Finite Group Databasebeta

← #27686

Subgroups — #27686

Order 256.

399 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 399 conjugacy classes of subgroups — too many to draw. Only its 135 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
2C2×#382C211yesmaximal, characteristic
3#2101C212yesmaximal, complemented
4#2097C212yesmaximal, complemented
5#1949C212yesmaximal, complemented
6#1872C211yesmaximal, characteristic
7#1872C211yesmaximal, characteristic
8#1853C211yesmaximal, characteristic
9#1852C211yesmaximal, characteristic
10#1805C212yesmaximal, complemented
11#1797C211yesmaximal, characteristic
12C2×C22⋊Q8C2211yescharacteristic
13C2×C4×Q8C2211yescharacteristic
14C22×C4⋊C4C2211yescharacteristic
15C22×C4⋊C4C2211yescharacteristic
16C22×C42C2211yescharacteristic
17C2×#107C2212yes—
18C2×#106C2212yes—
19C2×C4⋊C8C2212yes—
20C2×#103C2212yes—
21C2×#103C2212yes—
22C2×C22⋊C8C2211yescharacteristic
23C2×C22⋊C8C2211yescharacteristic
24#391C2211yescharacteristic
25#389C2211yescharacteristic
26#387—21no—
27#385—21no—
28#382—21no—
29#382—21no—
30#382C2212yes—
31#382C2212yes—
32#380C2211yescharacteristic
33C4×C4⋊C4—21no—
34C4×C4⋊C4C2211yescharacteristic
35C4×C22⋊C4—21no—
36C2×#95—21no—
37C2×#95—21no—
38#340C2212yes—
39#340C2212yes—
40#340C2212yes—
41#340C2212yes—
42#336C2212yes—
43C22×Q8C2311yescharacteristic
44C23×C4—21no—
45C23×C4—21no—
46C23×C4D411yescharacteristic
47C23×C4D411yescharacteristic
48C23×C4C2311yescharacteristic
49C22×C8C2312yes—
50C22×C8C2312yes—
51C22⋊Q8—22no—
52C22⋊Q8Q814yes—
53C4×Q8—24no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—42no—
60C2×C4⋊C4—42no—
61C2×C4⋊C4C2312yes—
62C2×C4⋊C4C2312yes—
63C2×C4⋊C4C2311yescharacteristic
64C2×C4⋊C4C2311yescharacteristic
65C2×C4⋊C4C2311yescharacteristic
66C2×C22⋊C4—21no—
67C2×C22⋊C4—21no—
68C2×C42—24no—
69C2×C42—21no—
70C2×C42—21no—
71C2×C42D411yescharacteristic
72C2×C42D411yescharacteristic
73C2×C42C2311yescharacteristic
74C2×C42C2311yescharacteristic
75#107—42no—
76#106—42no—
77C4⋊C8—24no—
78#103—42no—
79#103—42no—
80C22⋊C8—22no—
81C22⋊C8—22no—
82#95—22no—
83#95—22no—
84#95—22no—
85#95—22no—
86#95—21no—
87#95—21no—
88#95—21no—
89#95—21no—
90#95—21no—
91C24C2×D411yescharacteristic
92C2×Q8—21no—
93C2×Q8—22no—
94C2×Q8—22no—
95C2×Q8C2×Q812yes—
96C22×C4—22no—
97C22×C4—22no—
98C22×C4—22no—
99C22×C4—22no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—21no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—21no—
108C22×C4—41no—
109C22×C4—41no—
110C22×C4—42no—
111C22×C4—42no—
112C22×C4—42no—
113C22×C4—42no—
114C22×C4—42no—
115C22×C4—42no—
116C22×C4C2×D411yescharacteristic
117C22×C4C2×D411yescharacteristic
118C22×C4C2×D411yescharacteristic
119C22×C4C2×D411yescharacteristic
120C22×C4C2×D411yescharacteristic
121C22×C4C2×Q812yes—
122C22×C4#4112yes—
123C22×C4C2411yescharacteristic
124C2×C8—22no—
125C2×C8—22no—
126C2×C8—22no—
127C2×C8—22no—
128C2×C8—22no—
129C2×C8—22no—
130C4⋊C4—24no—
131C4⋊C4—21no—
132C4⋊C4—22no—
133C4⋊C4—22no—
134C4⋊C4—22no—
135C4⋊C4—22no—
136C4⋊C4—22no—
137C4⋊C4—22no—
138C4⋊C4—42no—
139C4⋊C4—42no—
140C4⋊C4—42no—
141C4⋊C4—42no—
142C4⋊C4C2×Q812yes—
143C22⋊C4—24no—
144C22⋊C4—42no—
145C42—22no—
146C42—22no—
147C42—22no—
148C42—42no—
149C42—42no—
150C42#4114yes—
151C23—21no—
152C23—21no—
153C23—21no—
154C23—21no—
155C23C22⋊Q812yes—
156C23#13512yes—
157C23#13712yes—
158C23C22×D411yescharacteristic
159Q8—24no—
160Q8—42no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—24no—
168C2×C4—24no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—44no—
178C2×C4—44no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—84no—
196C2×C4—84no—
197C2×C4C22⋊Q812yes—
198C2×C4#12312yes—
199C2×C4#12312yes—
200C2×C4C22×Q811yescharacteristic
201C2×C4C2×#4111yescharacteristic
202C2×C4C2×#4112yes—
203C2×C4H(5,𝔽2)12yes—
204C8—42no—
205C8—42no—
206C22—22no—
207C22—22no—
208C22—22no—
209C22—22no—
210C22—22no—
211C22—22no—
212C22#47914yes—
213C22C2×C22⋊Q811yescharacteristic
214C22C2×#12311yescharacteristic
215C22#53511yescharacteristic
216C22C2×#13511yescharacteristic
217C22C2×#13711yescharacteristic
218C22#57511yescharacteristic
219C22#57511yescharacteristic
220C4—22no—
221C4—22no—
222C4—22no—
223C4—42no—
224C4—41no—
225C4—41no—
226C4—42no—
227C4—81no—
228C4—81no—
229C4—82no—
230C4—82no—
231C4#55312yes—
232C4#55412yes—
233C2—24no—
234C2#248711yescharacteristic
235C2C2×#47911yescharacteristic
236C2#308011yescharacteristic
237C2#309111yescharacteristic
238C2#309511yescharacteristic
239C2#321111yescharacteristic
240C2#321511yescharacteristic
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. #2101C2semidirect
4. #2097C2semidirect
5. #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
C2×#3822
#21012
#20972
#19492
#18722
#18532
#18522
#18052
#17972