Finite Group Databasebeta

← #23764

Subgroups — #23764

Order 256.

412 conjugacy classes of subgroups.

The subgroup lattice has 323 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#121C211yesmaximal, characteristic
2#2283C211yesmaximal, characteristic, complemented
3#2101C211yesmaximal, characteristic, complemented
4#2092C211yesmaximal, characteristic, complemented
5#2085C211yesmaximal, characteristic, complemented
6#2077C211yesmaximal, characteristic, complemented
7#2075C211yesmaximal, characteristic, complemented
8#2073C211yesmaximal, characteristic, complemented
9#2034C211yesmaximal, characteristic
10#2033C211yesmaximal, characteristic
11#1776C211yesmaximal, characteristic, complemented
12#1776C211yesmaximal, characteristic, complemented
13#1762C211yesmaximal, characteristic, complemented
14#1762C211yesmaximal, characteristic, complemented
15#1749C211yesmaximal, characteristic, complemented
16C2×#121C2211yescharacteristic
17C2×C4×D4—21no—
18C2×C4×D4C2211yescharacteristic
19C22×C42C2211yescharacteristic
20C2×#107C2211yescharacteristic
21C2×#107C2211yescharacteristic
22C2×#106C2211yescharacteristic
23C2×#106C2211yescharacteristic
24C2×#102C2211yescharacteristic
25C2×#102C2211yescharacteristic
26C2×#102C2211yescharacteristic
27C2×#102C2211yescharacteristic
28C2×C22⋊C8C2211yescharacteristic
29C2×C22⋊C8C2211yescharacteristic
30C2×#97C2211yescharacteristic
31C2×#97C2211yescharacteristic
32#390C2211yescharacteristic
33#389C2211yescharacteristic
34#388—21no—
35#386—21no—
36#383—21no—
37#382C2211yescharacteristic
38#382C2211yescharacteristic
39#381—21no—
40#380C2211yescharacteristic
41#380C2211yescharacteristic
42#380C2211yescharacteristic
43C4×C4⋊C4C2211yescharacteristic
44C4×C4⋊C4C2211yescharacteristic
45C4×C4⋊C4C2211yescharacteristic
46C4×C22⋊C4—21no—
47C4×C22⋊C4—21no—
48#376—21no—
49#340C2211yescharacteristic
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×D4—21no—
60C22×D4C2311yescharacteristic
61C23×C4—21no—
62C23×C4D411yescharacteristic
63C23×C4D411yescharacteristic
64C23×C4C2311yescharacteristic
65C22×C8C2311yescharacteristic
66C22×C8C2311yescharacteristic
67C22×C8C2311yescharacteristic
68C22×C8C2311yescharacteristic
69#121—22no—
70#121—22no—
71C4×D4—22no—
72C4×D4—22no—
73C4×D4—44no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4C2311yescharacteristic
76C2×C4⋊C4C2311yescharacteristic
77C2×C4⋊C4C2311yescharacteristic
78C2×C4⋊C4C2311yescharacteristic
79C2×C4⋊C4C2311yescharacteristic
80C2×C4⋊C4C2311yescharacteristic
81C2×C4⋊C4C2311yescharacteristic
82C2×C22⋊C4—21no—
83C2×C22⋊C4—21no—
84C2×C22⋊C4—21no—
85C2×C22⋊C4—21no—
86C2×C22⋊C4—21no—
87C2×C22⋊C4—21no—
88C2×C22⋊C4—21no—
89C2×C42—22no—
90C2×C42—22no—
91C2×C42—21no—
92C2×C42—21no—
93C2×C42—21no—
94C2×C42D411yescharacteristic
95C2×C42D411yescharacteristic
96C2×C42C2311yescharacteristic
97C2×C42C2311yescharacteristic
98#107—22no—
99#107—22no—
100#106—22no—
101#106—22no—
102#102—22no—
103#102—22no—
104#102—22no—
105#102—22no—
106C22⋊C8—22no—
107C22⋊C8—22no—
108#97—22no—
109#97—22no—
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—
122C24C2×D411yescharacteristic
123C2×D4—24no—
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4—21no—
128C2×D4—21no—
129C2×D4—42no—
130C2×D4—42no—
131C2×D4#4112yes—
132C22×C4—22no—
133C22×C4—21no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
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—42no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4C2×D411yescharacteristic
155C22×C4C2×D411yescharacteristic
156C22×C4C2×D411yescharacteristic
157C22×C4C2×D411yescharacteristic
158C22×C4C2×D411yescharacteristic
159C22×C4C2411yescharacteristic
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8#4112yes—
171C2×C8#4112yes—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—42no—
186C4⋊C4#4112yes—
187C22⋊C4—24no—
188C22⋊C4—42no—
189C22⋊C4—42no—
190C22⋊C4—42no—
191C22⋊C4—42no—
192C22⋊C4—42no—
193C22⋊C4—42no—
194C42—22no—
195C42—22no—
196C42—21no—
197C42—21no—
198C42—42no—
199C42—42no—
200C42—42no—
201C42#4112yes—
202C42#4112yes—
203C23—21no—
204C23—21no—
205C23—21no—
206C23—21no—
207C23—21no—
208C23—21no—
209C23—22no—
210C23—41no—
211C23—41no—
212C23—41no—
213C23—41no—
214C23—41no—
215C23—41no—
216C23AGL(1,ℤ/8)12yes—
217C23C22×D411yescharacteristic
218D4—22no—
219D4—22no—
220D4—48no—
221D4—41no—
222D4—41no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—44no—
238C2×C4—44no—
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×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—84no—
264C2×C4#12312yes—
265C2×C4#12312yes—
266C2×C4C2×#4111yescharacteristic
267C2×C4C2×#4111yescharacteristic
268C2×C4C2×#4111yescharacteristic
269C2×C4C2×#4111yescharacteristic
270C2×C4C2×#4111yescharacteristic
271C2×C4C2×#4111yescharacteristic
272C8—22no—
273C8—22no—
274C8—41no—
275C8—41no—
276C22—22no—
277C22—22no—
278C22—22no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—42no—
284C22—42no—
285C22—42no—
286C22—81no—
287C22—81no—
288C22—81no—
289C22—81no—
290C22C2×#12311yescharacteristic
291C22#52511yescharacteristic
292C22#53211yescharacteristic
293C22C2×AGL(1,ℤ/8)11yescharacteristic
294C22#57511yescharacteristic
295C22#57511yescharacteristic
296C22#57511yescharacteristic
297C4—22no—
298C4—22no—
299C4—41no—
300C4—41no—
301C4—42no—
302C4—41no—
303C4—81no—
304C4—81no—
305C4—81no—
306C4—81no—
307C4—81no—
308C4—81no—
309C4#52911yescharacteristic
310C4#52911yescharacteristic
311C4#52911yescharacteristic
312C4#52911yescharacteristic
313C2—42no—
314C2—82no—
315C2#239511yescharacteristic
316C2#309311yescharacteristic
317C2#309411yescharacteristic
318C2#311111yescharacteristic
319C2#311411yescharacteristic
320C2#318011yescharacteristic
321C2#318311yescharacteristic
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
2. #2283C2semidirect
3. #2101C2semidirect
4. #2092C2semidirect
5. #2085C2semidirect
6. #2077C2semidirect
7. #2075C2semidirect
8. #2073C2semidirect
11. #1776C2semidirect
12. #1776C2semidirect
13. #1762C2semidirect
14. #1762C2semidirect
15. #1749C2semidirect

Maximal subgroups

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

subgroupindex
C4×#1212
#22832
#21012
#20922
#20852
#20772
#20752
#20732
#20342
#20332
#17762
#17622
#17492