Finite Group Databasebeta

← #29372

Subgroups — #29372

Order 256.

427 conjugacy classes of subgroups.

The subgroup lattice has 337 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#124C211yesmaximal, characteristic, complemented
2#2293C211yesmaximal, characteristic, complemented
3#2101C211yesmaximal, characteristic, complemented
4#2092C211yesmaximal, characteristic, complemented
5#2042C211yesmaximal, characteristic, complemented
6#2012C211yesmaximal, characteristic
7#1928C211yesmaximal, characteristic, complemented
8#1926C211yesmaximal, characteristic, complemented
9#1880C211yesmaximal, characteristic, complemented
10#1879C211yesmaximal, characteristic
11#1879C211yesmaximal, characteristic
12#1877C211yesmaximal, characteristic, complemented
13#1865C211yesmaximal, characteristic, complemented
14#1862C211yesmaximal, characteristic, complemented
15#1839C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C22×D8C2211yescharacteristic
18C2×#124C2211yescharacteristic
19C2×C4×Q8C2211yescharacteristic
20C2×C4×D4C2211yescharacteristic
21C2×C4×D4C2211yescharacteristic
22C2×#117—21no—
23C2×#107C2211yescharacteristic
24C2×#106C2211yescharacteristic
25C2×#103C2211yescharacteristic
26C2×#103C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33#391C2211yescharacteristic
34#390C2211yescharacteristic
35#390C2211yescharacteristic
36#390C2211yescharacteristic
37#389C2211yescharacteristic
38#389C2211yescharacteristic
39#388—21no—
40#388—21no—
41#388—21no—
42#388—21no—
43#382C2211yescharacteristic
44C4×C4⋊C4C2211yescharacteristic
45C4×C22⋊C4—21no—
46C4×C22⋊C4—21no—
47C4×C22⋊C4—21no—
48C43C2211yescharacteristic
49#340C2211yescharacteristic
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#336C2211yescharacteristic
54#336C2211yescharacteristic
55#336C2211yescharacteristic
56#336C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8C2311yescharacteristic
60C22×D4C2311yescharacteristic
61C22×D4C2311yescharacteristic
62C23×C4—21no—
63C23×C4—21no—
64C2×SD16—22no—
65C2×SD16—22no—
66C2×SD16—41no—
67C2×D8—22no—
68C2×D8—22no—
69C2×D8—41no—
70C22×C8C2311yescharacteristic
71C22×C8C2311yescharacteristic
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74#124—22no—
75#124—22no—
76C4×Q8—41no—
77C4×Q8—41no—
78C4×D4—22no—
79C4×D4—22no—
80C4×D4—41no—
81C4×D4—41no—
82#117—44no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
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—21no—
103C2×C42D411yescharacteristic
104C2×C42D411yescharacteristic
105C2×C42D411yescharacteristic
106C2×C42D411yescharacteristic
107C2×C42C2311yescharacteristic
108C2×C42C2311yescharacteristic
109C2×C42C2311yescharacteristic
110#107—41no—
111#106—41no—
112#103—22no—
113#103—22no—
114#103—41no—
115#102—22no—
116#102—22no—
117#102—41no—
118#102—41no—
119#102—41no—
120#95—21no—
121#95—21no—
122#95—21no—
123#95—21no—
124#95—21no—
125#95—21no—
126#95—21no—
127C24—21no—
128C24—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8#4112yes—
135C2×D4—21no—
136C2×D4—21no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4#4112yes—
146C2×D4#4112yes—
147C22×C4—24no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4C2×D411yescharacteristic
160C22×C4C2×D411yescharacteristic
161C22×C4C2×D411yescharacteristic
162C22×C4C2×D411yescharacteristic
163C22×C4C2×D411yescharacteristic
164C22×C4C2×D411yescharacteristic
165C22×C4C2411yescharacteristic
166SD16—44no—
167D8—44no—
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—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—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—42no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4#4112yes—
193C22⋊C4—24no—
194C22⋊C4—24no—
195C22⋊C4—24no—
196C22⋊C4—42no—
197C22⋊C4—42no—
198C22⋊C4—42no—
199C22⋊C4—42no—
200C22⋊C4—42no—
201C22⋊C4—42no—
202C22⋊C4—42no—
203C42—22no—
204C42—22no—
205C42—21no—
206C42—21no—
207C42—21no—
208C42—21no—
209C42—21no—
210C42—21no—
211C42—41no—
212C42—41no—
213C42—42no—
214C42—42no—
215C23—22no—
216C23—22no—
217C23—41no—
218C23—41no—
219C23—41no—
220C23—41no—
221C23—41no—
222C23—41no—
223C23—41no—
224C23—41no—
225C23—41no—
226C23—41no—
227C23—41no—
228C23—41no—
229C23C22×D411yescharacteristic
230Q8—22no—
231Q8—22no—
232Q8—41no—
233Q8—41no—
234D4—22no—
235D4—22no—
236D4—22no—
237D4—22no—
238D4—41no—
239D4—41no—
240D4—41no—
241D4—41no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—48no—
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—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—84no—
278C2×C4#13512yes—
279C2×C4AGL(1,ℤ/8)12yes—
280C2×C4C2×#4111yescharacteristic
281C2×C4C2×#4111yescharacteristic
282C2×C4C2×#4111yescharacteristic
283C2×C4C2×#4111yescharacteristic
284C2×C4H(5,𝔽2)11yescharacteristic
285C2×C4H(5,𝔽2)11yescharacteristic
286C8—41no—
287C8—41no—
288C8—41no—
289C8—41no—
290C22—42no—
291C22—42no—
292C22—42no—
293C22—42no—
294C22—42no—
295C22—42no—
296C22—81no—
297C22—81no—
298C22—81no—
299C22—81no—
300C22—81no—
301C22—81no—
302C22—81no—
303C22—81no—
304C22#52511yescharacteristic
305C22#52511yescharacteristic
306C22#53511yescharacteristic
307C22C2×#13511yescharacteristic
308C22C2×AGL(1,ℤ/8)11yescharacteristic
309C22#57511yescharacteristic
310C22#57511yescharacteristic
311C4—22no—
312C4—22no—
313C4—41no—
314C4—42no—
315C4—42no—
316C4—41no—
317C4—41no—
318C4—41no—
319C4—81no—
320C4—81no—
321C4—81no—
322C4—81no—
323C4#55111yescharacteristic
324C4#55111yescharacteristic
325C4#55311yescharacteristic
326C4#55311yescharacteristic
327C2—82no—
328C2—82no—
329C2#259811yescharacteristic
330C2#310211yescharacteristic
331C2#310511yescharacteristic
332C2#311111yescharacteristic
333C2#311511yescharacteristic
334C2#321211yescharacteristic
335C2#321311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D8C22
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. C4×#124C2semidirect
2. #2293C2semidirect
3. #2101C2semidirect
4. #2092C2semidirect
5. #2042C2semidirect
7. #1928C2semidirect
8. #1926C2semidirect
9. #1880C2semidirect
12. #1877C2semidirect
13. #1865C2semidirect
14. #1862C2semidirect
15. #1839C2semidirect

Maximal subgroups

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

subgroupindex
C4×#1242
#22932
#21012
#20922
#20422
#20122
#19282
#19262
#18802
#18792
#18772
#18652
#18622
#18392