Finite Group Databasebeta

← #22944

Subgroups — #22944

Order 256.

451 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#482C211yesmaximal, characteristic
2#2361C211yesmaximal, characteristic
3#2347C211yesmaximal, characteristic, complemented
4#1943C211yesmaximal, characteristic, complemented
5#1934C211yesmaximal, characteristic, complemented
6#1926C211yesmaximal, characteristic, complemented
7#1920C211yesmaximal, characteristic, complemented
8#1918C211yesmaximal, characteristic, complemented
9#1895C211yesmaximal, characteristic
10#1864C211yesmaximal, characteristic, complemented
11#1853C211yesmaximal, characteristic, complemented
12C4×#106C211yesmaximal, characteristic, complemented
13#1762C211yesmaximal, characteristic, complemented
14#1753C211yesmaximal, characteristic, complemented
15#1736C211yesmaximal, characteristic, complemented
16C22×M16C2211yescharacteristic
17C2×#121C2211yescharacteristic
18C2×C4×D4—21no—
19C2×C4×D4C2211yescharacteristic
20C2×#117C2211yescharacteristic
21C22×C4⋊C4C2211yescharacteristic
22#482C418yescomplemented
23C2×#107C2211yescharacteristic
24C2×#106C2211yescharacteristic
25C2×#106C2211yescharacteristic
26C2×#106C2211yescharacteristic
27C2×#102C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×C22⋊C8C2211yescharacteristic
32C2×#97C2211yescharacteristic
33C2×C4×C8C2211yescharacteristic
34#390C2211yescharacteristic
35#389C2211yescharacteristic
36#389C2211yescharacteristic
37#389C2211yescharacteristic
38#388—21no—
39#388—21no—
40#388—21no—
41#386—21no—
42#385—21no—
43#383—21no—
44#380C2211yescharacteristic
45C4×C4⋊C4—21no—
46C4×C4⋊C4C2211yescharacteristic
47C4×C4⋊C4C2211yescharacteristic
48C4×C4⋊C4C2211yescharacteristic
49C4×C22⋊C4C2211yescharacteristic
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#336C2211yescharacteristic
59#336C2211yescharacteristic
60C22×D4—21no—
61C22×D4C2311yescharacteristic
62C23×C4—21no—
63C23×C4—21no—
64C23×C4C2311yescharacteristic
65C2×M16—22no—
66C2×M16—22no—
67C2×M16—41no—
68C22×C8C2311yescharacteristic
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71C22×C8C2311yescharacteristic
72#121—22no—
73#121C2×C414yes—
74C4×D4—22no—
75C4×D4—41no—
76C4×D4—44no—
77#117—22no—
78#117C2×C414yescomplemented
79C2×C4⋊C4—22no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4C2311yescharacteristic
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
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⋊C4D411yescharacteristic
99C2×C22⋊C4D411yescharacteristic
100C2×C42—21no—
101C2×C42—21no—
102C2×C42—21no—
103C2×C42—21no—
104C2×C42C2311yescharacteristic
105C2×C42C2311yescharacteristic
106#107C2×C414yescomplemented
107#106—22no—
108#106—41no—
109#106C2×C414yescomplemented
110#102—22no—
111#102—41no—
112#102C2×C414yescomplemented
113#102C2×C414yescomplemented
114C22⋊C8C2×C414yescomplemented
115#97—41no—
116C4×C8—22no—
117#95—21no—
118#95—21no—
119#95—21no—
120#95—21no—
121#95—21no—
122#95—21no—
123#95—21no—
124#95D411yescharacteristic
125#95D411yescharacteristic
126C24—21no—
127C24C2×D411yescharacteristic
128C2×D4—24no—
129C2×D4—21no—
130C2×D4—21no—
131C2×D4—21no—
132C2×D4—21no—
133C2×D4—21no—
134C2×D4—42no—
135C2×D4—42no—
136C2×D4C22×C412yes—
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—42no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—81no—
153C22×C4—81no—
154C22×C4C22×C412yes—
155C22×C4C2×D411yescharacteristic
156C22×C4C2×D411yescharacteristic
157C22×C4C2×D411yescharacteristic
158C22×C4C2×D411yescharacteristic
159C22×C4C2×D411yescharacteristic
160C22×C4#4112yes—
161C22×C4C2411yescharacteristic
162M16—44no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8C22×C412yes—
173C2×C8C22×C412yes—
174C2×C8#4112yes—
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—42no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4C22×C412yes—
190C4⋊C4C22×C412yes—
191C4⋊C4C22×C412yes—
192C4⋊C4#4112yes—
193C22⋊C4—24no—
194C22⋊C4—24no—
195C22⋊C4—22no—
196C22⋊C4—41no—
197C22⋊C4—42no—
198C22⋊C4—42no—
199C22⋊C4—81no—
200C22⋊C4—81no—
201C42—24no—
202C42—21no—
203C42—21no—
204C42—21no—
205C42—21no—
206C42—42no—
207C42—81no—
208C42—81no—
209C23—21no—
210C23—21no—
211C23—21no—
212C23—21no—
213C23—21no—
214C23—21no—
215C23—22no—
216C23—41no—
217C23—41no—
218C23—41no—
219C23—41no—
220C23—41no—
221C23—41no—
222C23C4×D412yes—
223C23C22×D411yescharacteristic
224D4—22no—
225D4—22no—
226D4—48no—
227D4—41no—
228D4—41no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—21no—
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—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—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—84no—
268C2×C4—84no—
269C2×C4C4×D412yes—
270C2×C4AGL(1,ℤ/8)12yes—
271C2×C4C23×C411yescharacteristic
272C2×C4C2×#4111yescharacteristic
273C2×C4C2×#4111yescharacteristic
274C2×C4C2×#4111yescharacteristic
275C2×C4H(5,𝔽2)11yescharacteristic
276C2×C42−1+411yescharacteristic
277C8—22no—
278C8—22no—
279C8—22no—
280C8—41no—
281C22—22no—
282C22—22no—
283C22—22no—
284C22—41no—
285C22—41no—
286C22—41no—
287C22—41no—
288C22—42no—
289C22—42no—
290C22—42no—
291C22—81no—
292C22—81no—
293C22—81no—
294C22—81no—
295C22C2×C4×D411yescharacteristic
296C22#54611yescharacteristic
297C22#54711yescharacteristic
298C22C2×AGL(1,ℤ/8)11yescharacteristic
299C22#57511yescharacteristic
300C22#57711yescharacteristic
301C22#57711yescharacteristic
302C4—22no—
303C4—22no—
304C4—41no—
305C4—42no—
306C4—42no—
307C4—42no—
308C4—41no—
309C4—42no—
310C4—41no—
311C4—81no—
312C4—81no—
313C4—81no—
314C4C4×#4111yescharacteristic
315C4#52011yescharacteristic
316C4#55111yescharacteristic
317C4#55211yescharacteristic
318C2—42no—
319C2—82no—
320C2#235911yescharacteristic
321C2C4×AGL(1,ℤ/8)11yescharacteristic
322C2#295211yescharacteristic
323C2#332411yescharacteristic
324C2#332611yescharacteristic
325C2#334511yescharacteristic
326C2#335011yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #2347C2semidirect
4. #1943C2semidirect
5. #1934C2semidirect
6. #1926C2semidirect
7. #1920C2semidirect
8. #1918C2semidirect
10. #1864C2semidirect
11. #1853C2semidirect
12. C4×#106C2semidirect
13. #1762C2semidirect
14. #1753C2semidirect
15. #1736C2semidirect
22. #482C4semidirect
78. #117C2×C4semidirect
106. #107C2×C4semidirect
109. #106C2×C4semidirect
112. #102C2×C4semidirect
113. #102C2×C4semidirect
114. C22⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#4822
#23612
#23472
#19432
#19342
#19262
#19202
#19182
#18952
#18642
#18532
C4×#1062
#17622
#17532
#17362