Finite Group Databasebeta

← #24212

Subgroups — #24212

Order 256.

715 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 715 conjugacy classes of subgroups — too many to draw. Only its 167 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
1C22×C4⋊Q8C211yesmaximal, characteristic
2C2×#462C212yesmaximal, complemented
3C2×#451C212yesmaximal
4C4×C22⋊Q8C211yesmaximal, characteristic
5#2043C212yesmaximal, complemented
6#2016C212yesmaximal, complemented
7#1981C212yesmaximal, complemented
8#1797C211yesmaximal, characteristic
9#1787C212yesmaximal
10C23×Q8C2212yes—
11C22×Q16C2214yes—
12C2×C4⋊Q8—28no—
13C2×C4⋊Q8—21no—
14C2×C4⋊Q8—21no—
15C2×C4⋊Q8C2212yes—
16C2×C4⋊Q8C2212yes—
17C2×C22⋊Q8C2211yescharacteristic
18C2×C4×Q8C2211yescharacteristic
19C22×C4⋊C4—21no—
20C22×C4⋊C4—21no—
21C22×C42C2211yescharacteristic
22#462—28no—
23#451—28no—
24C2×C4⋊C8C2212yes—
25C2×#103C2214yes—
26C2×#103C2214yes—
27C2×#103C2214yes—
28C2×C22⋊C8C2212yes—
29#391C2211yescharacteristic
30#389C2211yescharacteristic
31#387—21no—
32#385—21no—
33#380C2211yescharacteristic
34C4×C4⋊C4—21no—
35C4×C4⋊C4C2211yescharacteristic
36C4×C22⋊C4—21no—
37#336C2212yes—
38C22×Q8—28no—
39C22×Q8—22no—
40C22×Q8—24no—
41C22×Q8—24no—
42C22×Q8—44no—
43C22×Q8C2311yescharacteristic
44C22×Q8C2314yes—
45C23×C4—22no—
46C23×C4D411yescharacteristic
47C23×C4D411yescharacteristic
48C23×C4C2311yescharacteristic
49C2×Q16—28no—
50C2×Q16—28no—
51C2×Q16—44no—
52C22×C8C2314yes—
53C4⋊Q8—28no—
54C4⋊Q8—28no—
55C4⋊Q8—28no—
56C4⋊Q8—24no—
57C4⋊Q8D418yes—
58C22⋊Q8—22no—
59C22⋊Q8D414yes—
60C4×Q8—24no—
61C2×C4⋊C4—28no—
62C2×C4⋊C4—28no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4C2311yescharacteristic
71C2×C4⋊C4C2311yescharacteristic
72C2×C4⋊C4C2311yescharacteristic
73C2×C22⋊C4—21no—
74C2×C22⋊C4—21no—
75C2×C42—24no—
76C2×C42—21no—
77C2×C42—21no—
78C2×C42D411yescharacteristic
79C2×C42D411yescharacteristic
80C2×C42C2311yescharacteristic
81C2×C42C2311yescharacteristic
82C4⋊C8—24no—
83#103—28no—
84#103—28no—
85#103—44no—
86C22⋊C8—24no—
87#95—21no—
88#95—21no—
89#95—21no—
90#95—21no—
91#95—21no—
92C24C2×D411yescharacteristic
93C2×Q8—28no—
94C2×Q8—24no—
95C2×Q8—21no—
96C2×Q8—24no—
97C2×Q8—22no—
98C2×Q8—24no—
99C2×Q8—24no—
100C2×Q8—22no—
101C2×Q8—24no—
102C2×Q8—416no—
103C2×Q8—416no—
104C2×Q8—44no—
105C2×Q8—44no—
106C2×Q8C2×D418yes—
107C2×Q8C2×D412yes—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—28no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—24no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—44no—
122C22×C4—44no—
123C22×C4Q1614yescomplemented
124C22×C4C2×D411yescharacteristic
125C22×C4C2×D411yescharacteristic
126C22×C4C2×D412yes—
127C22×C4C2×D411yescharacteristic
128C22×C4C2×D411yescharacteristic
129C22×C4C2×D411yescharacteristic
130C22×C4#4112yes—
131C22×C4C2411yescharacteristic
132Q16—416no—
133C2×C8—24no—
134C2×C8—24no—
135C2×C8—24no—
136C4⋊C4—28no—
137C4⋊C4—28no—
138C4⋊C4—28no—
139C4⋊C4—28no—
140C4⋊C4—24no—
141C4⋊C4—21no—
142C4⋊C4—22no—
143C4⋊C4—22no—
144C4⋊C4—42no—
145C4⋊C4—42no—
146C4⋊C4C2×D412yes—
147C22⋊C4—24no—
148C22⋊C4—42no—
149C42—22no—
150C42—22no—
151C42—22no—
152C42—42no—
153C42—42no—
154C42C2×D414yes—
155C23—21no—
156C23—21no—
157C23—21no—
158C23—21no—
159C23#12112yes—
160C23C2×Q1612yes—
161C23#13712yes—
162C23C22×D411yescharacteristic
163Q8—24no—
164Q8—28no—
165Q8—28no—
166Q8—416no—
167Q8—44no—
168Q8—42no—
169Q8—44no—
170C2×C4—28no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—24no—
180C2×C4—24no—
181C2×C4—28no—
182C2×C4—28no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—416no—
186C2×C4—41no—
187C2×C4—44no—
188C2×C4—44no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4#12012yes—
199C2×C4#12012yes—
200C2×C4#12112yes—
201C2×C4C2×Q1612yes—
202C2×C4C2×Q1612yes—
203C2×C4#13712yes—
204C2×C4#13712yes—
205C2×C4C22×D411yescharacteristic
206C2×C4C22×D412yes—
207C2×C4C2×#4111yescharacteristic
208C2×C4H(5,𝔽2)12yes—
209C8—44no—
210C22—22no—
211C22—22no—
212C22—22no—
213C22—22no—
214C22—22no—
215C22—22no—
216C22#46214yescomplemented
217C22C2×#12011yescharacteristic
218C22C2×#12111yescharacteristic
219C22#53511yescharacteristic
220C22C22×Q1611yescharacteristic
221C22C2×#13711yescharacteristic
222C22C2×#13711yescharacteristic
223C22C2×#13711yescharacteristic
224C4—22no—
225C4—22no—
226C4—22no—
227C4—22no—
228C4—48no—
229C4—42no—
230C4—42no—
231C4—81no—
232C4—81no—
233C4#45114yescomplemented
234C4D4212yes—
235C4#54612yes—
236C2—24no—
237C2#240811yescharacteristic
238C2C2×#45111yescharacteristic
239C2#300411yescharacteristic
240C2C2×#46211yescharacteristic
241C2#303911yescharacteristic
242C2#320911yescharacteristic
243C2#321811yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
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. C2×#462C2semidirect
5. #2043C2semidirect
6. #2016C2semidirect
7. #1981C2semidirect
123. C22×C4Q16semidirect
216. C22#462semidirect
233. C4#451semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊Q82
C2×#4622
C2×#4512
C4×C22⋊Q82
#20432
#20162
#19812
#17972
#17872