Finite Group Databasebeta

← #22780

Subgroups — #22780

Order 256.

471 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#442C211yesmaximal, characteristic
2#2354C211yesmaximal, characteristic, complemented
3#2349C211yesmaximal, characteristic, complemented
4#1936C211yesmaximal, characteristic, complemented
5#1934C211yesmaximal, characteristic, complemented
6#1920C211yesmaximal, characteristic, complemented
7#1919C211yesmaximal, characteristic, complemented
8#1879C211yesmaximal, characteristic
9#1879C211yesmaximal, characteristic
10#1836C211yesmaximal, characteristic, complemented
11#1804C211yesmaximal, characteristic, complemented
12#1802C211yesmaximal, characteristic, complemented
13#1771C211yesmaximal, characteristic, complemented
14#1762C211yesmaximal, characteristic, complemented
15C4×#102C211yesmaximal, characteristic, complemented
16C22×D8C2211yescharacteristic
17C2×C4×D4C2211yescharacteristic
18C2×C4×D4C2211yescharacteristic
19C2×C4×D4C2211yescharacteristic
20C2×C4×D4C2211yescharacteristic
21C2×#117—21no—
22#442C418yescomplemented
23C2×#106C2211yescharacteristic
24C2×C4⋊C8C2211yescharacteristic
25C2×#102C2211yescharacteristic
26C2×#102C2211yescharacteristic
27C2×#102C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#97C2211yescharacteristic
32C2×#97C2211yescharacteristic
33C2×C4×C8C2211yescharacteristic
34#390C2211yescharacteristic
35#390C2211yescharacteristic
36#389C2211yescharacteristic
37#389C2211yescharacteristic
38#388—21no—
39#388—21no—
40#385—21no—
41#385—21no—
42#384C2211yescharacteristic
43#382C2211yescharacteristic
44#380—21no—
45#380—21no—
46C4×C4⋊C4C2211yescharacteristic
47C4×C4⋊C4C2211yescharacteristic
48C4×C4⋊C4C2211yescharacteristic
49C2×#95—21no—
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#336C2211yescharacteristic
59#336C2211yescharacteristic
60C22×D4C2311yescharacteristic
61C22×D4C2311yescharacteristic
62C23×C4—21no—
63C23×C4—21no—
64C23×C4—21no—
65C23×C4—21no—
66C2×D8—22no—
67C2×D8—22no—
68C2×D8C2×C414yes—
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71C22×C8C2311yescharacteristic
72C22×C8C2311yescharacteristic
73C4×D4—22no—
74C4×D4—22no—
75C4×D4—22no—
76C4×D4—22no—
77C4×D4—41no—
78C4×D4—41no—
79C4×D4C2×C414yes—
80C4×D4C2×C414yescomplemented
81#117—44no—
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×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C42—21no—
103C2×C42—21no—
104C2×C42C2311yescharacteristic
105C2×C42C2311yescharacteristic
106C2×C42C2311yescharacteristic
107#106C2×C414yescomplemented
108C4⋊C8—22no—
109#102—22no—
110#102—22no—
111#102—41no—
112#102—41no—
113#102C2×C414yes—
114#102C2×C414yescomplemented
115#97—41no—
116#97C2×C414yescomplemented
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—
127C24—21no—
128C2×D4—21no—
129C2×D4—21no—
130C2×D4—21no—
131C2×D4—21no—
132C2×D4—21no—
133C2×D4—21no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4C22×C412yes—
137C2×D4C22×C412yes—
138C2×D4#4112yes—
139C2×D4#4112yes—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—24no—
143C22×C4—24no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—81no—
155C22×C4—81no—
156C22×C4—81no—
157C22×C4—81no—
158C22×C4C2×D411yescharacteristic
159C22×C4C2×D411yescharacteristic
160C22×C4C2×D411yescharacteristic
161C22×C4C2×D411yescharacteristic
162C22×C4C2×D411yescharacteristic
163C22×C4C2×D411yescharacteristic
164C22×C4C2411yescharacteristic
165D8—28no—
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×C8C22×C412yes—
177C2×C8C22×C412yes—
178C4⋊C4—22no—
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—41no—
190C4⋊C4—42no—
191C4⋊C4—81no—
192C4⋊C4—81no—
193C4⋊C4—81no—
194C4⋊C4C22×C412yes—
195C4⋊C4C22×C412yes—
196C22⋊C4—24no—
197C22⋊C4—24no—
198C22⋊C4—42no—
199C22⋊C4—42no—
200C22⋊C4—42no—
201C22⋊C4—42no—
202C22⋊C4—81no—
203C22⋊C4—81no—
204C42—21no—
205C42—21no—
206C42—21no—
207C42—21no—
208C42—42no—
209C42—42no—
210C42C22×C412yes—
211C42#4112yes—
212C23—22no—
213C23—22no—
214C23—22no—
215C23—22no—
216C23—41no—
217C23—41no—
218C23—41no—
219C23—41no—
220C23—41no—
221C23—41no—
222C23—41no—
223C23—41no—
224C23—41no—
225C23—41no—
226C23C22×D411yescharacteristic
227D4—22no—
228D4—22no—
229D4—22no—
230D4—22no—
231D4—22no—
232D4—41no—
233D4—41no—
234D4#11714yescomplemented
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—48no—
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—84no—
270C2×C4—84no—
271C2×C4—84no—
272C2×C4C4×D412yes—
273C2×C4C4×D412yes—
274C2×C4AGL(1,ℤ/8)12yes—
275C2×C4C23×C411yescharacteristic
276C2×C4C2×#4111yescharacteristic
277C2×C4C2×#4111yescharacteristic
278C2×C4C2×#4111yescharacteristic
279C2×C4H(5,𝔽2)11yescharacteristic
280C2×C42−1+411yescharacteristic
281C8—22no—
282C8—22no—
283C8—41no—
284C8—41no—
285C22—24no—
286C22—42no—
287C22—42no—
288C22—42no—
289C22—42no—
290C22—42no—
291C22—42no—
292C22—42no—
293C22—42no—
294C22—42no—
295C22—81no—
296C22—81no—
297C22—81no—
298C22—81no—
299C22C2×C4×D411yescharacteristic
300C22#54611yescharacteristic
301C22#54811yescharacteristic
302C22C2×AGL(1,ℤ/8)11yescharacteristic
303C22#57511yescharacteristic
304C22#57611yescharacteristic
305C22#57711yescharacteristic
306C4—22no—
307C4—22no—
308C4—22no—
309C4—42no—
310C4—42no—
311C4—41no—
312C4—41no—
313C4—41no—
314C4—81no—
315C4—81no—
316C4—81no—
317C4—81no—
318C4C2×#11711yescharacteristic
319C4#52011yescharacteristic
320C4#55211yescharacteristic
321C4#55311yescharacteristic
322C2—44no—
323C2—82no—
324C2#235211yescharacteristic
325C2C4×AGL(1,ℤ/8)11yescharacteristic
326C2#294611yescharacteristic
327C2#330611yescharacteristic
328C2#331911yescharacteristic
329C2#336611yescharacteristic
330C2#337611yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC22×D8C22
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
2. #2354C2semidirect
3. #2349C2semidirect
4. #1936C2semidirect
5. #1934C2semidirect
6. #1920C2semidirect
7. #1919C2semidirect
10. #1836C2semidirect
11. #1804C2semidirect
12. #1802C2semidirect
13. #1771C2semidirect
14. #1762C2semidirect
15. C4×#102C2semidirect
22. #442C4semidirect
80. C4×D4C2×C4semidirect
107. #106C2×C4semidirect
114. #102C2×C4semidirect
116. #97C2×C4semidirect
234. D4#117semidirect

Maximal subgroups

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

subgroupindex
C2×#4422
#23542
#23492
#19362
#19342
#19202
#19192
#18792
#18362
#18042
#18022
#17712
#17622
C4×#1022