Finite Group Databasebeta

← #24299

Subgroups — #24299

Order 256.

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

normal subgroups only

This group has 689 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, complemented
2C2×#476C212yesmaximal, complemented
3C4×#121C211yesmaximal, characteristic
4#2058C212yesmaximal, complemented
5#2023C212yesmaximal, complemented
6#1987C212yesmaximal, complemented
7#1874C212yesmaximal
8#1853C212yesmaximal, complemented
9#1797C211yesmaximal, characteristic, complemented
10C23×Q8—21no—
11C2×C4⋊Q8—28no—
12C2×C4⋊Q8—21no—
13C2×C4⋊Q8—21no—
14C2×C4⋊Q8C2212yes—
15C2×C4⋊Q8C2212yes—
16C2×#121C2211yescharacteristic
17C2×C4×D4—21no—
18C2×C4×D4C2211yescharacteristic
19C22×C4⋊C4—21no—
20C22×C4⋊C4C2212yes—
21C22×C42C2211yescharacteristic
22#476—28no—
23C2×#106C2214yes—
24C2×C4⋊C8C2212yes—
25C2×#102C2214yes—
26C2×C22⋊C8C2212yes—
27#390C2211yescharacteristic
28#389C2211yescharacteristic
29#388—21no—
30#386—21no—
31#380C2211yescharacteristic
32C4×C4⋊C4C2211yescharacteristic
33C4×C22⋊C4—21no—
34#340C2214yes—
35#340C2214yes—
36#336C2212yes—
37C22×Q8—28no—
38C22×Q8—22no—
39C22×Q8—22no—
40C22×Q8—44no—
41C22×Q8—44no—
42C22×D4—21no—
43C22×D4C2311yescharacteristic
44C23×C4—22no—
45C23×C4—21no—
46C23×C4D411yescharacteristic
47C23×C4D411yescharacteristic
48C23×C4C2311yescharacteristic
49C22×C8C2314yes—
50C4⋊Q8—28no—
51C4⋊Q8—28no—
52C4⋊Q8—28no—
53C4⋊Q8—24no—
54C4⋊Q8D418yescomplemented
55#121—22no—
56#121Q814yescomplemented
57C4×D4—24no—
58C4×D4—44no—
59C2×C4⋊C4—28no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—28no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4C2311yescharacteristic
66C2×C4⋊C4C2311yescharacteristic
67C2×C4⋊C4C2314yes—
68C2×C4⋊C4C2311yescharacteristic
69C2×C22⋊C4—21no—
70C2×C22⋊C4—21no—
71C2×C22⋊C4—21no—
72C2×C22⋊C4—21no—
73C2×C22⋊C4—21no—
74C2×C42—24no—
75C2×C42—21no—
76C2×C42D411yescharacteristic
77C2×C42D411yescharacteristic
78C2×C42C2311yescharacteristic
79C2×C42C2311yescharacteristic
80#106—28no—
81C4⋊C8—24no—
82#102—28no—
83C22⋊C8—24no—
84#95—21no—
85#95—21no—
86#95—21no—
87C24—21no—
88C24C2×D411yescharacteristic
89C2×Q8—28no—
90C2×Q8—28no—
91C2×Q8—432no—
92C2×Q8—44no—
93C2×Q8—44no—
94C2×Q8—44no—
95C2×Q8—44no—
96C2×D4—24no—
97C2×D4—22no—
98C2×D4—21no—
99C2×D4—22no—
100C2×D4—42no—
101C2×D4—42no—
102C2×D4C2×Q812yescomplemented
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—28no—
111C22×C4—21no—
112C22×C4—24no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—44no—
116C22×C4—44no—
117C22×C4—42no—
118C22×C4—42no—
119C22×C4—42no—
120C22×C4SD1614yes—
121C22×C4C2×D411yescharacteristic
122C22×C4C2×D411yescharacteristic
123C22×C4C2×D411yescharacteristic
124C22×C4C2×D411yescharacteristic
125C22×C4C2×D411yescharacteristic
126C22×C4C2×Q812yes—
127C22×C4#4112yes—
128C22×C4C2411yescharacteristic
129C2×C8—24no—
130C2×C8—24no—
131C2×C8—24no—
132C4⋊C4—28no—
133C4⋊C4—28no—
134C4⋊C4—28no—
135C4⋊C4—24no—
136C4⋊C4—21no—
137C4⋊C4—22no—
138C4⋊C4—22no—
139C4⋊C4—42no—
140C4⋊C4C2×D418yes—
141C4⋊C4C2×Q812yescomplemented
142C22⋊C4—24no—
143C22⋊C4—42no—
144C22⋊C4—42no—
145C22⋊C4—42no—
146C22⋊C4—42no—
147C42—22no—
148C42—22no—
149C42—22no—
150C42—42no—
151C42C2×D414yes—
152C23—21no—
153C23—21no—
154C23—21no—
155C23—21no—
156C23—22no—
157C23—41no—
158C23—41no—
159C23—41no—
160C23—41no—
161C23—41no—
162C23—41no—
163C23C22⋊Q812yes—
164C23C2×SD1612yes—
165C23AGL(1,ℤ/8)12yes—
166C23C22×D411yescharacteristic
167Q8—416no—
168Q8—416no—
169D4—24no—
170D4—48no—
171D4—42no—
172C2×C4—28no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—24no—
182C2×C4—24no—
183C2×C4—28no—
184C2×C4—28no—
185C2×C4—22no—
186C2×C4—416no—
187C2×C4—41no—
188C2×C4—44no—
189C2×C4—44no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—84no—
198C2×C4#12012yes—
199C2×C4#12012yes—
200C2×C4C22⋊Q812yes—
201C2×C4C2×SD1612yes—
202C2×C4C2×SD1612yes—
203C2×C4AGL(1,ℤ/8)12yes—
204C2×C4#13712yes—
205C2×C4C22×D412yes—
206C2×C4C22×Q811yescharacteristic
207C2×C4C2×#4111yescharacteristic
208C2×C42−1+412yes—
209C8—44no—
210C22—22no—
211C22—22no—
212C22—22no—
213C22—22no—
214C22—22no—
215C22—22no—
216C22—42no—
217C22—42no—
218C22—42no—
219C22—81no—
220C22—81no—
221C22—81no—
222C22—81no—
223C22#47614yescomplemented
224C22C2×#12011yescharacteristic
225C22C2×C22⋊Q811yescharacteristic
226C22#53611yescharacteristic
227C22C22×SD1611yescharacteristic
228C22C2×AGL(1,ℤ/8)11yescharacteristic
229C22C2×AGL(1,ℤ/8)11yescharacteristic
230C22C2×#13711yescharacteristic
231C4—22no—
232C4—22no—
233C4—22no—
234C4—22no—
235C4—48no—
236C4—42no—
237C4—81no—
238C4—81no—
239C4#44814yes—
240C4#54812yes—
241C4D4×Q812yes—
242C2—24no—
243C2—82no—
244C2#240911yescharacteristic
245C2C2×#44811yescharacteristic
246C2#300311yescharacteristic
247C2C2×#47611yescharacteristic
248C2#307911yescharacteristic
249C2#322711yescharacteristic
250C2#323811yescharacteristic
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. C22×C4⋊Q8C2semidirect
2. C2×#476C2semidirect
4. #2058C2semidirect
5. #2023C2semidirect
6. #1987C2semidirect
8. #1853C2semidirect
9. #1797C2semidirect
54. C4⋊Q8D4semidirect
56. #121Q8semidirect
102. C2×D4C2×Q8semidirect
141. C4⋊C4C2×Q8semidirect
223. C22#476semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊Q82
C2×#4762
C4×#1212
#20582
#20232
#19872
#18742
#18532
#17972