Finite Group Databasebeta

← #29913

Subgroups — #29913

Order 256.

367 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 367 conjugacy classes of subgroups — too many to draw. Only its 119 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#2855C211yesmaximal, characteristic
2C2×#382C211yesmaximal, characteristic
3#2101C212yesmaximal, complemented
4#2097C212yesmaximal, complemented
5#2085C212yesmaximal, complemented
6#2074C212yesmaximal, complemented
7#1872C211yesmaximal, characteristic
8#1872C211yesmaximal, characteristic
9#1853C211yesmaximal, characteristic
10#1852C211yesmaximal, characteristic
11#1817C211yesmaximal, characteristic
12C2×C22⋊Q8C2211yescharacteristic
13C22×C4⋊C4C2211yescharacteristic
14C22×C4⋊C4C2211yescharacteristic
15C22×C42C2211yescharacteristic
16C2×#107C2212yes—
17C2×#106C2212yes—
18C2×#103C2212yes—
19C2×#103C2212yes—
20C2×C22⋊C8C2211yescharacteristic
21C2×C22⋊C8C2211yescharacteristic
22#391C2211yescharacteristic
23#391C2211yescharacteristic
24#389C2211yescharacteristic
25#389C2211yescharacteristic
26#387—21no—
27#387—21no—
28#385—21no—
29#382—21no—
30#382—21no—
31#382C2212yes—
32#382C2212yes—
33#381—21no—
34#380C2211yescharacteristic
35C2×#95—21no—
36C2×#95—21no—
37#340C2212yes—
38#340C2212yes—
39#340C2212yes—
40#340C2212yes—
41#336C2212yes—
42#336C2212yes—
43C22×Q8C2311yescharacteristic
44C23×C4—21no—
45C23×C4—21no—
46C23×C4D411yescharacteristic
47C23×C4D411yescharacteristic
48C23×C4C2311yescharacteristic
49C22×C8C2312yes—
50C22×C8C2312yes—
51C22⋊Q8—22no—
52C22⋊Q8—22no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—42no—
59C2×C4⋊C4—42no—
60C2×C4⋊C4C2312yes—
61C2×C4⋊C4C2312yes—
62C2×C4⋊C4C2311yescharacteristic
63C2×C4⋊C4C2311yescharacteristic
64C2×C4⋊C4C2311yescharacteristic
65C2×C22⋊C4—21no—
66C2×C22⋊C4—21no—
67C2×C42—22no—
68C2×C42—22no—
69C2×C42D411yescharacteristic
70C2×C42D411yescharacteristic
71C2×C42C2311yescharacteristic
72C2×C42C2311yescharacteristic
73#107—42no—
74#106—42no—
75#103—42no—
76#103—42no—
77C22⋊C8—22no—
78C22⋊C8—22no—
79#95—22no—
80#95—22no—
81#95—22no—
82#95—22no—
83#95—21no—
84#95—21no—
85#95—21no—
86#95—21no—
87#95—21no—
88#95—21no—
89#95—21no—
90C24C2×D411yescharacteristic
91C2×Q8—21no—
92C2×Q8—21no—
93C2×Q8—22no—
94C2×Q8—22no—
95C22×C4—22no—
96C22×C4—22no—
97C22×C4—22no—
98C22×C4—22no—
99C22×C4—22no—
100C22×C4—22no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—22no—
105C22×C4—22no—
106C22×C4—21no—
107C22×C4—41no—
108C22×C4—41no—
109C22×C4—42no—
110C22×C4—42no—
111C22×C4—42no—
112C22×C4—42no—
113C22×C4—42no—
114C22×C4—42no—
115C22×C4C2×D411yescharacteristic
116C22×C4C2×D411yescharacteristic
117C22×C4C2×D411yescharacteristic
118C22×C4C2×D411yescharacteristic
119C22×C4C2×D411yescharacteristic
120C22×C4#4112yes—
121C22×C4#4112yes—
122C22×C4C2411yescharacteristic
123C2×C8—22no—
124C2×C8—22no—
125C2×C8—22no—
126C2×C8—22no—
127C2×C8—22no—
128C2×C8—22no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4—22no—
132C4⋊C4—22no—
133C4⋊C4—22no—
134C4⋊C4—22no—
135C4⋊C4—22no—
136C4⋊C4—22no—
137C4⋊C4—42no—
138C4⋊C4—42no—
139C4⋊C4—42no—
140C4⋊C4—42no—
141C4⋊C4—42no—
142C22⋊C4—42no—
143C22⋊C4—42no—
144C42—22no—
145C42—22no—
146C42—21no—
147C42—21no—
148C42—21no—
149C42—21no—
150C23—21no—
151C23—21no—
152C23—21no—
153C23—21no—
154C23#12312yes—
155C23#13512yes—
156C23#13712yes—
157C23C22×D411yescharacteristic
158Q8—42no—
159Q8—42no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—24no—
163C2×C4—24no—
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—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—44no—
177C2×C4—44no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—84no—
197C2×C4—84no—
198C2×C4#12312yes—
199C2×C4C2×#4111yescharacteristic
200C2×C4C2×#4111yescharacteristic
201C2×C4H(5,𝔽2)12yes—
202C2×C42−1+412yes—
203C8—42no—
204C8—42no—
205C22—22no—
206C22—22no—
207C22—22no—
208C22—22no—
209C22—22no—
210C22—22no—
211C22#48514yes—
212C22C2×#12311yescharacteristic
213C22#53511yescharacteristic
214C22#53611yescharacteristic
215C22C2×#13511yescharacteristic
216C22C2×#13711yescharacteristic
217C22#57511yescharacteristic
218C22#57511yescharacteristic
219C4—22no—
220C4—41no—
221C4—41no—
222C4—41no—
223C4—41no—
224C4—81no—
225C4—81no—
226C4—81no—
227C4—82no—
228C4—82no—
229C4—81no—
230C4#53912yes—
231C4#53912yes—
232C2—24no—
233C2#262311yescharacteristic
234C2C2×#48511yescharacteristic
235C2#309311yescharacteristic
236C2#321111yescharacteristic
237C2#321511yescharacteristic
238C2#323211yescharacteristic
239C2#323611yescharacteristic
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. #2085C2semidirect
6. #2074C2semidirect

Maximal subgroups

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

subgroupindex
#28552
C2×#3822
#21012
#20972
#20852
#20742
#18722
#18532
#18522
#18172