Finite Group Databasebeta

← #8428

Subgroups — #8428

Order 256.

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

normal subgroups only

This group has 510 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#3496C211yesmaximal, characteristic
2#1804C211yesmaximal, characteristic, complemented
3#1540C211yesmaximal, characteristic, complemented
4#1517C211yesmaximal, characteristic, complemented
5#1517C211yesmaximal, characteristic, complemented
6#1480C211yesmaximal, characteristic, complemented
7#1478C211yesmaximal, characteristic, complemented
8C2×H(5,𝔽2)—21no—
9#547—21no—
10#547C412yescomplemented
11#546—22no—
12#546—22no—
13D42—21no—
14D42C412yescomplemented
15#537—21no—
16#534—22no—
17#532C2211yescharacteristic
18#532C2211yescharacteristic
19C2×#121—22no—
20C2×C4×D4C2211yescharacteristic
21#431C2211yescharacteristic, complemented
22#431C2211yescharacteristic, complemented
23C2×C4⋊C8C2211yescharacteristic, complemented
24C2×#102—21no—
25#382C2211yescharacteristic, complemented
26#340—21no—
27#332—21no—
28#331—21no—
29#327—22no—
30#327—22no—
31D4⋊C8—22no—
32D4⋊C8—22no—
33H(5,𝔽2)—42no—
34H(5,𝔽2)—42no—
35H(5,𝔽2)—42no—
36H(5,𝔽2)—42no—
37C2×#41—22no—
38C2×#41—22no—
39C2×#41—21no—
40C2×#41—21no—
41C22×D4—24no—
42C22×D4—22no—
43C22×D4—22no—
44C22×D4—22no—
45C22×D4D411yescharacteristic
46C23×C4—21no—
47C22×C8—21no—
48C4⋊Q8C2×C411yescharacteristic
49C4⋊Q8D411yescharacteristic
50#127C2×C411yescharacteristic
51#127D411yescharacteristic
52#124—21no—
53#124—21no—
54#123—24no—
55#123—24no—
56C22⋊Q8—22no—
57C22⋊Q8—22no—
58#121—24no—
59#121—24no—
60#121—22no—
61#121—22no—
62#121—21no—
63#121—21no—
64#121—22no—
65#121—22no—
66#121D412yes—
67#121D412yes—
68#120—24no—
69#120—24no—
70C4×D4—21no—
71C4×D4—21no—
72C4×D4—21no—
73C4×D4—21no—
74C4×D4C2×C412yes—
75C4×D4C2×C412yes—
76C4×D4D412yes—
77C4×D4D412yes—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4D411yescharacteristic
81C2×C22⋊C4—24no—
82C2×C22⋊C4—21no—
83C2×C42C2311yescharacteristic
84C4⋊C8—21no—
85C4⋊C8—21no—
86C4⋊C8—21no—
87C4⋊C8—21no—
88#102—42no—
89C22⋊C8—21no—
90C22⋊C8—21no—
91C4×C8—21no—
92C4×C8—21no—
93#95—21no—
94#95—21no—
95C24—21no—
96C24—24no—
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—21no—
110C2×Q8—21no—
111C2×D4—24no—
112C2×D4—24no—
113C2×D4—22no—
114C2×D4—22no—
115C2×D4—24no—
116C2×D4—21no—
117C2×D4—21no—
118C2×D4—21no—
119C2×D4—21no—
120C2×D4—21no—
121C2×D4—21no—
122C2×D4—21no—
123C2×D4—44no—
124C2×D4—44no—
125C2×D4—44no—
126C2×D4—48no—
127C2×D4—48no—
128C2×D4—44no—
129C2×D4—84no—
130C2×D4C22⋊C412yes—
131C2×D4C22⋊C412yes—
132C2×D4C22⋊C412yes—
133C2×D4D812yes—
134C2×D4SD1612yes—
135C22×C4—24no—
136C22×C4—21no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—41no—
142C22×C4—82no—
143C22×C4C2×D411yescharacteristic
144C22×C4C2×D411yescharacteristic
145C22×C4C2×D411yescharacteristic
146C2×C8—41no—
147C2×C8—41no—
148C2×C8—41no—
149C2×C8—41no—
150C2×C8—41no—
151C4⋊C4—24no—
152C4⋊C4—21no—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—41no—
156C4⋊C4—41no—
157C4⋊C4C22⋊C412yes—
158C4⋊C4C22⋊C412yes—
159C4⋊C4C22⋊C412yes—
160C22⋊C4—24no—
161C22⋊C4—24no—
162C22⋊C4—22no—
163C22⋊C4—22no—
164C22⋊C4—22no—
165C22⋊C4—22no—
166C22⋊C4—22no—
167C22⋊C4—22no—
168C42C22×C411yescharacteristic
169C42C2×D411yescharacteristic
170C42C2×D411yescharacteristic
171C42C2×D411yescharacteristic
172C23—22no—
173C23—22no—
174C23—22no—
175C23—22no—
176C23—22no—
177C23—48no—
178C23—81no—
179C23—81no—
180C23—84no—
181C23—84no—
182C23#12011yescharacteristic
183Q8—41no—
184Q8—41no—
185Q8—41no—
186Q8—41no—
187D4—22no—
188D4—22no—
189D4—22no—
190D4—42no—
191D4—42no—
192D4—42no—
193D4—48no—
194D4—48no—
195D4—42no—
196D4—42no—
197D4—42no—
198D4—42no—
199D4—41no—
200D4—41no—
201D4—41no—
202D4—41no—
203D4—81no—
204D4—84no—
205D4#10214yes—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—44no—
212C2×C4—48no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—82no—
218C2×C4—81no—
219C2×C4—81no—
220C2×C4—81no—
221C2×C4—81no—
222C2×C4C2×C22⋊C411yescharacteristic
223C2×C4C2×C22⋊C411yescharacteristic
224C2×C4C2×C22⋊C411yescharacteristic
225C2×C4#12011yescharacteristic
226C2×C4#12011yescharacteristic
227C2×C4#12011yescharacteristic
228C2×C4C2×D811yescharacteristic
229C2×C4C2×SD1611yescharacteristic
230C2×C4AGL(1,ℤ/8)11yescharacteristic
231C2×C4AGL(1,ℤ/8)11yescharacteristic
232C8—81no—
233C8—81no—
234C8—81no—
235C22—21no—
236C22—21no—
237C22—24no—
238C22—42no—
239C22—42no—
240C22—42no—
241C22—42no—
242C22—42no—
243C22—42no—
244C22—42no—
245C22—42no—
246C22—82no—
247C22—82no—
248C22—168no—
249C22#37911yescharacteristic
250C22#44711yescharacteristic
251C22#45011yescharacteristic
252C4—21no—
253C4—21no—
254C4—21no—
255C4—21no—
256C4—42no—
257C4—42no—
258C4—42no—
259C4—161no—
260C4C2×#10211yescharacteristic
261C4#41811yescharacteristic
262C4#44711yescharacteristic
263C4#45011yescharacteristic
264C2—21no—
265C2—21no—
266C2—44no—
267C2—82no—
268C2—82no—
269C2#178611yescharacteristic
270C2#179411yescharacteristic
271C2#179511yescharacteristic
subgroupG/H
center, socleC22#379
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radical#3496C2
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. #1804C2semidirect
3. #1540C2semidirect
4. #1517C2semidirect
5. #1517C2semidirect
6. #1480C2semidirect
7. #1478C2semidirect
10. #547C4semidirect
14. D42C4semidirect
21. #431C22semidirect
22. #431C22semidirect
23. C2×C4⋊C8C22semidirect
25. #382C22semidirect

Maximal subgroups

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

subgroupindex
#34962
#18042
#15402
#15172
#14802
#14782