Finite Group Databasebeta

← #22580

Subgroups — #22580

Order 256.

575 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#418C211yesmaximal, characteristic
2#2347C211yesmaximal, characteristic, complemented
3D4×C22⋊C4C211yesmaximal, characteristic
4#1936C211yesmaximal, characteristic, complemented
5#1934C211yesmaximal, characteristic, complemented
6#1926C211yesmaximal, characteristic, complemented
7#1925C211yesmaximal, characteristic, complemented
8#1864C211yesmaximal, characteristic, complemented
9#1862C211yesmaximal, characteristic, complemented
10#1825C211yesmaximal, characteristic, complemented
11#1809C211yesmaximal, characteristic, complemented
12#1786C211yesmaximal, characteristic
13#1762C211yesmaximal, characteristic, complemented
14C4×#102C211yesmaximal, characteristic, complemented
15#1753C211yesmaximal, characteristic, complemented
16C23×D4C2211yescharacteristic
17C22×M16C2211yescharacteristic
18C2×C4×D4C2211yescharacteristic
19C2×C4×D4C2211yescharacteristic
20C2×#117C2211yescharacteristic
21C22×C22⋊C4—21no—
22C2×#107C2211yescharacteristic, complemented
23C2×#106C2211yescharacteristic, complemented
24#418C418yescomplemented
25C2×#102C2211yescharacteristic
26C2×#102C2211yescharacteristic
27C2×#102C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×C22⋊C8C2211yescharacteristic
34C2×#97C2211yescharacteristic, complemented
35C2×C4×C8C2211yescharacteristic, complemented
36#390C2211yescharacteristic
37#390C2211yescharacteristic
38#389C2211yescharacteristic, complemented
39#389C2211yescharacteristic, complemented
40#388—21no—
41#388—21no—
42#386—21no—
43#386—21no—
44#383—21no—
45#380C2211yescharacteristic
46#380C2211yescharacteristic
47#379—21no—
48C4×C4⋊C4—21no—
49C4×C4⋊C4C2211yescharacteristic, complemented
50C4×C4⋊C4C2211yescharacteristic, complemented
51C4×C22⋊C4C2211yescharacteristic
52#340C2211yescharacteristic, complemented
53#340C2211yescharacteristic, complemented
54#340C2211yescharacteristic, complemented
55#340C2211yescharacteristic, complemented
56#340C2211yescharacteristic, complemented
57#340C2211yescharacteristic, complemented
58#336C2211yescharacteristic, complemented
59#336C2211yescharacteristic, complemented
60C25—21no—
61C22×D4—22no—
62C22×D4—22no—
63C22×D4—21no—
64C22×D4—22no—
65C22×D4—41no—
66C22×D4—41no—
67C22×D4C2×C414yes—
68C22×D4C2311yescharacteristic
69C22×D4C2311yescharacteristic
70C23×C4—21no—
71C23×C4—21no—
72C23×C4C2311yescharacteristic
73C2×M16—41no—
74C2×M16—41no—
75C2×M16C2×C414yescomplemented
76C22×C8C2311yescharacteristic
77C22×C8C2311yescharacteristic
78C22×C8C2311yescharacteristic
79C22×C8C2311yescharacteristic
80C4×D4—22no—
81C4×D4—22no—
82C4×D4—41no—
83C4×D4—41no—
84#117—22no—
85#117C2×C414yescomplemented
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
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×C22⋊C4—21no—
103C2×C22⋊C4—44no—
104C2×C22⋊C4D411yescharacteristic
105C2×C22⋊C4D411yescharacteristic
106C2×C42—21no—
107C2×C42—21no—
108C2×C42—21no—
109C2×C42C2311yescharacteristic
110C2×C42C2311yescharacteristic
111#107—41no—
112#106—41no—
113#102—22no—
114#102—22no—
115#102—41no—
116#102—41no—
117#102C2×C414yes—
118#102C2×C414yes—
119#102C2×C414yescomplemented
120#102C2×C414yescomplemented
121C22⋊C8—22no—
122#97—41no—
123C4×C8—41no—
124#95—21no—
125#95—21no—
126#95—21no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95D411yescharacteristic
131#95D411yescharacteristic
132C24—21no—
133C24—21no—
134C24—24no—
135C24—42no—
136C24—42no—
137C24—42no—
138C24—42no—
139C24—81no—
140C24—81no—
141C24C2×D411yescharacteristic
142C2×D4—28no—
143C2×D4—24no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—44no—
153C2×D4—44no—
154C2×D4—44no—
155C2×D4—42no—
156C2×D4—42no—
157C2×D4C22×C412yes—
158C2×D4C22×C412yes—
159C2×D4#4112yes—
160C2×D4#4112yes—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—81no—
176C22×C4—81no—
177C22×C4C22×C412yes—
178C22×C4C2×D411yescharacteristic
179C22×C4C2×D411yescharacteristic
180C22×C4C2×D411yescharacteristic
181C22×C4C2×D411yescharacteristic
182C22×C4C2×D411yescharacteristic
183C22×C4#4112yes—
184C22×C4C2411yescharacteristic
185M16—44no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8C22×C412yes—
197C2×C8C22×C412yes—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—21no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4C22×C412yes—
211C4⋊C4C22×C412yes—
212C22⋊C4—24no—
213C22⋊C4—22no—
214C22⋊C4—41no—
215C22⋊C4—42no—
216C22⋊C4—42no—
217C22⋊C4—42no—
218C22⋊C4—42no—
219C22⋊C4—42no—
220C22⋊C4—81no—
221C22⋊C4—81no—
222C22⋊C4—81no—
223C42—24no—
224C42—21no—
225C42—21no—
226C42—21no—
227C42—21no—
228C42—81no—
229C42—81no—
230C23—22no—
231C23—22no—
232C23—22no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—22no—
240C23—48no—
241C23—48no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—84no—
253C23—84no—
254C23—84no—
255C23—84no—
256C23—84no—
257C23C4×D412yes—
258C23C22×D411yescharacteristic
259D4—22no—
260D4—22no—
261D4—22no—
262D4—22no—
263D4—22no—
264D4—48no—
265D4—41no—
266D4—41no—
267D4#11714yes—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—84no—
304C2×C4—84no—
305C2×C4C4×D412yes—
306C2×C4AGL(1,ℤ/8)12yes—
307C2×C4C23×C411yescharacteristic
308C2×C4C2×#4111yescharacteristic
309C2×C4C2×#4111yescharacteristic
310C2×C4C2×#4111yescharacteristic
311C2×C4H(5,𝔽2)11yescharacteristic
312C2×C4H(5,𝔽2)11yescharacteristic
313C8—22no—
314C8—22no—
315C8—41no—
316C8—41no—
317C22—22no—
318C22—24no—
319C22—22no—
320C22—22no—
321C22—42no—
322C22—42no—
323C22—41no—
324C22—42no—
325C22—42no—
326C22—41no—
327C22—41no—
328C22—42no—
329C22—42no—
330C22—41no—
331C22—42no—
332C22—42no—
333C22—42no—
334C22—816no—
335C22—81no—
336C22—81no—
337C22—81no—
338C22—81no—
339C22C2×C4×D411yescharacteristic
340C22#54611yescharacteristic
341C22#54611yescharacteristic
342C22C2×AGL(1,ℤ/8)11yescharacteristic
343C22#57511yescharacteristic
344C22#57611yescharacteristic
345C22#57711yescharacteristic
346C4—22no—
347C4—22no—
348C4—42no—
349C4—42no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—81no—
354C4—81no—
355C4—81no—
356C4—81no—
357C4C2×#11711yescharacteristic
358C4#51811yescharacteristic
359C4#55111yescharacteristic
360C4#55111yescharacteristic
361C2—42no—
362C2—44no—
363C2—82no—
364C2#234211yescharacteristic
365C2C4×AGL(1,ℤ/8)11yescharacteristic
366C2#294611yescharacteristic
367C2#331111yescharacteristic
368C2#331811yescharacteristic
369C2#332211yescharacteristic
370C2#332411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC23×D4C22
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
2. #2347C2semidirect
4. #1936C2semidirect
5. #1934C2semidirect
6. #1926C2semidirect
7. #1925C2semidirect
8. #1864C2semidirect
9. #1862C2semidirect
10. #1825C2semidirect
11. #1809C2semidirect
13. #1762C2semidirect
14. C4×#102C2semidirect
15. #1753C2semidirect
22. C2×#107C22semidirect
23. C2×#106C22semidirect
24. #418C4semidirect
34. C2×#97C22semidirect
35. C2×C4×C8C22semidirect
38. #389C22semidirect
39. #389C22semidirect
49. C4×C4⋊C4C22semidirect
50. C4×C4⋊C4C22semidirect
52. #340C22semidirect
53. #340C22semidirect
54. #340C22semidirect
55. #340C22semidirect
56. #340C22semidirect
57. #340C22semidirect
58. #336C22semidirect
59. #336C22semidirect
75. C2×M16C2×C4semidirect
85. #117C2×C4semidirect
119. #102C2×C4semidirect
120. #102C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#4182
#23472
D4×C22⋊C42
#19362
#19342
#19262
#19252
#18642
#18622
#18252
#18092
#17862
#17622
C4×#1022
#17532