Finite Group Databasebeta

← #29805

Subgroups — #29805

Order 256.

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

normal subgroups only

This group has 419 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×#464C211yesmaximal, characteristic
2C2×#462C211yesmaximal, characteristic, complemented
3#2658C211yesmaximal, characteristic
4#2320C211yesmaximal, characteristic, complemented
5#2043C212yesmaximal, complemented
6#2042C212yesmaximal
7#2016C212yesmaximal, complemented
8#2015C212yesmaximal
9#1983C211yesmaximal, characteristic
10#1981C211yesmaximal, characteristic, complemented
11#1832C211yesmaximal, characteristic, complemented
12C22×Q16C2212yes—
13C22×Q16C2211yescharacteristic
14C22×SD16C2211yescharacteristic
15C2×C4⋊Q8C2211yescharacteristic
16C2×C4⋊Q8C2212yes—
17C2×#124C2211yescharacteristic
18C2×#124C2212yes—
19C2×C4×Q8C2211yescharacteristic
20#464—24no—
21#462—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#391C2212yes—
35#389C2212yes—
36#387—22no—
37#383—21no—
38#382—21no—
39C4×C4⋊C4C2211yescharacteristic
40#376C2211yescharacteristic
41#336C2212yes—
42#336C2212yes—
43C22×Q8C2311yescharacteristic
44C22×Q8C2312yes—
45C22×Q8C2311yescharacteristic
46C22×D4C2311yescharacteristic
47C2×Q16—24no—
48C2×Q16—22no—
49C2×Q16—42no—
50C2×Q16—42no—
51C2×Q16—42no—
52C2×SD16—22no—
53C2×SD16—42no—
54C22×C8C2311yescharacteristic
55C22×C8C2311yescharacteristic
56C22×C8C2312yes—
57C4⋊Q8—22no—
58C4⋊Q8—22no—
59C4⋊Q8—42no—
60C4⋊Q8—42no—
61#124—22no—
62#124—22no—
63#124—42no—
64#124—42no—
65C4×Q8—22no—
66C4×Q8D414yes—
67C2×C4⋊C4—21no—
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—22no—
76C2×C22⋊C4—22no—
77C2×C22⋊C4—21no—
78C2×C22⋊C4—21no—
79C2×C42—21no—
80C2×C42—21no—
81C2×C42C2312yes—
82C2×C42C2311yescharacteristic
83C4⋊C8—22no—
84C4⋊C8—22no—
85#103—24no—
86#103—22no—
87#103—42no—
88#103—42no—
89#103—42no—
90#102—22no—
91#102—42no—
92#95—21no—
93#95—22no—
94#95—22no—
95#95—22no—
96#95D411yescharacteristic
97#95D411yescharacteristic
98#95D412yes—
99C24—21no—
100C2×Q8—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×Q8—22no—
112C2×Q8—22no—
113C2×Q8C2×D412yes—
114C2×D4—21no—
115C2×D4—21no—
116C2×D4—22no—
117C2×D4—22no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—22no—
121C22×C4—21no—
122C22×C4—22no—
123C22×C4C2×D412yes—
124C22×C4C2×D412yes—
125C22×C4C2×D411yescharacteristic
126C22×C4C2×D411yescharacteristic
127C22×C4C2411yescharacteristic
128Q16—44no—
129Q16—48no—
130SD16—44no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—22no—
134C2×C8—22no—
135C2×C8—22no—
136C2×C8—22no—
137C2×C8—22no—
138C4⋊C4—24no—
139C4⋊C4—21no—
140C4⋊C4—22no—
141C4⋊C4—22no—
142C4⋊C4—44no—
143C4⋊C4—44no—
144C4⋊C4—42no—
145C4⋊C4—42no—
146C4⋊C4—82no—
147C4⋊C4C2×D412yes—
148C22⋊C4—44no—
149C22⋊C4—44no—
150C22⋊C4—42no—
151C22⋊C4—42no—
152C42—24no—
153C42—22no—
154C42—22no—
155C42—81no—
156C42C2×D412yes—
157C42#4112yes—
158C23—41no—
159C23—42no—
160C23—41no—
161C23—41no—
162C23—42no—
163C23C22×D411yescharacteristic
164Q8—22no—
165Q8—22no—
166Q8—42no—
167Q8—41no—
168Q8—42no—
169Q8—41no—
170Q8—42no—
171Q8—42no—
172Q8—42no—
173Q8—42no—
174D4—41no—
175D4—41no—
176D4—42no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—42no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4#12112yes—
200C2×C4#12112yes—
201C2×C4C22×D411yescharacteristic
202C2×C4C2×#4111yescharacteristic
203C2×C4H(5,𝔽2)12yes—
204C2×C4H(5,𝔽2)12yes—
205C8—41no—
206C8—41no—
207C8—42no—
208C22—81no—
209C22—82no—
210C22—82no—
211C22—81no—
212C22—81no—
213C22C2×#12111yescharacteristic
214C22#53512yes—
215C22#57711yescharacteristic
216C22#57811yescharacteristic
217C22#57812yes—
218C4—22no—
219C4—22no—
220C4—42no—
221C4—42no—
222C4—42no—
223C4—42no—
224C4—81no—
225C4—82no—
226C4—82no—
227C4#53511yescharacteristic
228C4#53511yescharacteristic
229C4#54012yes—
230C2—161no—
231C2#261911yescharacteristic
232C2#304411yescharacteristic
233C2#304611yescharacteristic
234C2#322312yes—
235C2#322512yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
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×#462C2semidirect
4. #2320C2semidirect
5. #2043C2semidirect
7. #2016C2semidirect
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×#4642
C2×#4622
#26582
#23202
#20432
#20422
#20162
#20152
#19832
#19812
#18322