Finite Group Databasebeta

← #28168

Subgroups — #28168

Order 256.

433 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 433 conjugacy classes of subgroups — too many to draw. Only its 123 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×#462C211yesmaximal, characteristic, complemented
2C2×#461C211yesmaximal, characteristic
3#2327C211yesmaximal, characteristic, complemented
4#2326C211yesmaximal, characteristic
5#2072C212yesmaximal, complemented
6#2015C212yesmaximal
7#1880C212yesmaximal
8#1878C212yesmaximal, complemented
9#1832C211yesmaximal, characteristic, complemented
10#1805C211yesmaximal, characteristic, complemented
11#1804C211yesmaximal, characteristic
12C22×Q16C2212yes—
13C22×SD16C2212yes—
14C2×C4⋊Q8C2211yescharacteristic
15C2×#124C2211yescharacteristic
16C2×C4×Q8C2211yescharacteristic
17C2×C4×Q8C2212yes—
18C2×C4×D4C2211yescharacteristic
19#462—24no—
20#461—24no—
21C2×C4⋊C8C2211yescharacteristic
22C2×C4⋊C8C2211yescharacteristic
23C2×#103C2212yes—
24C2×#103C2212yes—
25C2×#103C2212yes—
26C2×#102C2212yes—
27#391C2211yescharacteristic
28#391C2212yes—
29#390C2211yescharacteristic
30#388—22no—
31#387—22no—
32#387—22no—
33#385—22no—
34#384C2211yescharacteristic
35#382C2211yescharacteristic
36#376C2211yescharacteristic
37#340C2212yes—
38#340C2212yes—
39#336C2212yes—
40#336C2212yes—
41C22×Q8C2312yes—
42C22×Q8C2311yescharacteristic
43C22×D4C2311yescharacteristic
44C23×C4—21no—
45C2×Q16—24no—
46C2×Q16—24no—
47C2×Q16—42no—
48C2×SD16—24no—
49C2×SD16—24no—
50C2×SD16—42no—
51C22×C8C2312yes—
52C22×C8C2312yes—
53C4⋊Q8—22no—
54C4⋊Q8D414yes—
55#124—22no—
56#124—22no—
57C4×Q8—22no—
58C4×Q8—22no—
59C4×Q8—42no—
60C4×Q8—42no—
61C4×D4—22no—
62C4×D4—22no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4C2311yescharacteristic
69C2×C4⋊C4C2312yes—
70C2×C4⋊C4C2311yescharacteristic
71C2×C22⋊C4—22no—
72C2×C22⋊C4—21no—
73C2×C22⋊C4—22no—
74C2×C42—22no—
75C2×C42—21no—
76C2×C42C2311yescharacteristic
77C2×C42C2311yescharacteristic
78C2×C42C2311yescharacteristic
79C4⋊C8—22no—
80C4⋊C8—22no—
81#103—24no—
82#103—24no—
83#103—42no—
84#102—42no—
85#95—21no—
86#95—21no—
87#95—22no—
88#95—24no—
89#95D412yes—
90#95D412yes—
91C24—21no—
92C2×Q8—21no—
93C2×Q8—22no—
94C2×Q8—21no—
95C2×Q8—21no—
96C2×Q8—22no—
97C2×Q8—22no—
98C2×Q8—21no—
99C2×Q8—22no—
100C2×Q8—22no—
101C2×Q8—21no—
102C2×Q8C2×D414yes—
103C2×Q8#4112yes—
104C2×D4—21no—
105C2×D4—21no—
106C2×D4—21no—
107C2×D4—21no—
108C2×D4—21no—
109C2×D4#4112yes—
110C22×C4—21no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—42no—
116C22×C4—82no—
117C22×C4C2×D412yes—
118C22×C4C2×D412yes—
119C22×C4C2×D411yescharacteristic
120C22×C4C2×D411yescharacteristic
121C22×C4C2411yescharacteristic
122Q16—48no—
123SD16—48no—
124C2×C8—22no—
125C2×C8—22no—
126C2×C8—22no—
127C2×C8—22no—
128C2×C8—22no—
129C2×C8—22no—
130C4⋊C4—24no—
131C4⋊C4—24no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—21no—
135C4⋊C4—21no—
136C4⋊C4—22no—
137C4⋊C4—22no—
138C4⋊C4—44no—
139C4⋊C4—42no—
140C4⋊C4—82no—
141C22⋊C4—28no—
142C22⋊C4—42no—
143C22⋊C4—82no—
144C42—21no—
145C42—21no—
146C42—21no—
147C42—21no—
148C42—44no—
149C42—42no—
150C42C2×D412yes—
151C42#4112yes—
152C23—22no—
153C23—41no—
154C23—41no—
155C23—42no—
156C23—42no—
157C23C22×D411yescharacteristic
158Q8—24no—
159Q8—22no—
160Q8—24no—
161Q8—22no—
162Q8—41no—
163Q8—41no—
164Q8—42no—
165Q8—42no—
166D4—22no—
167D4—22no—
168D4—41no—
169D4—41no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—24no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—41no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—84no—
192C2×C4#12112yes—
193C2×C4#12112yes—
194C2×C4C22×D411yescharacteristic
195C2×C4C2×#4111yescharacteristic
196C2×C4C2×#4111yescharacteristic
197C2×C4C2×#4111yescharacteristic
198C2×C4H(5,𝔽2)11yescharacteristic
199C2×C4H(5,𝔽2)11yescharacteristic
200C8—42no—
201C8—42no—
202C22—44no—
203C22—42no—
204C22—82no—
205C22—81no—
206C22—81no—
207C22C2×#12111yescharacteristic
208C22#52511yescharacteristic
209C22#53511yescharacteristic
210C22#57712yes—
211C22#57812yes—
212C4—22no—
213C4—22no—
214C4—42no—
215C4—42no—
216C4—44no—
217C4—42no—
218C4—82no—
219C4—81no—
220C4—81no—
221C4#54611yescharacteristic
222C4#54611yescharacteristic
223C4#55311yescharacteristic
224C4#55311yescharacteristic
225C2—82no—
226C2#253511yescharacteristic
227C2#304311yescharacteristic
228C2#304411yescharacteristic
229C2#312012yes—
230C2#322512yes—
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
1. C2×#462C2semidirect
3. #2327C2semidirect
5. #2072C2semidirect
8. #1878C2semidirect
9. #1832C2semidirect
10. #1805C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4622
C2×#4612
#23272
#23262
#20722
#20152
#18802
#18782
#18322
#18052
#18042