Finite Group Databasebeta

← #8429

Subgroups — #8429

Order 256.

468 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 87 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#3497C211yesmaximal, characteristic
2#1805C211yesmaximal, characteristic, complemented
3#1540C211yesmaximal, characteristic, complemented
4#1517C211yesmaximal, characteristic, complemented
5#1517C211yesmaximal, characteristic, complemented
6#1480C211yesmaximal, characteristic, complemented
7#1479C211yesmaximal, characteristic, complemented
8C2×2−1+4—21no—
9#550—21no—
10#550C412yescomplemented
11D4×Q8—21no—
12D4×Q8C412yescomplemented
13#548—22no—
14#548—22no—
15#537—22no—
16#537—21no—
17#532—22no—
18#532C2211yescharacteristic
19#532C2211yescharacteristic
20C2×C4×Q8C2211yescharacteristic
21#431C2211yescharacteristic
22#431C2211yescharacteristic
23C2×C4⋊C8C2211yescharacteristic
24C2×#103—21no—
25#382C2211yescharacteristic
26#340—21no—
27#332—21no—
28#331—21no—
29#327—22no—
30#327—22no—
31Q8⋊C8—22no—
32D4⋊C8—22no—
332−1+4—42no—
342−1+4—42no—
352−1+4—42no—
362−1+4—42no—
37C2×#41—24no—
38C2×#41—22no—
39C2×#41—22no—
40C2×#41—21no—
41C2×#41—21no—
42C22×Q8—22no—
43C22×Q8—22no—
44C22×Q8D411yescharacteristic
45C22×C8—21no—
46C4⋊Q8—22no—
47C4⋊Q8C2×C411yescharacteristic
48C4⋊Q8D411yescharacteristic
49#127—22no—
50#127C2×C411yescharacteristic
51#127D411yescharacteristic
52#124—24no—
53#124—21no—
54#124—21no—
55C22⋊Q8—22no—
56C22⋊Q8—24no—
57C22⋊Q8—22no—
58C22⋊Q8—24no—
59#121—24no—
60#121—22no—
61#121—24no—
62#121—24no—
63#121—21no—
64#121—21no—
65#121—22no—
66#121D412yes—
67#121D412yes—
68C4×Q8—21no—
69C4×Q8—21no—
70C4×Q8C2×C412yes—
71C4×Q8D412yes—
72C4×D4—24no—
73C4×D4—22no—
74C4×D4—22no—
75C4×D4—21no—
76C4×D4—21no—
77C4×D4C2×C412yes—
78C4×D4D412yes—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4D411yescharacteristic
82C2×C42—21no—
83C2×C42C2311yescharacteristic
84C4⋊C8—21no—
85C4⋊C8—21no—
86C4⋊C8—21no—
87C4⋊C8—21no—
88#103—42no—
89C22⋊C8—21no—
90C22⋊C8—21no—
91C4×C8—21no—
92C4×C8—21no—
93#95—21no—
94#95—21no—
95#41—48no—
96#41—48no—
97#41—44no—
98#41—44no—
99#41—44no—
100#41—44no—
101#41—41no—
102#41—41no—
103#41—41no—
104#41—41no—
105#41—41no—
106#41—41no—
107#41—41no—
108#41—41no—
109C2×Q8—24no—
110C2×Q8—21no—
111C2×Q8—21no—
112C2×Q8—21no—
113C2×Q8—21no—
114C2×Q8—21no—
115C2×Q8—44no—
116C2×Q8—44no—
117C2×Q8—44no—
118C2×Q8—44no—
119C2×Q8C22⋊C412yes—
120C2×Q8D812yes—
121C2×Q8SD1612yes—
122C2×D4—24no—
123C2×D4—22no—
124C2×D4—22no—
125C2×D4—24no—
126C2×D4—21no—
127C2×D4—21no—
128C2×D4—21no—
129C2×D4—21no—
130C2×D4C22⋊C412yes—
131C2×D4C22⋊C412yes—
132C22×C4—21no—
133C22×C4—24no—
134C22×C4—24no—
135C22×C4—21no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—41no—
139C22×C4C2×D411yescharacteristic
140C22×C4C2×D411yescharacteristic
141C22×C4C2×D411yescharacteristic
142C2×C8—41no—
143C2×C8—41no—
144C2×C8—41no—
145C2×C8—41no—
146C2×C8—41no—
147C4⋊C4—24no—
148C4⋊C4—22no—
149C4⋊C4—21no—
150C4⋊C4—22no—
151C4⋊C4—21no—
152C4⋊C4—21no—
153C4⋊C4—41no—
154C4⋊C4—41no—
155C4⋊C4C22⋊C412yes—
156C4⋊C4C22⋊C412yes—
157C4⋊C4C22⋊C412yes—
158C22⋊C4—24no—
159C22⋊C4—24no—
160C22⋊C4—22no—
161C22⋊C4—22no—
162C22⋊C4—22no—
163C22⋊C4—22no—
164C42—22no—
165C42—22no—
166C42C22×C411yescharacteristic
167C42C2×D411yescharacteristic
168C42C2×D411yescharacteristic
169C42C2×D411yescharacteristic
170C23—22no—
171C23—22no—
172C23#12011yescharacteristic
173Q8—22no—
174Q8—22no—
175Q8—22no—
176Q8—48no—
177Q8—41no—
178Q8—41no—
179Q8—41no—
180Q8—41no—
181Q8#10214yes—
182D4—42no—
183D4—42no—
184D4—42no—
185D4—48no—
186D4—42no—
187D4—42no—
188D4—42no—
189D4—42no—
190D4—41no—
191D4—41no—
192D4—41no—
193D4—41no—
194D4—81no—
195D4—84no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—48no—
205C2×C4—48no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—81no—
211C2×C4—81no—
212C2×C4—81no—
213C2×C4—81no—
214C2×C4C2×C22⋊C411yescharacteristic
215C2×C4C2×C22⋊C411yescharacteristic
216C2×C4C2×C22⋊C411yescharacteristic
217C2×C4#12011yescharacteristic
218C2×C4#12011yescharacteristic
219C2×C4#12011yescharacteristic
220C2×C4C2×D811yescharacteristic
221C2×C4C2×SD1611yescharacteristic
222C2×C4AGL(1,ℤ/8)11yescharacteristic
223C2×C4AGL(1,ℤ/8)11yescharacteristic
224C8—81no—
225C8—81no—
226C8—81no—
227C22—21no—
228C22—21no—
229C22—42no—
230C22—42no—
231C22—42no—
232C22—42no—
233C22—42no—
234C22—42no—
235C22#37911yescharacteristic
236C22#44711yescharacteristic
237C22#45011yescharacteristic
238C4—21no—
239C4—21no—
240C4—21no—
241C4—21no—
242C4—24no—
243C4—42no—
244C4—42no—
245C4—42no—
246C4—161no—
247C4C2×#10211yescharacteristic
248C4#41811yescharacteristic
249C4#44711yescharacteristic
250C4#45011yescharacteristic
251C2—21no—
252C2—21no—
253C2—82no—
254C2—82no—
255C2#178611yescharacteristic
256C2#179411yescharacteristic
257C2#179611yescharacteristic
subgroupG/H
center, socleC22#379
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radical#3497C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1805C2semidirect
3. #1540C2semidirect
4. #1517C2semidirect
5. #1517C2semidirect
6. #1480C2semidirect
7. #1479C2semidirect
10. #550C4semidirect
12. D4×Q8C4semidirect

Maximal subgroups

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

subgroupindex
#34972
#18052
#15402
#15172
#14802
#14792