Finite Group Databasebeta

← #30112

Subgroups — #30112

Order 256.

509 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 509 conjugacy classes of subgroups — too many to draw. Only its 115 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#2856C211yesmaximal, characteristic, complemented
2C4×#121C211yesmaximal, characteristic, complemented
3#2100C211yesmaximal, characteristic, complemented
4#2094C211yesmaximal, characteristic, complemented
5#2041C211yesmaximal, characteristic, complemented
6#2034C212yesmaximal, complemented
7#2011C211yesmaximal, characteristic, complemented
8#1999C212yesmaximal, complemented
9#1926C212yesmaximal, complemented
10#1879C212yesmaximal, complemented
11#1817C211yesmaximal, characteristic, complemented
12C22×D8C2212yes—
13C2×#127C2211yescharacteristic
14C2×#121—21no—
15C2×#121—21no—
16C2×#121C2211yescharacteristic
17C2×#121C2212yes—
18C2×C4×D4—21no—
19C2×C4×D4C2211yescharacteristic
20C22×C42C2211yescharacteristic
21C2×#106C2212yes—
22C2×#102C2212yes—
23C2×#102C2212yes—
24C2×#102C2212yes—
25C2×#102C2212yes—
26C2×C22⋊C8C2212yes—
27#390—21no—
28#390—21no—
29#390C2212yes—
30#390C2211yescharacteristic
31#389C2211yescharacteristic
32#388—21no—
33#386—21no—
34#384C2211yescharacteristic
35#380C2211yescharacteristic
36C4×C4⋊C4C2211yescharacteristic
37C4×C22⋊C4—21no—
38#340C2212yes—
39#340C2212yes—
40#336C2211yescharacteristic
41#336C2211yescharacteristic
42#336C2212yes—
43C22×D4—21no—
44C22×D4—22no—
45C22×D4—21no—
46C22×D4—21no—
47C22×D4C2312yes—
48C22×D4C2311yescharacteristic
49C23×C4—21no—
50C23×C4D411yescharacteristic
51C23×C4D411yescharacteristic
52C23×C4C2311yescharacteristic
53C2×D8—24no—
54C2×D8—42no—
55C2×D8—42no—
56C22×C8C2312yes—
57C22×C8C2312yes—
58#127—42no—
59#121—22no—
60#121—22no—
61#121—42no—
62#121—42no—
63#121—44no—
64#121—44no—
65C4×D4—42no—
66C4×D4—44no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4C2311yescharacteristic
71C2×C4⋊C4C2312yes—
72C2×C4⋊C4C2311yescharacteristic
73C2×C4⋊C4C2311yescharacteristic
74C2×C22⋊C4—21no—
75C2×C22⋊C4—21no—
76C2×C22⋊C4—21no—
77C2×C22⋊C4—21no—
78C2×C22⋊C4—22no—
79C2×C22⋊C4—22no—
80C2×C22⋊C4—22no—
81C2×C22⋊C4—21no—
82C2×C42—21no—
83C2×C42—42no—
84C2×C42D411yescharacteristic
85C2×C42D411yescharacteristic
86C2×C42C2311yescharacteristic
87C2×C42C2311yescharacteristic
88#106—42no—
89#102—24no—
90#102—42no—
91#102—42no—
92#102—42no—
93C22⋊C8—42no—
94#95—21no—
95#95—21no—
96#95—21no—
97C24—21no—
98C24—22no—
99C24C2×D411yescharacteristic
100C2×D4—24no—
101C2×D4—21no—
102C2×D4—22no—
103C2×D4—22no—
104C2×D4—22no—
105C2×D4—22no—
106C2×D4—22no—
107C2×D4—22no—
108C2×D4—22no—
109C2×D4—22no—
110C2×D4—42no—
111C2×D4—42no—
112C2×D4—42no—
113C2×D4—42no—
114C2×D4—42no—
115C2×D4—42no—
116C2×D4—44no—
117C2×D4—44no—
118C2×D4—44no—
119C2×D4—42no—
120C2×D4—42no—
121C2×D4#4112yes—
122C22×C4—22no—
123C22×C4—21no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—22no—
134C22×C4—41no—
135C22×C4—41no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4—42no—
139C22×C4C2×D411yescharacteristic
140C22×C4C2×D411yescharacteristic
141C22×C4C2×D411yescharacteristic
142C22×C4C2×D411yescharacteristic
143C22×C4C2×D411yescharacteristic
144C22×C4C2411yescharacteristic
145D8—48no—
146C2×C8—22no—
147C2×C8—22no—
148C2×C8—22no—
149C2×C8—22no—
150C2×C8—22no—
151C2×C8—22no—
152C4⋊C4—21no—
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—22no—
156C4⋊C4—22no—
157C4⋊C4—42no—
158C4⋊C4—42no—
159C4⋊C4—42no—
160C4⋊C4#4112yes—
161C22⋊C4—24no—
162C22⋊C4—42no—
163C22⋊C4—44no—
164C22⋊C4—44no—
165C22⋊C4—42no—
166C22⋊C4—42no—
167C22⋊C4—42no—
168C22⋊C4—44no—
169C42—22no—
170C42—22no—
171C42—41no—
172C42—41no—
173C42—42no—
174C23—21no—
175C23—21no—
176C23—21no—
177C23—21no—
178C23—21no—
179C23—21no—
180C23—22no—
181C23—41no—
182C23—41no—
183C23—41no—
184C23—41no—
185C23—41no—
186C23—41no—
187C23—42no—
188C23—42no—
189C23—42no—
190C23—42no—
191C23—42no—
192C23—42no—
193C23—42no—
194C23AGL(1,ℤ/8)12yes—
195C23C22×D411yescharacteristic
196D4—24no—
197D4—48no—
198D4—42no—
199D4—42no—
200D4—42no—
201D4—42no—
202D4—42no—
203D4—84no—
204D4—84no—
205D4—88no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—44no—
220C2×C4—44no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—41no—
226C2×C4—42no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—41no—
234C2×C4—84no—
235C2×C4#13512yes—
236C2×C4AGL(1,ℤ/8)12yes—
237C2×C4AGL(1,ℤ/8)12yes—
238C2×C4C2×#4111yescharacteristic
239C2×C4C2×#4111yescharacteristic
240C2×C4H(5,𝔽2)12yes—
241C2×C4H(5,𝔽2)12yes—
242C8—42no—
243C8—42no—
244C22—22no—
245C22—22no—
246C22—22no—
247C22—41no—
248C22—41no—
249C22—41no—
250C22—41no—
251C22—42no—
252C22—42no—
253C22—42no—
254C22—81no—
255C22—81no—
256C22—81no—
257C22—81no—
258C22—82no—
259C22—82no—
260C22—82no—
261C22—82no—
262C22—82no—
263C22—82no—
264C22—82no—
265C22#52511yescharacteristic
266C22#53411yescharacteristic
267C22#53511yescharacteristic
268C22C2×#13511yescharacteristic
269C22C2×AGL(1,ℤ/8)11yescharacteristic
270C22C2×AGL(1,ℤ/8)11yescharacteristic
271C22C2×AGL(1,ℤ/8)11yescharacteristic
272C4—22no—
273C4—22no—
274C4—22no—
275C4—42no—
276C4—41no—
277C4—41no—
278C4—81no—
279C4—81no—
280C4—82no—
281C4#53812yes—
282C4#54012yes—
283C2—42no—
284C2—82no—
285C2—162no—
286C2#263511yescharacteristic
287C2#310611yescharacteristic
288C2#310911yescharacteristic
289C2#318811yescharacteristic
290C2#319211yescharacteristic
291C2#320711yescharacteristic
292C2#321711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2856C2semidirect
2. C4×#121C2semidirect
3. #2100C2semidirect
4. #2094C2semidirect
5. #2041C2semidirect
6. #2034C2semidirect
7. #2011C2semidirect
8. #1999C2semidirect
9. #1926C2semidirect
10. #1879C2semidirect
11. #1817C2semidirect

Maximal subgroups

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

subgroupindex
#28562
C4×#1212
#21002
#20942
#20412
#20342
#20112
#19992
#19262
#18792
#18172