Finite Group Databasebeta

← #33034

Subgroups — #33034

Order 256.

459 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 459 conjugacy classes of subgroups — too many to draw. Only its 131 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×#476C211yesmaximal, characteristic, complemented
2C2×#463C211yesmaximal, characteristic
3#2640C211yesmaximal, characteristic, complemented
4#2275C211yesmaximal, characteristic, complemented
5#1988C211yesmaximal, characteristic, complemented
6#1983C211yesmaximal, characteristic, complemented
7#1927C212yesmaximal, complemented
8#1926C212yesmaximal, complemented
9#1879C212yesmaximal
10#1877C212yesmaximal, complemented
11#1832C211yesmaximal, characteristic, complemented
12C22×SD16C2211yescharacteristic
13C22×D8C2211yescharacteristic
14C2×C4⋊Q8C2211yescharacteristic
15C2×#124C2211yescharacteristic
16C2×C4×D4C2212yes—
17C2×C4×D4C2211yescharacteristic
18#476—24no—
19#463—24no—
20C2×#106C2212yes—
21C2×C4⋊C8C2211yescharacteristic
22C2×C4⋊C8C2211yescharacteristic
23C2×#103C2212yes—
24C2×#103C2211yescharacteristic
25C2×#102C2212yes—
26C2×#102C2212yes—
27C2×#102C2211yescharacteristic
28#397—21no—
29#397—21no—
30#393—22no—
31#391C2212yes—
32#390C2212yes—
33C4×C4⋊C4C2212yes—
34C4×C4⋊C4C2211yescharacteristic
35C4×C22⋊C4—22no—
36C4×C22⋊C4—21no—
37#376C2211yescharacteristic
38C2×#95—21no—
39#340C2212yes—
40#340C2212yes—
41#336C2212yes—
42#336C2212yes—
43C22×Q8—21no—
44C22×Q8C2311yescharacteristic
45C22×D4C2311yescharacteristic
46C22×D4C2311yescharacteristic
47C23×C4—22no—
48C23×C4—21no—
49C2×SD16—22no—
50C2×SD16—24no—
51C2×D8—22no—
52C2×D8—24no—
53C22×C8C2311yescharacteristic
54C22×C8C2311yescharacteristic
55C22×C8C2312yes—
56C4⋊Q8—22no—
57C4⋊Q8—22no—
58#124—22no—
59#124—22no—
60C4×D4—24no—
61C4×D4—24no—
62C4×D4—22no—
63C4×D4D414yes—
64C2×C4⋊C4—21no—
65C2×C4⋊C4C2312yes—
66C2×C4⋊C4C2311yescharacteristic
67C2×C4⋊C4C2312yes—
68C2×C22⋊C4—22no—
69C2×C22⋊C4—21no—
70C2×C22⋊C4—21no—
71C2×C22⋊C4—21no—
72C2×C22⋊C4—22no—
73C2×C22⋊C4—22no—
74C2×C42—22no—
75C2×C42—21no—
76C2×C42C2312yes—
77C2×C42C2311yescharacteristic
78#106—24no—
79C4⋊C8—22no—
80C4⋊C8—22no—
81#103—22no—
82#103—24no—
83#102—24no—
84#102—22no—
85#102—24no—
86#95—22no—
87#95—21no—
88#95—22no—
89#95—22no—
90#95D411yescharacteristic
91#95D412yes—
92#95D411yescharacteristic
93C24—21no—
94C24—21no—
95C2×Q8—21no—
96C2×Q8—22no—
97C2×Q8—21no—
98C2×Q8—22no—
99C2×Q8—42no—
100C2×Q8—81no—
101C2×Q8—81no—
102C2×D4—21no—
103C2×D4—21no—
104C2×D4—22no—
105C2×D4—22no—
106C2×D4—21no—
107C2×D4—22no—
108C2×D4C2×D412yes—
109C2×D4#4114yes—
110C22×C4—21no—
111C22×C4—22no—
112C22×C4—28no—
113C22×C4—24no—
114C22×C4—21no—
115C22×C4—22no—
116C22×C4—82no—
117C22×C4—82no—
118C22×C4—82no—
119C22×C4C2×D412yes—
120C22×C4C2×D412yes—
121C22×C4C2×D411yescharacteristic
122C22×C4C2×D411yescharacteristic
123C22×C4C2411yescharacteristic
124SD16—44no—
125D8—44no—
126C2×C8—22no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—22no—
130C2×C8—22no—
131C2×C8—22no—
132C2×C8—22no—
133C4⋊C4—21no—
134C4⋊C4—22no—
135C4⋊C4—22no—
136C4⋊C4—22no—
137C4⋊C4—42no—
138C4⋊C4C2×D412yes—
139C4⋊C4#4114yes—
140C22⋊C4—28no—
141C22⋊C4—24no—
142C22⋊C4—42no—
143C22⋊C4—42no—
144C22⋊C4—82no—
145C22⋊C4—82no—
146C42—22no—
147C42—22no—
148C42—81no—
149C42—82no—
150C42C2×D412yes—
151C42#4112yes—
152C23—22no—
153C23—24no—
154C23—41no—
155C23—42no—
156C23—41no—
157C23—41no—
158C23—42no—
159C23—41no—
160C23—41no—
161C23—42no—
162C23C22×D411yescharacteristic
163Q8—41no—
164Q8—42no—
165Q8—41no—
166Q8—84no—
167D4—24no—
168D4—22no—
169D4—41no—
170D4—42no—
171D4—41no—
172D4#12414yes—
173C2×C4—21no—
174C2×C4—22no—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—24no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—41no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—84no—
193C2×C4—88no—
194C2×C4#12112yes—
195C2×C4#12112yes—
196C2×C4C22×D411yescharacteristic
197C2×C4C2×#4112yes—
198C2×C4C2×#4111yescharacteristic
199C2×C4C2×#4112yes—
200C8—41no—
201C8—41no—
202C8—42no—
203C22—24no—
204C22—44no—
205C22—44no—
206C22—42no—
207C22—42no—
208C22—81no—
209C22—82no—
210C22—82no—
211C22—81no—
212C22—81no—
213C22C2×#12111yescharacteristic
214C22#52512yes—
215C22#57611yescharacteristic
216C22#57712yes—
217C22#57711yescharacteristic
218C4—22no—
219C4—22no—
220C4—42no—
221C4—44no—
222C4—42no—
223C4—42no—
224C4—81no—
225C4—82no—
226C4C2×#12411yescharacteristic
227C4#52912yes—
228C4#53211yescharacteristic
229C2—44no—
230C2—161no—
231C2#285011yescharacteristic
232C2#304211yescharacteristic
233C2#304511yescharacteristic
234C2#311812yes—
235C2#312512yes—
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
1. C2×#476C2semidirect
3. #2640C2semidirect
4. #2275C2semidirect
5. #1988C2semidirect
6. #1983C2semidirect
7. #1927C2semidirect
8. #1926C2semidirect
10. #1877C2semidirect
11. #1832C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4762
C2×#4632
#26402
#22752
#19882
#19832
#19272
#19262
#18792
#18772
#18322