Finite Group Databasebeta

← #22781

Subgroups — #22781

Order 256.

433 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#440C211yesmaximal, characteristic
2#2354C211yesmaximal, characteristic
3#2351C211yesmaximal, characteristic, complemented
4#1936C211yesmaximal, characteristic
5#1934C211yesmaximal, characteristic
6#1921C211yesmaximal, characteristic, complemented
7#1918C211yesmaximal, characteristic, complemented
8#1880C211yesmaximal, characteristic
9#1880C211yesmaximal, characteristic
10#1836C211yesmaximal, characteristic, complemented
11#1804C211yesmaximal, characteristic
12#1803C211yesmaximal, characteristic, complemented
13#1771C211yesmaximal, characteristic, complemented
14#1763C211yesmaximal, characteristic, complemented
15C4×#103C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C2×C4×Q8C2211yescharacteristic
18C2×C4×Q8C2211yescharacteristic
19C2×C4×Q8C2211yescharacteristic
20C2×C4×D4C2211yescharacteristic
21C2×#117—21no—
22#440C418yescomplemented
23C2×#107C2211yescharacteristic
24C2×C4⋊C8C2211yescharacteristic
25C2×#103C2211yescharacteristic
26C2×#103C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#97C2211yescharacteristic
32C2×#97C2211yescharacteristic
33C2×C4×C8C2211yescharacteristic
34#390C2211yescharacteristic
35#390C2211yescharacteristic
36#389—21no—
37#389—21no—
38#389C2211yescharacteristic
39#389C2211yescharacteristic
40#388—21no—
41#388—21no—
42#384C2211yescharacteristic
43#382—21no—
44#382C2211yescharacteristic
45#380—21no—
46C4×C4⋊C4—21no—
47C4×C4⋊C4C2211yescharacteristic
48C4×C4⋊C4C2211yescharacteristic
49C4×C4⋊C4C2211yescharacteristic
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#336C2211yescharacteristic
59#336C2211yescharacteristic
60C22×Q8C2311yescharacteristic
61C22×D4C2311yescharacteristic
62C23×C4—21no—
63C2×SD16—22no—
64C2×SD16—22no—
65C2×SD16C2×C414yes—
66C22×C8C2311yescharacteristic
67C22×C8C2311yescharacteristic
68C22×C8C2311yescharacteristic
69C22×C8C2311yescharacteristic
70C4×Q8—22no—
71C4×Q8—22no—
72C4×Q8—22no—
73C4×Q8—41no—
74C4×Q8—41no—
75C4×Q8C2×C414yes—
76C4×D4—22no—
77C4×D4C2×C414yes—
78#117—44no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4D411yescharacteristic
87C2×C4⋊C4D411yescharacteristic
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×C42—21no—
100C2×C42—21no—
101C2×C42—21no—
102C2×C42—21no—
103C2×C42—21no—
104C2×C42C2311yescharacteristic
105C2×C42C2311yescharacteristic
106C2×C42C2311yescharacteristic
107#107C2×C414yes—
108C4⋊C8—22no—
109#103—22no—
110#103—22no—
111#103C2×C414yes—
112#102—41no—
113#102—41no—
114#102C2×C414yes—
115#97—41no—
116#97C2×C414yes—
117C4×C8—41no—
118#95—21no—
119#95—21no—
120#95—21no—
121#95—21no—
122#95—21no—
123#95—21no—
124#95D411yescharacteristic
125#95D411yescharacteristic
126C24—21no—
127C2×Q8—21no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8C22×C412yes—
131C2×Q8#4112yes—
132C2×Q8#4112yes—
133C2×D4—21no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—21no—
137C2×D4—21no—
138C2×D4C22×C412yes—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—24no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4C2×D411yescharacteristic
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D411yescharacteristic
152C22×C4C2×D411yescharacteristic
153C22×C4C2×D411yescharacteristic
154C22×C4C2×D411yescharacteristic
155C22×C4C2411yescharacteristic
156SD16—28no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8C22×C412yes—
168C2×C8C22×C412yes—
169C4⋊C4—24no—
170C4⋊C4—22no—
171C4⋊C4—21no—
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—41no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—81no—
188C4⋊C4C22×C412yes—
189C4⋊C4C22×C412yes—
190C22⋊C4—24no—
191C22⋊C4—42no—
192C22⋊C4—42no—
193C22⋊C4—81no—
194C22⋊C4—81no—
195C42—24no—
196C42—21no—
197C42—21no—
198C42—21no—
199C42—21no—
200C42—42no—
201C42—42no—
202C42—42no—
203C42—42no—
204C42C22×C412yes—
205C42#4112yes—
206C23—22no—
207C23—41no—
208C23—41no—
209C23—41no—
210C23—41no—
211C23—41no—
212C23—41no—
213C23C22×D411yescharacteristic
214Q8—22no—
215Q8—22no—
216Q8—22no—
217Q8#11714yescomplemented
218D4—22no—
219D4—22no—
220D4—41no—
221D4—41no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—48no—
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×C4C4×D412yes—
260C2×C4C4×D412yes—
261C2×C4AGL(1,ℤ/8)12yes—
262C2×C4C23×C411yescharacteristic
263C2×C4C2×#4111yescharacteristic
264C2×C4C2×#4111yescharacteristic
265C2×C4C2×#4111yescharacteristic
266C2×C4H(5,𝔽2)11yescharacteristic
267C2×C42−1+411yescharacteristic
268C8—22no—
269C8—22no—
270C8—41no—
271C8—41no—
272C22—42no—
273C22—42no—
274C22—42no—
275C22—81no—
276C22—81no—
277C22—81no—
278C22—81no—
279C22C2×C4×D411yescharacteristic
280C22#54611yescharacteristic
281C22#54811yescharacteristic
282C22C2×AGL(1,ℤ/8)11yescharacteristic
283C22#57511yescharacteristic
284C22#57711yescharacteristic
285C22#57811yescharacteristic
286C4—24no—
287C4—22no—
288C4—22no—
289C4—22no—
290C4—42no—
291C4—42no—
292C4—41no—
293C4—41no—
294C4—41no—
295C4—81no—
296C4—81no—
297C4—81no—
298C4—81no—
299C4C2×#11711yescharacteristic
300C4#52011yescharacteristic
301C4#55211yescharacteristic
302C4#55311yescharacteristic
303C2—82no—
304C2#235211yescharacteristic
305C2C4×AGL(1,ℤ/8)11yescharacteristic
306C2#294711yescharacteristic
307C2#330811yescharacteristic
308C2#332111yescharacteristic
309C2#336811yescharacteristic
310C2#337711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #2351C2semidirect
6. #1921C2semidirect
7. #1918C2semidirect
10. #1836C2semidirect
12. #1803C2semidirect
13. #1771C2semidirect
14. #1763C2semidirect
15. C4×#103C2semidirect
22. #440C4semidirect
217. Q8#117semidirect

Maximal subgroups

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

subgroupindex
C2×#4402
#23542
#23512
#19362
#19342
#19212
#19182
#18802
#18362
#18042
#18032
#17712
#17632
C4×#1032