Finite Group Databasebeta

← #32718

Subgroups — #32718

Order 256.

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

normal subgroups only

This group has 476 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
2#2854C211yesmaximal, characteristic, complemented
3#2099C211yesmaximal, characteristic, complemented
4#2093C211yesmaximal, characteristic, complemented
5#2067C212yesmaximal, complemented
6#2054C212yesmaximal, complemented
7#2041C211yesmaximal, characteristic, complemented
8#2033C212yesmaximal, complemented
9#2011C211yesmaximal, characteristic, complemented
10#1999C212yesmaximal, complemented
11#1797C211yesmaximal, characteristic, complemented
12C22×D8C2212yes—
13C2×#127C2211yescharacteristic
14C2×#123—21no—
15C2×#121—21no—
16C2×#121—21no—
17C2×#121C2211yescharacteristic
18C2×#121C2212yes—
19C22×C42C2211yescharacteristic
20C2×#107C2212yes—
21C2×C4⋊C8C2212yes—
22C2×#102C2212yes—
23C2×#102C2212yes—
24C2×#102C2212yes—
25C2×#102C2212yes—
26C2×C22⋊C8C2212yes—
27#390—21no—
28#390—21no—
29#390C2211yescharacteristic
30#390C2212yes—
31#390C2211yescharacteristic
32#389C2211yescharacteristic
33#389C2211yescharacteristic
34#388—21no—
35#388—21no—
36#386—21no—
37#384C2211yescharacteristic
38#380C2211yescharacteristic
39#340C2212yes—
40#340C2212yes—
41#336C2211yescharacteristic
42#336C2211yescharacteristic
43C22×D4—22no—
44C22×D4—21no—
45C22×D4—21no—
46C22×D4—21no—
47C22×D4C2312yes—
48C22×D4C2311yescharacteristic
49C23×C4D411yescharacteristic
50C23×C4D411yescharacteristic
51C23×C4C2311yescharacteristic
52C2×D8—42no—
53C2×D8—42no—
54C2×D8—42no—
55C22×C8C2312yes—
56C22×C8C2312yes—
57#127—42no—
58#123—44no—
59#121—42no—
60#121—42no—
61#121—42no—
62#121—44no—
63#121—44no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4C2312yes—
69C2×C4⋊C4C2311yescharacteristic
70C2×C4⋊C4C2311yescharacteristic
71C2×C4⋊C4C2311yescharacteristic
72C2×C22⋊C4—21no—
73C2×C22⋊C4—22no—
74C2×C22⋊C4—22no—
75C2×C22⋊C4—21no—
76C2×C22⋊C4—22no—
77C2×C22⋊C4—21no—
78C2×C22⋊C4—21no—
79C2×C22⋊C4—21no—
80C2×C22⋊C4—21no—
81C2×C42—42no—
82C2×C42D411yescharacteristic
83C2×C42D411yescharacteristic
84C2×C42C2311yescharacteristic
85C2×C42C2311yescharacteristic
86#107—42no—
87C4⋊C8—42no—
88#102—42no—
89#102—42no—
90#102—42no—
91#102—42no—
92C22⋊C8—42no—
93#95—21no—
94#95—21no—
95#95—21no—
96C24—21no—
97C24—22no—
98C24C2×D411yescharacteristic
99C2×D4—22no—
100C2×D4—22no—
101C2×D4—22no—
102C2×D4—22no—
103C2×D4—22no—
104C2×D4—22no—
105C2×D4—22no—
106C2×D4—21no—
107C2×D4—22no—
108C2×D4—21no—
109C2×D4—42no—
110C2×D4—42no—
111C2×D4—42no—
112C2×D4—42no—
113C2×D4—42no—
114C2×D4—44no—
115C2×D4—44no—
116C2×D4—42no—
117C2×D4—44no—
118C2×D4—42no—
119C2×D4—42no—
120C2×D4—42no—
121C22×C4—22no—
122C22×C4—21no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—22no—
130C22×C4—41no—
131C22×C4—41no—
132C22×C4—41no—
133C22×C4—41no—
134C22×C4D814yescomplemented
135C22×C4C2×D411yescharacteristic
136C22×C4C2×D411yescharacteristic
137C22×C4C2×D411yescharacteristic
138C22×C4C2×D411yescharacteristic
139C22×C4C2×D411yescharacteristic
140C22×C4C2411yescharacteristic
141D8—48no—
142C2×C8—22no—
143C2×C8—22no—
144C2×C8—22no—
145C2×C8—22no—
146C2×C8—22no—
147C2×C8—22no—
148C4⋊C4—21no—
149C4⋊C4—22no—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—22no—
153C4⋊C4—21no—
154C4⋊C4—42no—
155C4⋊C4—42no—
156C4⋊C4—42no—
157C4⋊C4—42no—
158C22⋊C4—42no—
159C22⋊C4—42no—
160C22⋊C4—44no—
161C22⋊C4—44no—
162C22⋊C4—42no—
163C22⋊C4—44no—
164C22⋊C4—42no—
165C22⋊C4—42no—
166C22⋊C4—42no—
167C42—22no—
168C42—22no—
169C42—41no—
170C42—41no—
171C23—21no—
172C23—21no—
173C23—21no—
174C23—21no—
175C23—21no—
176C23—21no—
177C23—41no—
178C23—42no—
179C23—42no—
180C23—41no—
181C23—42no—
182C23—41no—
183C23—41no—
184C23—42no—
185C23—42no—
186C23—41no—
187C23—41no—
188C23—42no—
189C23—41no—
190C23—42no—
191C23C2×D812yes—
192C23C22×D411yescharacteristic
193D4—42no—
194D4—42no—
195D4—42no—
196D4—42no—
197D4—42no—
198D4—42no—
199D4—84no—
200D4—84no—
201D4—84no—
202D4—88no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—44no—
216C2×C4—44no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—41no—
220C2×C4—42no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4C2×D812yes—
232C2×C4C2×D812yes—
233C2×C4AGL(1,ℤ/8)12yes—
234C2×C4H(5,𝔽2)11yescharacteristic
235C2×C4H(5,𝔽2)12yes—
236C2×C4H(5,𝔽2)12yes—
237C2×C42−1+411yescharacteristic
238C8—42no—
239C8—42no—
240C22—22no—
241C22—22no—
242C22—22no—
243C22—41no—
244C22—41no—
245C22—41no—
246C22—41no—
247C22—82no—
248C22—81no—
249C22—82no—
250C22—81no—
251C22—82no—
252C22—81no—
253C22—82no—
254C22—81no—
255C22—81no—
256C22—82no—
257C22—82no—
258C22—81no—
259C22—81no—
260C22—82no—
261C22#53411yescharacteristic
262C22#53511yescharacteristic
263C22#53711yescharacteristic
264C22C22×D811yescharacteristic
265C22C2×AGL(1,ℤ/8)11yescharacteristic
266C22#57511yescharacteristic
267C22#57511yescharacteristic
268C4—22no—
269C4—22no—
270C4—22no—
271C4—41no—
272C4—41no—
273C4—81no—
274C4—81no—
275C4—82no—
276C4—81no—
277C4#56012yes—
278C4#56212yes—
279C2—42no—
280C2—162no—
281C2—161no—
282C2#281311yescharacteristic
283C2#318611yescharacteristic
284C2#319411yescharacteristic
285C2#320511yescharacteristic
286C2#321911yescharacteristic
287C2#324511yescharacteristic
288C2#325111yescharacteristic
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. #2854C2semidirect
3. #2099C2semidirect
4. #2093C2semidirect
5. #2067C2semidirect
6. #2054C2semidirect
7. #2041C2semidirect
8. #2033C2semidirect
9. #2011C2semidirect
10. #1999C2semidirect
11. #1797C2semidirect
134. C22×C4D8semidirect

Maximal subgroups

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

subgroupindex
#28562
#28542
#20992
#20932
#20672
#20542
#20412
#20332
#20112
#19992
#17972