups only

This group has 448 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×#465C211yesmaximal, characteristic
2C22×C4⋊C8C211yesmaximal, characteristic, complemented
3#2855C211yesmaximal, characteristic, complemented
4#2854C211yesmaximal, characteristic
5#2088C212yesmaximal, complemented
6#2066C212yesmaximal, complemented
7#2065C212yesmaximal, complemented
8#2055C212yesmaximal, complemented
9#1893C211yesmaximal, characteristic
10#1892C211yesmaximal, characteristic, complemented
11#1807C211yesmaximal, characteristic, complemented
12C22×SD16C2211yescharacteristic
13C23×C8C2211yescharacteristic
14C23×C8C2211yescharacteristic
15C2×#123—21no—
16C2×C22⋊Q8C2211yescharacteristic
17C2×#121C2211yescharacteristic
18C22×C42C2211yescharacteristic
19#465—24no—
20C2×#106C2211yescharacteristic
21C2×#106C2212yes—
22C2×C4⋊C8—24no—
23C2×C4⋊C8C2212yes—
24C2×C4⋊C8C2212yes—
25C2×#103C2211yescharacteristic
26C2×#103C2212yes—
27C2×#102C2211yescharacteristic
28C2×#102C2212yes—
29#391C2212yes—
30#390C2212yes—
31#389C2212yes—
32#389C2212yes—
33#388—22no—
34#387—22no—
35#386—21no—
36#385—21no—
37#381—21no—
38#380C2211yescharacteristic
39#380C2211yescharacteristic
40#340C2212yes—
41#340C2212yes—
42#340C2212yes—
43C22×Q8C2311yescharacteristic
44C22×D4—21no—
45C22×D4C2311yescharacteristic
46C23×C4D411yescharacteristic
47C23×C4D411yescharacteristic
48C23×C4C2311yescharacteristic
49C2×SD16—22no—
50C2×SD16—42no—
51C22×C8—22no—
52C22×C8—42no—
53C22×C8—42no—
54C22×C8D414yes—
55C22×C8C2312yes—
56C22×C8C2311yescharacteristic
57C22×C8C2311yescharacteristic
58#123—44no—
59C22⋊Q8—22no—
60C22⋊Q8—22no—
61#121—22no—
62#121—22no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4C2311yescharacteristic
67C2×C4⋊C4C2311yescharacteristic
68C2×C4⋊C4C2312yes—
69C2×C4⋊C4C2312yes—
70C2×C22⋊C4—21no—
71C2×C22⋊C4—21no—
72C2×C22⋊C4—21no—
73C2×C22⋊C4—21no—
74C2×C22⋊C4—22no—
75C2×C22⋊C4—22no—
76C2×C42—22no—
77C2×C42—22no—
78C2×C42D412yes—
79C2×C42C2312yes—
80#106—22no—
81#106—42no—
82C4⋊C8—24no—
83C4⋊C8—24no—
84#103—22no—
85#103—42no—
86#102—22no—
87#102—42no—
88#95—21no—
89#95—21no—
90#95—22no—
91#95—22no—
92#95—22no—
93#95—22no—
94C24—21no—
95C24C2×D411yescharacteristic
96C2×Q8—21no—
97C2×Q8—21no—
98C2×Q8—22no—
99C2×Q8—22no—
100C2×D4—21no—
101C2×D4—21no—
102C2×D4—22no—
103C2×D4—22no—
104C2×D4—42no—
105C2×D4—42no—
106C2×D4—42no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—21no—
114C22×C4—22no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—42no—
118C22×C4SD1614yescomplemented
119C22×C4C2×D412yes—
120C22×C4C2×D411yescharacteristic
121C22×C4C2×D412yes—
122C22×C4C2×D412yes—
123C22×C4#4112yes—
124C22×C4C2411yescharacteristic
125SD16—44no—
126C2×C8—22no—
127C2×C8—22no—
128C2×C8—22no—
129C2×C8—22no—
130C2×C8—22no—
131C2×C8—44no—
132C2×C8—44no—
133C2×C8C2×D412yes—
134C2×C8C2×D412yes—
135C4⋊C4—21no—
136C4⋊C4—21no—
137C4⋊C4—21no—
138C4⋊C4—21no—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4—22no—
142C4⋊C4—22no—
143C4⋊C4—44no—
144C4⋊C4—42no—
145C4⋊C4—44no—
146C22⋊C4—44no—
147C22⋊C4—42no—
148C22⋊C4—42no—
149C22⋊C4—42no—
150C22⋊C4—42no—
151C22⋊C4—44no—
152C42—24no—
153C42—22no—
154C42—22no—
155C23—22no—
156C23—22no—
157C23—41no—
158C23—42no—
159C23—41no—
160C23—41no—
161C23—42no—
162C23#12112yes—
163C23C2×SD1612yes—
164C23#13512yes—
165C23C22×D411yescharacteristic
166Q8—41no—
167Q8—41no—
168Q8—42no—
169D4—41no—
170D4—41no—
171D4—42no—
172D4—84no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—28no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—24no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—44no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—41no—
192C2×C4—42no—
193C2×C4—41no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4#12112yes—
198C2×C4C2×SD1614yes—
199C2×C4C22×D411yescharacteristic
200C2×C4C2×#4111yescharacteristic
201C2×C4H(5,𝔽2)12yes—
202C2×C42−1+412yes—
203C8—22no—
204C8—22no—
205C8—42no—
206C22—22no—
207C22—24no—
208C22—24no—
209C22—22no—
210C22—81no—
211C22—82no—
212C22—82no—
213C22—81no—
214C22—81no—
215C22#46514yes—
216C22C2×#12111yescharacteristic
217C22#53712yes—
218C22C22×SD1611yescharacteristic
219C22C2×#13511yescharacteristic
220C22#57512yes—
221C4—22no—
222C4—24no—
223C4—42no—
224C4—81no—
225C4—82no—
226C4—82no—
227C4—81no—
228C4—81no—
229C4#53411yescharacteristic
230C4#53611yescharacteristic
231C4#53912yes—
232C2—24no—
233C2—161no—
234C2#261811yescharacteristic
235C2C2×#46511yescharacteristic
236C2#305411yescharacteristic
237C2#324612yes—
238C2#325012yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×C4⋊C8C2semidirect
3. #2855C2semidirect
5. #2088C2semidirect
6. #2066C2semidirect
7. #2065C2semidirect
8. #2055C2semidirect
10. #1892C2semidirect
11. #1807C2semidirect
118. C22×C4SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#4652
C22×C4⋊C82
#28552
#28542
#20882
#20662
#20652
#20552
#18932
#18922
#18072