Finite Group Databasebeta

← #29167

Subgroups — #29167

Order 256.

422 conjugacy classes of subgroups.

The subgroup lattice has 331 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2851C211yesmaximal, characteristic, complemented
2C4×#121C211yesmaximal, characteristic
3#2101C211yesmaximal, characteristic, complemented
4#2092C211yesmaximal, characteristic, complemented
5#2059C211yesmaximal, characteristic, complemented
6#2025C211yesmaximal, characteristic, complemented
7#1920C211yesmaximal, characteristic, complemented
8#1918C211yesmaximal, characteristic, complemented
9#1895C211yesmaximal, characteristic
10#1895C211yesmaximal, characteristic
11#1864C211yesmaximal, characteristic, complemented
12#1862C211yesmaximal, characteristic, complemented
13#1809C211yesmaximal, characteristic, complemented
14#1809C211yesmaximal, characteristic, complemented
15#1728C211yesmaximal, characteristic, complemented
16C22×M16C2211yescharacteristic
17C22×M16C2211yescharacteristic
18C2×#125C2211yescharacteristic
19C2×#121C2211yescharacteristic
20C2×C4×D4—21no—
21C2×C4×D4C2211yescharacteristic
22C22×C42C2211yescharacteristic
23C2×#107C2211yescharacteristic
24C2×#107C2211yescharacteristic
25C2×#106C2211yescharacteristic
26C2×#106C2211yescharacteristic
27C2×#102C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31#390C2211yescharacteristic
32#389C2211yescharacteristic
33#389C2211yescharacteristic
34#389C2211yescharacteristic
35#388—21no—
36#387—21no—
37#387—21no—
38#386—21no—
39#385—21no—
40#385—21no—
41#382C2211yescharacteristic
42#380C2211yescharacteristic
43#380C2211yescharacteristic
44#380C2211yescharacteristic
45C4×C4⋊C4C2211yescharacteristic
46C4×C22⋊C4—21no—
47#340C2211yescharacteristic
48#340C2211yescharacteristic
49#340C2211yescharacteristic
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#336C2211yescharacteristic
56#336C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×D4—21no—
60C22×D4C2311yescharacteristic
61C23×C4—21no—
62C23×C4D411yescharacteristic
63C23×C4D411yescharacteristic
64C23×C4C2311yescharacteristic
65C2×M16—22no—
66C2×M16—22no—
67C2×M16—22no—
68C2×M16—22no—
69C2×M16—41no—
70C2×M16—41no—
71C22×C8C2311yescharacteristic
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75#125—41no—
76#125—41no—
77#121—22no—
78#121Q814yes—
79C4×D4—41no—
80C4×D4—41no—
81C4×D4—44no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C42—21no—
102C2×C42—41no—
103C2×C42—41no—
104C2×C42D411yescharacteristic
105C2×C42D411yescharacteristic
106C2×C42C2311yescharacteristic
107C2×C42C2311yescharacteristic
108#107—22no—
109#107—41no—
110#106—22no—
111#106—41no—
112#102—22no—
113#102—22no—
114#102—41no—
115#102—41no—
116#95—21no—
117#95—21no—
118#95—21no—
119#95—21no—
120#95—21no—
121#95—21no—
122#95—21no—
123#95—21no—
124#95—21no—
125C24—21no—
126C24C2×D411yescharacteristic
127C2×D4—24no—
128C2×D4—21no—
129C2×D4—21no—
130C2×D4—21no—
131C2×D4—21no—
132C2×D4—21no—
133C2×D4—42no—
134C2×D4—42no—
135C2×D4C2×Q812yes—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—22no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—41no—
152C22×C4—41no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4C2×D411yescharacteristic
157C22×C4C2×D411yescharacteristic
158C22×C4C2×D411yescharacteristic
159C22×C4C2×D411yescharacteristic
160C22×C4C2×D411yescharacteristic
161C22×C4C2×Q812yes—
162C22×C4#4112yes—
163C22×C4C2411yescharacteristic
164M16—44no—
165M16—44no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8#4112yes—
177C2×C8#4112yes—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4C2×Q812yes—
197C22⋊C4—24no—
198C22⋊C4—42no—
199C22⋊C4—42no—
200C22⋊C4—42no—
201C22⋊C4—42no—
202C22⋊C4—42no—
203C22⋊C4—42no—
204C42—21no—
205C42—21no—
206C42—21no—
207C42—21no—
208C42—41no—
209C42—41no—
210C42—42no—
211C23—21no—
212C23—21no—
213C23—21no—
214C23—21no—
215C23—21no—
216C23—21no—
217C23—22no—
218C23—41no—
219C23—41no—
220C23—41no—
221C23—41no—
222C23—41no—
223C23—41no—
224C23C22⋊Q812yes—
225C23C22×D411yescharacteristic
226D4—22no—
227D4—22no—
228D4—48no—
229D4—41no—
230D4—41no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—44no—
250C2×C4—44no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—84no—
272C2×C4C22⋊Q812yes—
273C2×C4AGL(1,ℤ/8)12yes—
274C2×C4C22×Q811yescharacteristic
275C2×C4C2×#4111yescharacteristic
276C2×C4C2×#4111yescharacteristic
277C2×C4C2×#4111yescharacteristic
278C2×C4H(5,𝔽2)11yescharacteristic
279C2×C4H(5,𝔽2)11yescharacteristic
280C8—22no—
281C8—22no—
282C8—41no—
283C8—41no—
284C22—22no—
285C22—22no—
286C22—22no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—42no—
292C22—42no—
293C22—42no—
294C22—81no—
295C22—81no—
296C22—81no—
297C22—81no—
298C22C2×C22⋊Q811yescharacteristic
299C22#53211yescharacteristic
300C22#53411yescharacteristic
301C22C2×AGL(1,ℤ/8)11yescharacteristic
302C22#57511yescharacteristic
303C22#57511yescharacteristic
304C22#57511yescharacteristic
305C4—22no—
306C4—22no—
307C4—42no—
308C4—41no—
309C4—41no—
310C4—41no—
311C4—81no—
312C4—81no—
313C4—81no—
314C4—81no—
315C4—81no—
316C4—81no—
317C4#55111yescharacteristic
318C4#55111yescharacteristic
319C4#55411yescharacteristic
320C4#55411yescharacteristic
321C2—42no—
322C2—82no—
323C2#256411yescharacteristic
324C2#307711yescharacteristic
325C2#308011yescharacteristic
326C2#317111yescharacteristic
327C2#318311yescharacteristic
328C2#319411yescharacteristic
329C2#319711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
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. #2851C2semidirect
3. #2101C2semidirect
4. #2092C2semidirect
5. #2059C2semidirect
6. #2025C2semidirect
7. #1920C2semidirect
8. #1918C2semidirect
11. #1864C2semidirect
12. #1862C2semidirect
13. #1809C2semidirect
14. #1809C2semidirect
15. #1728C2semidirect

Maximal subgroups

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

subgroupindex
#28512
C4×#1212
#21012
#20922
#20592
#20252
#19202
#19182
#18952
#18642
#18622
#18092
#17282