Finite Group Databasebeta

← #29116

Subgroups — #29116

Order 256.

481 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 481 conjugacy classes of subgroups — too many to draw. Only its 115 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
1#2628C211yesmaximal, characteristic, complemented
2#2309C211yesmaximal, characteristic
3#2073C212yesmaximal, complemented
4#2023C212yesmaximal, complemented
5#1936C212yesmaximal, complemented
6#1934C212yesmaximal, complemented
7#1825C212yesmaximal, complemented
8#1786C212yesmaximal
9#1750C211yesmaximal, characteristic, complemented
10C23×D4C2211yescharacteristic
11C2×C4⋊Q8C2211yescharacteristic, complemented
12C2×#117C2211yescharacteristic
13C2×#117C2211yescharacteristic
14C2×#102C2212yes—
15C2×#102C2212yes—
16C2×#102C2212yes—
17C2×#102C2212yes—
18C2×C22⋊C8C2212yes—
19C2×#97C2211yescharacteristic, complemented
20C2×C4×C8C2211yescharacteristic, complemented
21#400—21no—
22#399—21no—
23#395—21no—
24#393—21no—
25#390—21no—
26#390—21no—
27#390C2212yes—
28#390C2212yes—
29#389C2212yescomplemented
30#382C2211yescharacteristic, complemented
31#380C2212yes—
32#379—21no—
33#379—21no—
34#340C2212yescomplemented
35#340C2212yescomplemented
36#340C2212yescomplemented
37#340C2212yescomplemented
38#336C2212yescomplemented
39C25—21no—
40C22×Q8—21no—
41C22×D4—24no—
42C22×D4—21no—
43C22×D4—24no—
44C22×D4—41no—
45C22×D4—41no—
46C22×D4C2312yes—
47C23×C4C2311yescharacteristic
48C22×C8C2312yes—
49C22×C8C2312yes—
50C4⋊Q8—41no—
51C4⋊Q8—41no—
52#117—24no—
53#117—41no—
54#117—41no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4D411yescharacteristic
58C2×C4⋊C4D411yescharacteristic
59C2×C4⋊C4C2312yes—
60C2×C4⋊C4C2312yes—
61C2×C4⋊C4C2312yes—
62C2×C22⋊C4—21no—
63C2×C22⋊C4—22no—
64C2×C22⋊C4—22no—
65C2×C22⋊C4—22no—
66C2×C22⋊C4—22no—
67C2×C22⋊C4—22no—
68C2×C22⋊C4D411yescharacteristic
69C2×C22⋊C4D411yescharacteristic
70C2×C42—21no—
71C2×C42C2311yescharacteristic
72C2×C42C2311yescharacteristic
73#102—24no—
74#102—24no—
75#102—42no—
76#102—42no—
77C22⋊C8—24no—
78#97—41no—
79C4×C8—41no—
80#95—22no—
81#95—21no—
82#95—21no—
83#95—22no—
84C24—24no—
85C24—22no—
86C24—42no—
87C24—42no—
88C24—82no—
89C24—82no—
90C24C2×D411yescharacteristic
91C2×Q8—42no—
92C2×Q8—82no—
93C2×D4—24no—
94C2×D4—22no—
95C2×D4—22no—
96C2×D4—22no—
97C2×D4—22no—
98C2×D4—22no—
99C2×D4—48no—
100C2×D4—48no—
101C2×D4—42no—
102C2×D4—42no—
103C2×D4#4114yes—
104C22×C4—22no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4C2×D411yescharacteristic
111C22×C4C2×D411yescharacteristic
112C22×C4C2×D411yescharacteristic
113C22×C4C2×D411yescharacteristic
114C22×C4C2×D411yescharacteristic
115C22×C4#4114yes—
116C22×C4C2411yescharacteristic
117C2×C8—22no—
118C2×C8—22no—
119C2×C8—22no—
120C2×C8—22no—
121C2×C8—22no—
122C2×C8—22no—
123C4⋊C4—22no—
124C4⋊C4—22no—
125C4⋊C4—24no—
126C4⋊C4—22no—
127C4⋊C4—22no—
128C4⋊C4—41no—
129C4⋊C4—41no—
130C4⋊C4—42no—
131C4⋊C4—82no—
132C22⋊C4—41no—
133C22⋊C4—41no—
134C22⋊C4—42no—
135C22⋊C4—82no—
136C22⋊C4—82no—
137C22⋊C4—82no—
138C22⋊C4—82no—
139C22⋊C4—82no—
140C42—21no—
141C42—21no—
142C42—21no—
143C42—21no—
144C42—42no—
145C23—21no—
146C23—21no—
147C23—22no—
148C23—22no—
149C23—24no—
150C23—48no—
151C23—42no—
152C23—42no—
153C23—42no—
154C23—42no—
155C23—42no—
156C23—42no—
157C23—88no—
158C23—88no—
159C23—84no—
160C23—84no—
161C23#12412yes—
162C23C22×D411yescharacteristic
163Q8—84no—
164D4—24no—
165D4—24no—
166D4—48no—
167D4—42no—
168D4—42no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—24no—
172C2×C4—24no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4#12412yes—
191C2×C4C2×#4112yes—
192C2×C4C2×#4112yes—
193C2×C4H(5,𝔽2)11yescharacteristic
194C2×C4H(5,𝔽2)11yescharacteristic
195C8—42no—
196C8—42no—
197C22—22no—
198C22—24no—
199C22—42no—
200C22—41no—
201C22—41no—
202C22—44no—
203C22—44no—
204C22—44no—
205C22—816no—
206C22—82no—
207C22—82no—
208C22—82no—
209C22—82no—
210C22#52511yescharacteristic
211C22C2×#12411yescharacteristic
212C22#53411yescharacteristic
213C22#57612yes—
214C22#57712yes—
215C4—22no—
216C4—41no—
217C4—41no—
218C4—41no—
219C4—41no—
220C4—82no—
221C4—82no—
222C4—82no—
223C4#55112yes—
224C4#55112yes—
225C2—42no—
226C2—84no—
227C2#256111yescharacteristic
228C2#311812yes—
229C2#313612yes—
230C2#319911yescharacteristic
231C2#320011yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC23×D4C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2628C2semidirect
3. #2073C2semidirect
4. #2023C2semidirect
5. #1936C2semidirect
6. #1934C2semidirect
7. #1825C2semidirect
9. #1750C2semidirect
11. C2×C4⋊Q8C22semidirect
19. C2×#97C22semidirect
20. C2×C4×C8C22semidirect
29. #389C22semidirect
30. #382C22semidirect
34. #340C22semidirect
35. #340C22semidirect
36. #340C22semidirect
37. #340C22semidirect
38. #336C22semidirect

Maximal subgroups

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

subgroupindex
#26282
#23092
#20732
#20232
#19362
#19342
#18252
#17862
#17502