Finite Group Databasebeta

← #30130

Subgroups — #30130

Order 256.

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

normal subgroups only

This group has 378 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#2860C211yesmaximal, characteristic
2C4×C4⋊Q8C211yesmaximal, characteristic, complemented
3#2099C211yesmaximal, characteristic, complemented
4#2093C211yesmaximal, characteristic
5#2068C212yesmaximal, complemented
6#2055C212yesmaximal
7#2043C211yesmaximal, characteristic, complemented
8#2013C211yesmaximal, characteristic
9#1880C212yesmaximal
10#1863C212yesmaximal, complemented
11#1831C211yesmaximal, characteristic, complemented
12C22×SD16C2212yes—
13C2×C4⋊Q8C2211yescharacteristic
14C2×#126—21no—
15C2×C4×Q8C2212yes—
16C2×#107C2212yes—
17C2×C4⋊C8C2212yes—
18C2×#103C2212yes—
19C2×#103C2212yes—
20C2×#103C2212yes—
21C2×#102C2212yes—
22#391C2212yes—
23#390C2211yescharacteristic
24#390C2211yescharacteristic
25#390C2211yescharacteristic
26#389—21no—
27#389—21no—
28#389C2211yescharacteristic
29#389C2211yescharacteristic
30#389C2211yescharacteristic
31#388—21no—
32#388—21no—
33#388—21no—
34#384C2211yescharacteristic
35C4×C4⋊C4—21no—
36C4×C4⋊C4—21no—
37C43C2211yescharacteristic
38#340C2212yes—
39#340C2212yes—
40#336C2211yescharacteristic
41#336C2211yescharacteristic
42#336C2212yes—
43C22×Q8C2312yes—
44C22×D4C2311yescharacteristic
45C2×SD16—24no—
46C2×SD16—42no—
47C2×SD16—42no—
48C22×C8C2312yes—
49C22×C8C2312yes—
50C4⋊Q8—24no—
51#126—44no—
52C4×Q8—42no—
53C4×Q8—42no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4C2312yes—
63C2×C4⋊C4C2311yescharacteristic
64C2×C4⋊C4C2311yescharacteristic
65C2×C4⋊C4C2311yescharacteristic
66C2×C22⋊C4—21no—
67C2×C22⋊C4—21no—
68C2×C22⋊C4—21no—
69C2×C22⋊C4—21no—
70C2×C22⋊C4—21no—
71C2×C22⋊C4—21no—
72C2×C42—22no—
73C2×C42D411yescharacteristic
74C2×C42D411yescharacteristic
75C2×C42D411yescharacteristic
76C2×C42D411yescharacteristic
77C2×C42C2311yescharacteristic
78C2×C42C2311yescharacteristic
79C2×C42C2311yescharacteristic
80#107—42no—
81C4⋊C8—42no—
82#103—24no—
83#103—42no—
84#103—42no—
85#102—42no—
86#95—22no—
87#95—22no—
88#95—21no—
89#95—21no—
90#95—21no—
91C24—21no—
92C2×Q8—22no—
93C2×Q8—22no—
94C2×Q8—22no—
95C2×Q8—22no—
96C2×Q8—22no—
97C2×Q8#4114yes—
98C2×D4—22no—
99C2×D4—22no—
100C2×D4—21no—
101C2×D4—21no—
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4C2×D411yescharacteristic
108C22×C4C2×D411yescharacteristic
109C22×C4C2×D411yescharacteristic
110C22×C4C2×D411yescharacteristic
111C22×C4C2×D411yescharacteristic
112C22×C4C2×D411yescharacteristic
113C22×C4C2411yescharacteristic
114SD16—48no—
115C2×C8—22no—
116C2×C8—22no—
117C2×C8—22no—
118C2×C8—22no—
119C2×C8—22no—
120C2×C8—22no—
121C4⋊C4—24no—
122C4⋊C4—24no—
123C4⋊C4—22no—
124C4⋊C4—22no—
125C4⋊C4—21no—
126C4⋊C4—22no—
127C4⋊C4—22no—
128C4⋊C4—21no—
129C4⋊C4—44no—
130C4⋊C4—42no—
131C4⋊C4—42no—
132C4⋊C4—42no—
133C4⋊C4—42no—
134C4⋊C4—42no—
135C22⋊C4—42no—
136C22⋊C4—42no—
137C22⋊C4—42no—
138C22⋊C4—42no—
139C22⋊C4—42no—
140C22⋊C4—42no—
141C42—22no—
142C42—22no—
143C42—22no—
144C42—22no—
145C42—22no—
146C42—21no—
147C42—22no—
148C42—21no—
149C42—44no—
150C23—41no—
151C23—41no—
152C23—41no—
153C23—41no—
154C23—41no—
155C23—41no—
156C23—41no—
157C23C22×D411yescharacteristic
158Q8—24no—
159Q8—24no—
160Q8—42no—
161Q8—42no—
162D4—42no—
163D4—42no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—24no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—41no—
182C2×C4—42no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—42no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—42no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4#13512yes—
194C2×C4AGL(1,ℤ/8)12yes—
195C2×C4#13712yes—
196C2×C4#13712yes—
197C2×C4C2×#4112yes—
198C2×C4H(5,𝔽2)11yescharacteristic
199C2×C4H(5,𝔽2)12yes—
200C2×C42−1+411yescharacteristic
201C8—42no—
202C8—42no—
203C22—81no—
204C22—81no—
205C22—81no—
206C22—81no—
207C22—81no—
208C22—81no—
209C22—81no—
210C22#52511yescharacteristic
211C22#53511yescharacteristic
212C22#53711yescharacteristic
213C22C2×#13511yescharacteristic
214C22C2×AGL(1,ℤ/8)11yescharacteristic
215C22C2×#13711yescharacteristic
216C22C2×#13711yescharacteristic
217C4—22no—
218C4—22no—
219C4—22no—
220C4—22no—
221C4—44no—
222C4—41no—
223C4—41no—
224C4—81no—
225C4—82no—
226C4—81no—
227C4—81no—
228C4#53812yes—
229C4#53912yes—
230C2—161no—
231C2#263611yescharacteristic
232C2#310811yescharacteristic
233C2#311211yescharacteristic
234C2#320711yescharacteristic
235C2#321811yescharacteristic
236C2#324811yescharacteristic
237C2#325411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×C4⋊Q8C2semidirect
3. #2099C2semidirect
5. #2068C2semidirect
7. #2043C2semidirect
10. #1863C2semidirect
11. #1831C2semidirect

Maximal subgroups

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

subgroupindex
#28602
C4×C4⋊Q82
#20992
#20932
#20682
#20552
#20432
#20132
#18802
#18632
#18312