Finite Group Databasebeta

← #29804

Subgroups — #29804

Order 256.

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

normal subgroups only

This group has 443 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
1C2×#463C211yesmaximal, characteristic
2C2×#461C211yesmaximal, characteristic, complemented
3#2658C211yesmaximal, characteristic, complemented
4#2319C211yesmaximal, characteristic, complemented
5#2043C212yesmaximal, complemented
6#2042C212yesmaximal, complemented
7#2013C212yesmaximal, complemented
8#2012C212yesmaximal
9#1983C211yesmaximal, characteristic, complemented
10#1981C211yesmaximal, characteristic, complemented
11#1832C211yesmaximal, characteristic, complemented
12C22×SD16C2212yes—
13C22×SD16C2211yescharacteristic
14C22×D8C2211yescharacteristic
15C2×C4⋊Q8C2211yescharacteristic
16C2×C4⋊Q8C2212yes—
17C2×#124C2211yescharacteristic
18C2×#124C2212yes—
19C2×C4×D4C2211yescharacteristic
20#463—24no—
21#461—24no—
22C2×C4⋊C8C2211yescharacteristic
23C2×C4⋊C8C2211yescharacteristic
24C2×#103C2212yes—
25C2×#103C2211yescharacteristic
26C2×#103C2212yes—
27C2×#103C2212yes—
28C2×#103C2212yes—
29C2×#102C2211yescharacteristic
30C2×#102C2212yes—
31#399—21no—
32#399—21no—
33#399—22no—
34#390C2212yes—
35#389C2212yes—
36#388—22no—
37#381—21no—
38#380—21no—
39C4×C4⋊C4C2211yescharacteristic
40#376C2211yescharacteristic
41#336C2212yes—
42#336C2212yes—
43C22×Q8C2311yescharacteristic
44C22×Q8C2312yes—
45C22×D4C2311yescharacteristic
46C22×D4C2311yescharacteristic
47C23×C4—21no—
48C2×SD16—24no—
49C2×SD16—22no—
50C2×SD16—42no—
51C2×SD16—42no—
52C2×SD16—42no—
53C2×D8—22no—
54C2×D8—42no—
55C22×C8C2311yescharacteristic
56C22×C8C2311yescharacteristic
57C22×C8C2312yes—
58C4⋊Q8—22no—
59C4⋊Q8—22no—
60C4⋊Q8—42no—
61C4⋊Q8—42no—
62#124—22no—
63#124—22no—
64#124—42no—
65#124—42no—
66C4×D4—22no—
67C4×D4D414yes—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4C2312yes—
74C2×C4⋊C4C2311yescharacteristic
75C2×C22⋊C4—21no—
76C2×C22⋊C4—22no—
77C2×C22⋊C4—22no—
78C2×C22⋊C4—21no—
79C2×C22⋊C4—21no—
80C2×C22⋊C4—22no—
81C2×C22⋊C4—22no—
82C2×C42—21no—
83C2×C42C2312yes—
84C2×C42C2311yescharacteristic
85C4⋊C8—22no—
86C4⋊C8—22no—
87#103—24no—
88#103—22no—
89#103—42no—
90#103—42no—
91#103—42no—
92#102—22no—
93#102—42no—
94#95—21no—
95#95—22no—
96#95D411yescharacteristic
97#95D411yescharacteristic
98#95D412yes—
99C24—21no—
100C24—21no—
101C2×Q8—21no—
102C2×Q8—22no—
103C2×Q8—21no—
104C2×Q8—22no—
105C2×Q8—22no—
106C2×Q8—22no—
107C2×Q8—22no—
108C2×Q8—22no—
109C2×Q8—22no—
110C2×Q8—22no—
111C2×D4—21no—
112C2×D4—21no—
113C2×D4—21no—
114C2×D4—22no—
115C2×D4—22no—
116C2×D4—22no—
117C2×D4—22no—
118C2×D4C2×D412yes—
119C22×C4—21no—
120C22×C4—24no—
121C22×C4—22no—
122C22×C4—21no—
123C22×C4—22no—
124C22×C4—82no—
125C22×C4C2×D412yes—
126C22×C4C2×D412yes—
127C22×C4C2×D411yescharacteristic
128C22×C4C2×D411yescharacteristic
129C22×C4C2411yescharacteristic
130SD16—44no—
131SD16—48no—
132D8—44no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—22no—
136C2×C8—22no—
137C2×C8—22no—
138C2×C8—22no—
139C2×C8—22no—
140C4⋊C4—21no—
141C4⋊C4—22no—
142C4⋊C4—22no—
143C4⋊C4—44no—
144C4⋊C4—44no—
145C4⋊C4—42no—
146C4⋊C4—42no—
147C4⋊C4—82no—
148C4⋊C4C2×D412yes—
149C22⋊C4—24no—
150C22⋊C4—44no—
151C22⋊C4—44no—
152C22⋊C4—42no—
153C22⋊C4—42no—
154C22⋊C4—82no—
155C22⋊C4—82no—
156C42—22no—
157C42—22no—
158C42—81no—
159C42C2×D412yes—
160C42#4112yes—
161C23—22no—
162C23—42no—
163C23—41no—
164C23—42no—
165C23—41no—
166C23—41no—
167C23—42no—
168C23—41no—
169C23—42no—
170C23—41no—
171C23C22×D411yescharacteristic
172Q8—41no—
173Q8—42no—
174Q8—41no—
175Q8—42no—
176Q8—42no—
177Q8—42no—
178Q8—42no—
179D4—22no—
180D4—22no—
181D4—41no—
182D4—41no—
183D4—42no—
184D4—42no—
185C2×C4—21no—
186C2×C4—22no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—42no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—84no—
205C2×C4#12112yes—
206C2×C4#12112yes—
207C2×C4C22×D411yescharacteristic
208C2×C4C2×#4111yescharacteristic
209C2×C4H(5,𝔽2)12yes—
210C2×C4H(5,𝔽2)12yes—
211C8—41no—
212C8—41no—
213C8—42no—
214C22—42no—
215C22—42no—
216C22—42no—
217C22—82no—
218C22—82no—
219C22—81no—
220C22—82no—
221C22—82no—
222C22—81no—
223C22—81no—
224C22C2×#12111yescharacteristic
225C22#53512yes—
226C22#57711yescharacteristic
227C22#57712yes—
228C22#57811yescharacteristic
229C4—22no—
230C4—22no—
231C4—42no—
232C4—42no—
233C4—42no—
234C4—81no—
235C4—82no—
236C4—82no—
237C4#53511yescharacteristic
238C4#53511yescharacteristic
239C4#54012yes—
240C2—82no—
241C2—161no—
242C2#261911yescharacteristic
243C2#304211yescharacteristic
244C2#304611yescharacteristic
245C2#322112yes—
246C2#322512yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D8C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#461C2semidirect
3. #2658C2semidirect
4. #2319C2semidirect
5. #2043C2semidirect
6. #2042C2semidirect
7. #2013C2semidirect
9. #1983C2semidirect
10. #1981C2semidirect
11. #1832C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4632
C2×#4612
#26582
#23192
#20432
#20422
#20132
#20122
#19832
#19812
#18322