Finite Group Databasebeta

← #26464

Subgroups — #26464

Order 256.

475 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 475 conjugacy classes of subgroups — too many to draw. Only its 147 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×#476C212yesmaximal, complemented
2#2551C211yesmaximal, characteristic, complemented
3#2366C211yesmaximal, characteristic
4#2065C212yesmaximal, complemented
5#2023C212yesmaximal, complemented
6#1982C211yesmaximal, characteristic
7#1957C211yesmaximal, characteristic
8#1948C211yesmaximal, characteristic, complemented
9#1939C211yesmaximal, characteristic, complemented
10#1864C212yesmaximal, complemented
11C4×#106C211yesmaximal, characteristic, complemented
12C2×C4⋊Q8—21no—
13C2×C4⋊Q8C2212yes—
14C2×#127C2211yescharacteristic
15C2×C4×D4—21no—
16C2×C4×D4C2212yes—
17#476—28no—
18C2×#106C2211yescharacteristic
19C2×#106C2212yes—
20C2×#106C2211yescharacteristic
21C2×C4⋊C8C2212yes—
22C2×C4⋊C8C2211yescharacteristic
23C2×#102C2212yes—
24C2×#102C2212yes—
25C2×#102C2212yes—
26C2×#102C2212yes—
27C2×C4×C8C2211yescharacteristic
28#395—22no—
29#390—21no—
30#390C2212yes—
31#389—21no—
32#389C2212yes—
33#386—22no—
34#384C2211yescharacteristic
35C4×C4⋊C4C2211yescharacteristic
36C4×C4⋊C4C2211yescharacteristic
37C4×C4⋊C4C2211yescharacteristic
38#340C2212yes—
39#340C2211yescharacteristic
40#340C2211yescharacteristic
41#336C2212yes—
42C22×Q8—22no—
43C22×D4—21no—
44C22×D4—21no—
45C22×D4C2312yes—
46C23×C4—22no—
47C22×C8C2311yescharacteristic
48C22×C8C2312yes—
49C22×C8C2311yescharacteristic
50C4⋊Q8—24no—
51C4⋊Q8—42no—
52C4⋊Q8—44no—
53#127—22no—
54#127Q814yescomplemented
55C4×D4—24no—
56C4×D4—42no—
57C4×D4—44no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4D411yescharacteristic
63C2×C4⋊C4D411yescharacteristic
64C2×C4⋊C4C2312yes—
65C2×C4⋊C4C2311yescharacteristic
66C2×C4⋊C4C2311yescharacteristic
67C2×C4⋊C4C2311yescharacteristic
68C2×C4⋊C4C2311yescharacteristic
69C2×C22⋊C4—22no—
70C2×C22⋊C4—22no—
71C2×C22⋊C4—22no—
72C2×C42—21no—
73C2×C42—21no—
74C2×C42C2312yes—
75C2×C42C2311yescharacteristic
76#106—24no—
77#106—22no—
78#106D414yescomplemented
79C4⋊C8—24no—
80C4⋊C8—22no—
81#102—24no—
82#102—24no—
83#102—24no—
84#102—42no—
85C4×C8—22no—
86#95—21no—
87#95—21no—
88#95—22no—
89#95D412yes—
90C24—22no—
91C2×Q8—44no—
92C2×Q8—44no—
93C2×Q8—82no—
94C2×D4—24no—
95C2×D4—24no—
96C2×D4—22no—
97C2×D4—22no—
98C2×D4—22no—
99C2×D4—22no—
100C2×D4—22no—
101C2×D4—42no—
102C2×D4—42no—
103C2×D4—81no—
104C2×D4—81no—
105C2×D4C2×Q814yes—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—22no—
111C22×C4—44no—
112C22×C4—44no—
113C22×C4—82no—
114C22×C4C2×D412yes—
115C22×C4C2×D411yescharacteristic
116C22×C4C2×D412yes—
117C22×C4C2×D411yescharacteristic
118C22×C4C2411yescharacteristic
119C2×C8—21no—
120C2×C8—21no—
121C2×C8—22no—
122C2×C8—21no—
123C2×C8—22no—
124C2×C8—22no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8C2×D412yes—
128C4⋊C4—22no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—22no—
133C4⋊C4—21no—
134C4⋊C4—22no—
135C4⋊C4—21no—
136C4⋊C4—44no—
137C4⋊C4—44no—
138C4⋊C4—42no—
139C4⋊C4—81no—
140C4⋊C4SD1614yes—
141C4⋊C4C2×D412yes—
142C4⋊C4C2×D412yes—
143C4⋊C4#4112yes—
144C22⋊C4—44no—
145C22⋊C4—44no—
146C22⋊C4—82no—
147C42—22no—
148C42—22no—
149C42—42no—
150C42—42no—
151C42C2×Q812yes—
152C42#4112yes—
153C23—24no—
154C23—42no—
155C23—42no—
156C23—42no—
157C23—42no—
158C23—42no—
159C23—42no—
160C23C22×D411yescharacteristic
161Q8—88no—
162D4—24no—
163D4—24no—
164D4—42no—
165D4—42no—
166D4—48no—
167D4—84no—
168C2×C4—22no—
169C2×C4—21no—
170C2×C4—22no—
171C2×C4—21no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—42no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—42no—
188C2×C4—41no—
189C2×C4—42no—
190C2×C4—88no—
191C2×C4C22⋊Q812yes—
192C2×C4C22⋊Q812yes—
193C2×C4C2×SD1614yes—
194C2×C4C2×SD1612yes—
195C2×C4AGL(1,ℤ/8)12yes—
196C2×C4C22×D411yescharacteristic
197C2×C4C22×Q811yescharacteristic
198C2×C4C2×#4111yescharacteristic
199C2×C4C2×#4111yescharacteristic
200C2×C4H(5,𝔽2)11yescharacteristic
201C2×C42−1+411yescharacteristic
202C8—22no—
203C8—42no—
204C8—41no—
205C22—44no—
206C22—44no—
207C22—44no—
208C22—82no—
209C22—82no—
210C22—82no—
211C22—82no—
212C22C2×C22⋊Q811yescharacteristic
213C22D4211yescharacteristic
214C22#54811yescharacteristic
215C22C22×SD1611yescharacteristic
216C22C2×AGL(1,ℤ/8)11yescharacteristic
217C22#57611yescharacteristic
218C22#57711yescharacteristic
219C4—22no—
220C4—24no—
221C4—42no—
222C4—42no—
223C4—42no—
224C4—42no—
225C4—81no—
226C4—82no—
227C4#47614yescomplemented
228C4#53211yescharacteristic
229C4#54211yescharacteristic
230C4D4×Q811yescharacteristic
231C4#55411yescharacteristic
232C2—84no—
233C2#245211yescharacteristic
234C2C2×#47611yescharacteristic
235C2#308111yescharacteristic
236C2#327411yescharacteristic
237C2D4×SD1611yescharacteristic
238C2#336211yescharacteristic
239C2#337311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#476C2semidirect
2. #2551C2semidirect
4. #2065C2semidirect
5. #2023C2semidirect
8. #1948C2semidirect
9. #1939C2semidirect
10. #1864C2semidirect
11. C4×#106C2semidirect
54. #127Q8semidirect
78. #106D4semidirect
227. C4#476semidirect

Maximal subgroups

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

subgroupindex
C2×#4762
#25512
#23662
#20652
#20232
#19822
#19572
#19482
#19392
#18642
C4×#1062