Finite Group Databasebeta

← #30244

Subgroups — #30244

Order 256.

508 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 139 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×#467C212yesmaximal, complemented
2C2×#465C212yesmaximal
3#2857C211yesmaximal, characteristic, complemented
4#2850C211yesmaximal, characteristic
5#2097C212yesmaximal, complemented
6#2096C211yesmaximal, characteristic, complemented
7#2093C211yesmaximal, characteristic, complemented
8#2016C211yesmaximal, characteristic, complemented
9#2015C212yesmaximal, complemented
10#2013C211yesmaximal, characteristic, complemented
11C2×#336C211yesmaximal, characteristic, complemented
12C22×Q16C2212yes—
13C22×SD16C2212yes—
14C23×C8C2212yes—
15C2×C4⋊Q8C2211yescharacteristic
16C2×#124C2211yescharacteristic
17C2×C22⋊Q8C2211yescharacteristic
18C2×C22⋊Q8C2212yes—
19C2×#121C2211yescharacteristic
20C22×C42C2211yescharacteristic
21#467—28no—
22#465—28no—
23C2×#107C2212yes—
24C2×#106C2212yes—
25C2×#103C2212yes—
26C2×#103C2212yes—
27C2×#103C2212yes—
28C2×#102C2212yes—
29#391C2211yescharacteristic
30#391C2212yes—
31#390C2211yescharacteristic
32#389—22no—
33#388—22no—
34#386—22no—
35#384C2211yescharacteristic
36#382C2211yescharacteristic
37#380—22no—
38#336C2211yescharacteristic
39#336C2211yescharacteristic
40#336C2212yes—
41C22×Q8C2311yescharacteristic
42C22×Q8C2312yes—
43C22×D4—21no—
44C22×D4C2311yescharacteristic
45C23×C4D412yes—
46C23×C4C2311yescharacteristic
47C2×Q16—24no—
48C2×Q16—42no—
49C2×Q16—42no—
50C2×SD16—24no—
51C2×SD16—42no—
52C2×SD16—42no—
53C22×C8—42no—
54C22×C8—42no—
55C22×C8D418yes—
56C22×C8C2312yes—
57C22×C8C2312yes—
58C4⋊Q8—42no—
59#124—42no—
60C22⋊Q8—24no—
61C22⋊Q8—24no—
62C22⋊Q8—24no—
63#121—24no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4C2311yescharacteristic
69C2×C4⋊C4C2311yescharacteristic
70C2×C4⋊C4C2312yes—
71C2×C22⋊C4—22no—
72C2×C22⋊C4—21no—
73C2×C22⋊C4—21no—
74C2×C22⋊C4—22no—
75C2×C22⋊C4—22no—
76C2×C42—42no—
77C2×C42D412yes—
78C2×C42C2311yescharacteristic
79C2×C42C2311yescharacteristic
80#107—24no—
81#106—24no—
82#103—24no—
83#103—24no—
84#103—24no—
85#102—24no—
86#95—24no—
87#95—22no—
88#95—22no—
89C24—21no—
90C24C2×D411yescharacteristic
91C2×Q8—22no—
92C2×Q8—22no—
93C2×Q8—22no—
94C2×Q8—22no—
95C2×Q8—22no—
96C2×Q8—22no—
97C2×Q8—22no—
98C2×Q8—21no—
99C2×Q8—21no—
100C2×Q8—22no—
101C2×D4—22no—
102C2×D4—22no—
103C2×D4—21no—
104C2×D4—21no—
105C2×D4—42no—
106C2×D4—44no—
107C22×C4—22no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—24no—
111C22×C4—24no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—42no—
118C22×C4C2×D412yes—
119C22×C4C2×D411yescharacteristic
120C22×C4C2×D414yes—
121C22×C4C2×D412yes—
122C22×C4C2411yescharacteristic
123Q16—48no—
124SD16—48no—
125C2×C8—22no—
126C2×C8—22no—
127C2×C8—22no—
128C2×C8—22no—
129C2×C8—48no—
130C2×C8C2×D414yes—
131C2×C8C2×D414yes—
132C4⋊C4—22no—
133C4⋊C4—22no—
134C4⋊C4—22no—
135C4⋊C4—22no—
136C4⋊C4—44no—
137C4⋊C4—42no—
138C4⋊C4—44no—
139C4⋊C4—44no—
140C22⋊C4—44no—
141C22⋊C4—44no—
142C22⋊C4—42no—
143C22⋊C4—42no—
144C22⋊C4—44no—
145C42—22no—
146C42—22no—
147C42—42no—
148C23—21no—
149C23—21no—
150C23—22no—
151C23—41no—
152C23—41no—
153C23—41no—
154C23—42no—
155C23—42no—
156C23#12712yes—
157C23#13714yes—
158C23C22×D411yescharacteristic
159Q8—42no—
160Q8—42no—
161Q8—42no—
162Q8—42no—
163Q8—42no—
164Q8—42no—
165D4—42no—
166D4—42no—
167D4—84no—
168C2×C4—22no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—22no—
172C2×C4—24no—
173C2×C4—24no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—48no—
178C2×C4—42no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—42no—
183C2×C4—41no—
184C2×C4—42no—
185C2×C4—41no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—44no—
189C2×C4—42no—
190C2×C4#12712yes—
191C2×C4#13514yes—
192C2×C4C22×D412yes—
193C2×C4H(5,𝔽2)11yescharacteristic
194C2×C4H(5,𝔽2)12yes—
195C2×C42−1+411yescharacteristic
196C8—24no—
197C8—24no—
198C22—22no—
199C22—22no—
200C22—24no—
201C22—24no—
202C22—81no—
203C22—81no—
204C22—81no—
205C22—82no—
206C22—82no—
207C22#49714yes—
208C22C2×#12711yescharacteristic
209C22#53411yescharacteristic
210C22#53711yescharacteristic
211C22C2×#13512yes—
212C22C2×#13712yes—
213C4—22no—
214C4—24no—
215C4—42no—
216C4—81no—
217C4—82no—
218C4—81no—
219C4—81no—
220C4—82no—
221C4#53412yes—
222C4#53712yes—
223C2—24no—
224C2—161no—
225C2#265611yescharacteristic
226C2#314711yescharacteristic
227C2C2×#49711yescharacteristic
228C2#319012yes—
229C2#324812yes—
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
1. C2×#467C2semidirect
3. #2857C2semidirect
5. #2097C2semidirect
6. #2096C2semidirect
7. #2093C2semidirect
8. #2016C2semidirect
9. #2015C2semidirect
10. #2013C2semidirect
11. C2×#336C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4672
C2×#4652
#28572
#28502
#20972
#20962
#20932
#20162
#20152
#20132
C2×#3362