Finite Group Databasebeta

← #27776

Subgroups — #27776

Order 256.

421 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 421 conjugacy classes of subgroups — too many to draw. Only its 131 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×#489C212yesmaximal
2#2551C211yesmaximal, characteristic, complemented
3#2547C211yesmaximal, characteristic
4#2042C212yesmaximal
5#2024C212yesmaximal, complemented
6#1981C211yesmaximal, characteristic, complemented
7#1956C211yesmaximal, characteristic, complemented
8#1935C212yesmaximal, complemented
9#1802C211yesmaximal, characteristic
10#1770C211yesmaximal, characteristic, complemented
11C4×#102C211yesmaximal, characteristic
12C2×C4⋊Q8—21no—
13C2×C4⋊Q8C2211yescharacteristic
14C2×C4⋊Q8C2212yes—
15C2×#124C2212yes—
16C2×C4×D4C2211yescharacteristic
17#489—28no—
18C2×C4⋊C8C2211yescharacteristic
19C2×#103C2212yes—
20C2×#103C2212yes—
21C2×#103C2212yes—
22C2×#103C2212yes—
23C2×#102C2212yes—
24C2×#102C2211yescharacteristic
25C2×#102C2211yescharacteristic
26C2×#97C2212yes—
27C2×C4×C8C2211yescharacteristic
28#397—22no—
29#395—22no—
30#391C2212yes—
31#389—21no—
32#389C2212yes—
33#384C2211yescharacteristic
34#380—21no—
35#380—21no—
36C4×C4⋊C4C2211yescharacteristic
37C4×C4⋊C4C2211yescharacteristic
38#340C2212yes—
39#340C2211yescharacteristic
40#340C2211yescharacteristic
41#336C2212yes—
42C22×Q8—22no—
43C22×Q8C2312yes—
44C22×D4C2311yescharacteristic
45C23×C4—21no—
46C22×C8C2311yescharacteristic
47C22×C8C2312yes—
48C22×C8C2311yescharacteristic
49C4⋊Q8—24no—
50C4⋊Q8—22no—
51C4⋊Q8—22no—
52C4⋊Q8—42no—
53C4⋊Q8—44no—
54#124—24no—
55#124—42no—
56C4×D4—22no—
57C4×D4—22no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4D411yescharacteristic
65C2×C4⋊C4D411yescharacteristic
66C2×C4⋊C4C2311yescharacteristic
67C2×C4⋊C4C2312yes—
68C2×C4⋊C4C2311yescharacteristic
69C2×C4⋊C4C2311yescharacteristic
70C2×C22⋊C4—21no—
71C2×C22⋊C4—22no—
72C2×C22⋊C4—22no—
73C2×C42—21no—
74C2×C42C2312yes—
75C2×C42C2311yescharacteristic
76C4⋊C8—22no—
77#103—24no—
78#103—24no—
79#103—24no—
80#103—42no—
81#102—24no—
82#102—22no—
83#102D414yes—
84#97—24no—
85C4×C8—22no—
86#95—21no—
87#95—22no—
88#95—22no—
89#95—22no—
90#95D412yes—
91C24—21no—
92C2×Q8—22no—
93C2×Q8—22no—
94C2×Q8—22no—
95C2×Q8—22no—
96C2×Q8—22no—
97C2×Q8—22no—
98C2×Q8—44no—
99C2×Q8—44no—
100C2×Q8—82no—
101C2×D4—21no—
102C2×D4—21no—
103C2×D4—21no—
104C2×D4—21no—
105C2×D4—21no—
106C2×D4C2×D412yes—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—42no—
113C22×C4—42no—
114C22×C4—42no—
115C22×C4C2×D412yes—
116C22×C4C2×D412yes—
117C22×C4C2×D411yescharacteristic
118C22×C4C2×D411yescharacteristic
119C22×C4C2411yescharacteristic
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—22no—
123C2×C8—21no—
124C2×C8—21no—
125C2×C8—22no—
126C2×C8—22no—
127C2×C8—21no—
128C2×C8C2×D412yes—
129C4⋊C4—22no—
130C4⋊C4—22no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—22no—
134C4⋊C4—21no—
135C4⋊C4—21no—
136C4⋊C4—22no—
137C4⋊C4—44no—
138C4⋊C4—44no—
139C4⋊C4—42no—
140C4⋊C4—42no—
141C4⋊C4—42no—
142C4⋊C4—81no—
143C4⋊C4C2×D412yes—
144C4⋊C4#4112yes—
145C22⋊C4—44no—
146C22⋊C4—44no—
147C22⋊C4—42no—
148C42—22no—
149C42—22no—
150C42—42no—
151C42#4112yes—
152C42#4112yes—
153C23—22no—
154C23—41no—
155C23—41no—
156C23—41no—
157C23—41no—
158C23—41no—
159C23—41no—
160C23C22×D411yescharacteristic
161Q8—42no—
162Q8—42no—
163Q8—42no—
164Q8—42no—
165Q8—88no—
166D4—22no—
167D4—22no—
168D4—41no—
169D4—41no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—22no—
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—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—42no—
193C2×C4—84no—
194C2×C4#12412yes—
195C2×C4#12412yes—
196C2×C4AGL(1,ℤ/8)12yes—
197C2×C4#13712yes—
198C2×C4C22×D411yescharacteristic
199C2×C4C2×#4111yescharacteristic
200C2×C4C2×#4111yescharacteristic
201C2×C4C2×#4111yescharacteristic
202C2×C4H(5,𝔽2)11yescharacteristic
203C2×C42−1+411yescharacteristic
204C8—22no—
205C8—42no—
206C8—41no—
207C22—42no—
208C22—42no—
209C22—42no—
210C22—81no—
211C22—81no—
212C22—81no—
213C22—81no—
214C22C2×#12411yescharacteristic
215C22D4211yescharacteristic
216C22#54811yescharacteristic
217C22C2×AGL(1,ℤ/8)11yescharacteristic
218C22C2×#13711yescharacteristic
219C22#57711yescharacteristic
220C22#57811yescharacteristic
221C4—22no—
222C4—42no—
223C4—42no—
224C4—42no—
225C4—42no—
226C4—81no—
227C4—82no—
228C4—82no—
229C4#48914yescomplemented
230C4#53211yescharacteristic
231C4#54211yescharacteristic
232C4#54711yescharacteristic
233C4#55511yescharacteristic
234C2—82no—
235C2#249111yescharacteristic
236C2C2×#48911yescharacteristic
237C2#313711yescharacteristic
238C2#327211yescharacteristic
239C2#327611yescharacteristic
240C2#336711yescharacteristic
241C2#336911yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
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
2. #2551C2semidirect
5. #2024C2semidirect
6. #1981C2semidirect
7. #1956C2semidirect
8. #1935C2semidirect
10. #1770C2semidirect
229. C4#489semidirect

Maximal subgroups

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

subgroupindex
C2×#4892
#25512
#25472
#20422
#20242
#19812
#19562
#19352
#18022
#17702
C4×#1022