Finite Group Databasebeta

← #27685

Subgroups — #27685

Order 256.

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

normal subgroups only

This group has 385 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
1C4×C22⋊Q8C211yesmaximal, characteristic
2#2285C211yesmaximal, characteristic
3#2100C211yesmaximal, characteristic, complemented
4#2099C211yesmaximal, characteristic, complemented
5#2096C211yesmaximal, characteristic, complemented
6#2095C211yesmaximal, characteristic, complemented
7#1949C212yesmaximal, complemented
8#1873C212yesmaximal
9#1854C212yesmaximal
10#1805C212yesmaximal, complemented
11#1797C211yesmaximal, characteristic
12C2×C22⋊Q8C2211yescharacteristic
13C2×C4×Q8C2211yescharacteristic
14C2×#117C2212yes—
15C22×C42C2211yescharacteristic
16C2×#107C2212yes—
17C2×#106C2212yes—
18C2×C4⋊C8C2212yes—
19C2×#103C2212yes—
20C2×#103C2212yes—
21C2×C22⋊C8C2212yes—
22#391C2211yescharacteristic
23#389C2211yescharacteristic
24#387—21no—
25#385—21no—
26#384C2211yescharacteristic
27#384C2211yescharacteristic
28#382C2212yes—
29#380—21no—
30#380—21no—
31#380C2211yescharacteristic
32C4×C4⋊C4—21no—
33C4×C4⋊C4—21no—
34C4×C4⋊C4—21no—
35C4×C4⋊C4C2211yescharacteristic
36C4×C22⋊C4—21no—
37#340C2212yes—
38#340C2212yes—
39#340C2212yes—
40#340C2212yes—
41#336C2211yescharacteristic
42#336C2211yescharacteristic
43C22×Q8C2311yescharacteristic
44C23×C4D411yescharacteristic
45C23×C4D411yescharacteristic
46C23×C4C2311yescharacteristic
47C22×C8C2312yes—
48C22×C8C2312yes—
49C22⋊Q8—22no—
50C22⋊Q8Q814yes—
51C4×Q8—24no—
52#117—42no—
53#117—42no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4C2312yes—
62C2×C4⋊C4C2312yes—
63C2×C4⋊C4C2311yescharacteristic
64C2×C4⋊C4C2311yescharacteristic
65C2×C4⋊C4C2311yescharacteristic
66C2×C22⋊C4—21no—
67C2×C22⋊C4—22no—
68C2×C22⋊C4—21no—
69C2×C42—24no—
70C2×C42—22no—
71C2×C42—21no—
72C2×C42—21no—
73C2×C42D411yescharacteristic
74C2×C42D411yescharacteristic
75C2×C42C2311yescharacteristic
76C2×C42C2311yescharacteristic
77#107—42no—
78#106—42no—
79C4⋊C8—24no—
80#103—42no—
81#103—42no—
82C22⋊C8—24no—
83#95—21no—
84#95—21no—
85#95—22no—
86#95—22no—
87#95—21no—
88#95—21no—
89#95—21no—
90C24C2×D411yescharacteristic
91C2×Q8—21no—
92C2×Q8—22no—
93C2×Q8—22no—
94C2×Q8C2×Q812yes—
95C22×C4—22no—
96C22×C4—22no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—22no—
100C22×C4—22no—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—41no—
108C22×C4—41no—
109C22×C4C2×D411yescharacteristic
110C22×C4C2×D411yescharacteristic
111C22×C4C2×D411yescharacteristic
112C22×C4C2×D411yescharacteristic
113C22×C4C2×D411yescharacteristic
114C22×C4C2×Q812yes—
115C22×C4#4112yes—
116C22×C4C2411yescharacteristic
117C2×C8—22no—
118C2×C8—22no—
119C2×C8—22no—
120C2×C8—22no—
121C2×C8—22no—
122C2×C8—22no—
123C4⋊C4—24no—
124C4⋊C4—21no—
125C4⋊C4—22no—
126C4⋊C4—22no—
127C4⋊C4—22no—
128C4⋊C4—22no—
129C4⋊C4—22no—
130C4⋊C4—22no—
131C4⋊C4—42no—
132C4⋊C4—42no—
133C4⋊C4—42no—
134C4⋊C4—42no—
135C4⋊C4—42no—
136C4⋊C4—42no—
137C4⋊C4C2×Q812yes—
138C22⋊C4—24no—
139C22⋊C4—42no—
140C22⋊C4—44no—
141C42—22no—
142C42—22no—
143C42—22no—
144C42—42no—
145C42—44no—
146C42—42no—
147C42#4114yes—
148C23—21no—
149C23—21no—
150C23—21no—
151C23—21no—
152C23—21no—
153C23—21no—
154C23C22⋊Q812yes—
155C23C22×D411yescharacteristic
156Q8—24no—
157Q8—42no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—24no—
169C2×C4—24no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—44no—
173C2×C4—44no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4C22⋊Q812yes—
191C2×C4#12312yes—
192C2×C4#12312yes—
193C2×C4#13512yes—
194C2×C4#13712yes—
195C2×C4C22×Q811yescharacteristic
196C2×C4C2×#4111yescharacteristic
197C2×C4C2×#4112yes—
198C2×C4H(5,𝔽2)12yes—
199C8—42no—
200C8—42no—
201C22—22no—
202C22—22no—
203C22—22no—
204C22—41no—
205C22—41no—
206C22—41no—
207C22—41no—
208C22C2×C22⋊Q811yescharacteristic
209C22C2×#12311yescharacteristic
210C22#53511yescharacteristic
211C22C2×#13511yescharacteristic
212C22C2×#13711yescharacteristic
213C22#57511yescharacteristic
214C22#57511yescharacteristic
215C4—22no—
216C4—22no—
217C4—22no—
218C4—22no—
219C4—22no—
220C4—42no—
221C4—42no—
222C4—81no—
223C4—82no—
224C4—82no—
225C4—81no—
226C4#55312yes—
227C4#55412yes—
228C2—42no—
229C2#248711yescharacteristic
230C2#307611yescharacteristic
231C2#307811yescharacteristic
232C2#309111yescharacteristic
233C2#309511yescharacteristic
234C2#321011yescharacteristic
235C2#321911yescharacteristic
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
3. #2100C2semidirect
4. #2099C2semidirect
5. #2096C2semidirect
6. #2095C2semidirect
7. #1949C2semidirect
10. #1805C2semidirect

Maximal subgroups

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

subgroupindex
C4×C22⋊Q82
#22852
#21002
#20992
#20962
#20952
#19492
#18732
#18542
#18052
#17972