Finite Group Databasebeta

← #29448

Subgroups — #29448

Order 256.

367 conjugacy classes of subgroups.

The subgroup lattice has 303 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2863C211yesmaximal, characteristic, complemented
2#2293C211yesmaximal, characteristic
3#2086C211yesmaximal, characteristic, complemented
4#2067C211yesmaximal, characteristic
5#2066C211yesmaximal, characteristic, complemented
6#2055C211yesmaximal, characteristic
7#1973C211yesmaximal, characteristic
8#1944C211yesmaximal, characteristic, complemented
9#1936C211yesmaximal, characteristic
10#1935C211yesmaximal, characteristic, complemented
11#1928C211yesmaximal, characteristic, complemented
12#1921C211yesmaximal, characteristic, complemented
13#1877C211yesmaximal, characteristic
14#1864C211yesmaximal, characteristic
15#1767C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C2×C4×D4C2211yescharacteristic
18C2×#117—21no—
19C2×#107C2211yescharacteristic
20C2×#107C2211yescharacteristic
21C2×#106C2211yescharacteristic
22C2×C4⋊C8C2211yescharacteristic
23C2×C4⋊C8C2211yescharacteristic
24C2×#103C2211yescharacteristic
25C2×#103C2211yescharacteristic
26C2×#103C2211yescharacteristic
27C2×#102C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#97C2211yescharacteristic
31C2×#97C2211yescharacteristic
32#391C2211yescharacteristic
33#391C2211yescharacteristic
34#391C2211yescharacteristic
35#390C2211yescharacteristic
36#390C2211yescharacteristic
37#389C2211yescharacteristic
38#389C2211yescharacteristic
39#389C2211yescharacteristic
40#389C2211yescharacteristic
41#389C2211yescharacteristic
42#388—21no—
43#388—21no—
44#387—21no—
45#387—21no—
46#387—21no—
47#383—21no—
48C4×C4⋊C4C2211yescharacteristic
49C4×C22⋊C4—21no—
50C43C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8C2311yescharacteristic
60C22×D4C2311yescharacteristic
61C23×C4—21no—
62C2×SD16—22no—
63C2×SD16—22no—
64C2×SD16—41no—
65C22×C8C2311yescharacteristic
66C22×C8C2311yescharacteristic
67C22×C8C2311yescharacteristic
68C22×C8C2311yescharacteristic
69C4×D4—22no—
70C4×D4—22no—
71#117—44no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4C2311yescharacteristic
78C2×C4⋊C4C2311yescharacteristic
79C2×C4⋊C4C2311yescharacteristic
80C2×C4⋊C4C2311yescharacteristic
81C2×C4⋊C4C2311yescharacteristic
82C2×C4⋊C4C2311yescharacteristic
83C2×C22⋊C4—21no—
84C2×C22⋊C4—21no—
85C2×C22⋊C4—21no—
86C2×C22⋊C4—21no—
87C2×C22⋊C4—21no—
88C2×C42—21no—
89C2×C42D411yescharacteristic
90C2×C42D411yescharacteristic
91C2×C42D411yescharacteristic
92C2×C42D411yescharacteristic
93C2×C42C2311yescharacteristic
94C2×C42C2311yescharacteristic
95C2×C42C2311yescharacteristic
96#107—22no—
97#107—41no—
98#106—22no—
99C4⋊C8—22no—
100C4⋊C8—22no—
101#103—22no—
102#103—22no—
103#103—41no—
104#102—22no—
105#102—22no—
106#102—41no—
107#97—41no—
108#97—41no—
109#95—21no—
110#95—21no—
111#95—21no—
112#95—21no—
113#95—21no—
114#95—21no—
115#95—21no—
116#95—21no—
117#95—21no—
118#95—21no—
119#95—21no—
120#95—21no—
121C24—21no—
122C2×Q8—21no—
123C2×Q8—21no—
124C2×Q8—21no—
125C2×Q8—21no—
126C2×Q8—21no—
127C2×Q8—21no—
128C2×D4—21no—
129C2×D4—21no—
130C2×D4—21no—
131C2×D4—21no—
132C2×D4—21no—
133C2×D4#4112yes—
134C22×C4—24no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4C2×D411yescharacteristic
145C22×C4C2×D411yescharacteristic
146C22×C4C2×D411yescharacteristic
147C22×C4C2×D411yescharacteristic
148C22×C4C2×D411yescharacteristic
149C22×C4C2×D411yescharacteristic
150C22×C4C2411yescharacteristic
151SD16—44no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8#4112yes—
163C2×C8#4112yes—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4#4112yes—
181C22⋊C4—24no—
182C22⋊C4—42no—
183C22⋊C4—42no—
184C22⋊C4—42no—
185C22⋊C4—42no—
186C42—22no—
187C42—22no—
188C42—21no—
189C42—21no—
190C42—21no—
191C42—21no—
192C42—21no—
193C42—21no—
194C42—41no—
195C42—41no—
196C42—42no—
197C23—22no—
198C23—41no—
199C23—41no—
200C23—41no—
201C23—41no—
202C23—41no—
203C23—41no—
204C23C22×D411yescharacteristic
205Q8—41no—
206Q8—41no—
207Q8—41no—
208Q8—41no—
209D4—22no—
210D4—22no—
211D4—41no—
212D4—41no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—48no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4AGL(1,ℤ/8)12yes—
252C2×C4#13712yes—
253C2×C4C2×#4111yescharacteristic
254C2×C4C2×#4111yescharacteristic
255C2×C4C2×#4111yescharacteristic
256C2×C4C2×#4111yescharacteristic
257C2×C4H(5,𝔽2)11yescharacteristic
258C2×C42−1+411yescharacteristic
259C8—22no—
260C8—22no—
261C8—41no—
262C8—41no—
263C22—42no—
264C22—42no—
265C22—42no—
266C22—81no—
267C22—81no—
268C22—81no—
269C22—81no—
270C22#52511yescharacteristic
271C22#53211yescharacteristic
272C22#53711yescharacteristic
273C22C2×AGL(1,ℤ/8)11yescharacteristic
274C22C2×#13711yescharacteristic
275C22#57511yescharacteristic
276C22#57511yescharacteristic
277C4—22no—
278C4—22no—
279C4—42no—
280C4—41no—
281C4—41no—
282C4—41no—
283C4—41no—
284C4—81no—
285C4—81no—
286C4—81no—
287C4—81no—
288C4—81no—
289C4—81no—
290C4#55111yescharacteristic
291C4#55211yescharacteristic
292C4#55311yescharacteristic
293C4#55611yescharacteristic
294C2—82no—
295C2#260111yescharacteristic
296C2#311011yescharacteristic
297C2#311111yescharacteristic
298C2#317211yescharacteristic
299C2#318011yescharacteristic
300C2#325111yescharacteristic
301C2#325211yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2863C2semidirect
3. #2086C2semidirect
5. #2066C2semidirect
8. #1944C2semidirect
10. #1935C2semidirect
11. #1928C2semidirect
12. #1921C2semidirect
15. #1767C2semidirect

Maximal subgroups

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

subgroupindex
#28632
#22932
#20862
#20672
#20662
#20552
#19732
#19442
#19362
#19352
#19282
#19212
#18772
#18642
#17672